text:
\section{\bf Introduction}
\label{sec:intro}

Continuous valuations are an alternative to measures, which are
popular in computer science, and notably in the semantics of
programming languages \cite{jones89,Jones:proba}.  The space of all
continuous valuations on a topological space $X$ is called the
probabilistic powerdomain $\Val X$ on $X$.  It is known that the
probabilistic powerdomain of a directed-complete partial order (dcpo)
is a dcpo again, in short, $\Val$ preserves dcpos; similarly, $\Val$
preserves continuous dcpos, but fails to preserve complete lattices
and bc-domains.  All that was proved by Jones
\cite{jones89,Jones:proba}.  It is unknown whether $\Val$ preserves
RB-domains or FS-domains, except in special cases
\cite{JT:troublesome}.  On the positive side, $\Val$ preserves stably
compact spaces \cite{JT:troublesome,AMJK:scs:prob}, QRB-domains
\cite{JGL:qrb,GLJ:QRB=QFS}, and coherent quasi-continuous dcpos
\cite{LK:prob}.  (The latter two results are equivalent, since
QRB-domains coincide with coherent quasi-continuous dcpos
\cite{LX:QRB=QFS,GLJ:QRB=QFS}, and also with Li and Xu's QFS-domains
\cite{LiXu:QFS}.)

Lyu and Kou \cite{LK:prob} asked whether coherence was required, in
other words, whether $\Val$ preserves quasi-continuous, not
necessarily coherent, dcpos.  We show that this is indeed the case,
and that, in this case, the Scott and weak topologies agree on the
probabilistic powerdomain.  We show this in
Section~\ref{sec:bf-main-theorem}, after a few preliminaries: general
preliminaries in Section~\ref{sec:bf-preliminaries}, some required
material due to Heckmann on so-called point-continuous valuations in
Section~\ref{sec:simple-valuations}, and a useful lemma on capacities
in Section~\ref{sec:capacities}.  We refine the result and we handle
the case of probability continuous valuations in
Section~\ref{sec:case-prob-cont}.

Since every continuous dcpo is quasi-continuous, the coincidence of
the Scott and weak topologies on $\Val X$, where $X$ is
quasi-continuous, generalizes a result of Kirch
\cite[Satz~8.6]{Kirch:bewertung}, see also
\cite[Satz~4.10]{Tix:bewertung}, according to which the Scott and weak
topologies on $\Val X$ agree for every continuous dcpo $X$.
Alvarez-Manilla, Jung and Keimel asked whether they agree on
$\Val_{\leq 1} X$ for every stably compact space $X$ \cite[Section~5,
second open problem]{AMJK:scs:prob}.  We will show that this is not
the case, through a simple, compact Hausdorff example in
Section~\ref{sec:scott-weak-topol}.  Hence the situation with
quasi-continuous domains is probably rather exceptional.

\section{\bf Preliminaries}
\label{sec:bf-preliminaries}

We refer to \cite{GHKLMS:contlatt,JGL-topology} on domain theory and
point-set topology, specially non-Hausdorff topology.  Compactness
does not involve separation.

A \emph{dcpo} (directed-complete partial order) is a poset $P$ in
which every directed family $D$ has a supremum $\sup D$.  A
\emph{Scott-open subset} of $P$ is a subset $U$ that is
\emph{upwards-closed} (for every $x \in U$ and every $y$ such that
$x \leq y$, $y$ is in $U$) and is such that, for every directed family
$D$ in $P$, if $\sup D \in U$ then $D$ intersects $U$.  The
Scott-open subsets of $P$ form a topology called the \emph{Scott
  topology}.

Every complete lattice is a dcpo.  For example,
$\creal \eqdef \Rp \cup \{\infty\}$, with the usual ordering that
places $\infty$ above all non-negative real numbers, is a dcpo.  The
family of open subsets $\Open X$ of a topological space is a dcpo,
too.

A \emph{Scott-continuous map} $f \colon P \to Q$ between dcpos is a
monotonic map that preserves suprema of directed families.  A map from
$P$ to $Q$ is Scott-continuous if and only if it is continuous with
respect to the Scott topologies on $P$ and $Q$.

A \emph{valuation} $\nu$ on a topological space $X$ is a strict,
modular, monotonic map from $\Open X$ to $\creal$.  That $\nu$ is
\emph{strict} means that $\nu (\emptyset)=0$.  That it is
\emph{modular} means that
$\nu (U) + \nu (V) = \nu (U \cup V) + \nu (U \cap V)$ for all open
subsets $U$ and $V$.  A \emph{continuous valuation} is a valuation
that is Scott-continuous from $\Open X$ to $\creal$.

Continuous valuations and measures are close cousins.  Every
$\tau$-smooth Borel measure defines a continuous valuation by
restricting it to $\Open X$; and every Borel measure on a hereditary
Lindel\"of space is $\tau$-smooth \cite{Adamski:measures}.  Conversely,
every continuous valuation on an LCS-complete space extends to a
measure on the Borel $\sigma$-algebra
\cite[Theorem~1.1]{dBGLJL:LCS}---an \emph{LCS-complete} space is any
subspace obtained as a $G_\delta$ subset of a locally compact sober
space.

We write $\Val X$ for the dcpo of all continuous valuations on $X$,
ordered by the \emph{stochastic ordering}: $\mu \leq \nu$ if and only
if $\mu (U) \leq \nu (U)$ for every $U \in \Open X$.
$\Val_{\leq 1} X$ (resp., $\Val_1 X$) is the subdcpo of all
\emph{subprobability} (resp., \emph{probability}) continuous
valuations $\nu$, namely those such that $\nu (X) \leq 1$ (resp.,
$\nu (X)=1$).  We will usually write $\Val_\ast X$ to denote any of
those dcpos, where $\ast$ stands for nothing, ``$\leq 1$'', or
``$1$''.

The \emph{weak topology} on $\Val_\ast X$ is the coarsest one that
makes
$[r \ll U]_\ast \eqdef \{\nu \in \Val_\ast X \mid r \ll \nu (U)\}$
open for every $r \in \Rp$ and every $U \in \Open X$.  Here $\ll$ is
the so-called way-below relation on $\creal$; we have $r \ll s$ if and
only if $r=0$ or $r < s$.  The sets $[U > r]_\ast$ with
$U \in \Open X$ and $r \in \Rp \diff \{0\}$ form another subbase of
the weak topology, since $[U > r]_\ast = [r \ll U]_\ast$ if
$r \neq 0$, and $[0 \ll U]_\ast = \Val_\ast X$.  We write
$\Val_{\ast,\wk} X$ for $\Val_\ast X$ with the weak topology.  The
weak topology is coarser than the Scott topology of the stochastic
ordering $\leq$.

Every topological space $X$ has a \emph{specialization preordering}
$\leq$, defined by $x \leq y$ if and only if every open neighborhood
of $x$ contains $y$.  A $T_0$ space is one such that $\leq$ is an
ordering.  As examples, for every dcpo $P$, ordered by $\leq$, the
specialization preordering of $P$ with its Scott topology is $\leq$;
and the specialization preordering of $\Val_\ast X$ is the stochastic
ordering.

For every point $x \in X$, the closure of $\{x\}$ coincides with the
downward closure $\dc x \eqdef \{y \in X \mid y \leq x\}$ of $x$ in
the specialization preordering.  In general, we write $\dc A$ for the
downward closure of a set $A$, so that $\dc x = \dc \{x\}$.

A subset $A$ of a space $X$ is \emph{saturated} if and only if it is
equal to the intersection of its open neighborhoods, equivalently if
it is upwards-closed with respect to the specialization preordering
$\leq$.  We write $\upc A$ for the upward closure of $A$.

For every compact subset $K$ of $X$, $\upc K$ is compact saturated.
This is the case in particular if $K$ is finite: we call the sets of
the form $\upc E$, with $E$ finite, \emph{finitary compact}.  A space
$X$ is \emph{locally finitary compact} if and only if it has a base
consisting of interiors $\interior {\upc E}$ of finitary compact sets.

The standard definition of a quasi-continuous dcpo is through the
notion of a so-called way-below relation between finite subsets.  We
will instead use the following characterization
\cite[Exercise~8.3.39]{JGL-topology}: the quasi-continuous dcpos are
exactly the locally finitary compact, sober spaces.  Notably, every
locally finitary compact, sober space is a quasi-continuous dcpo in
its specialization ordering $\leq$; also, the topology is exactly the
Scott topology of $\leq$.

We have mentioned sober spaces a few times already.  A closed subset
$C$ of a space $X$ is \emph{irreducible} if and only if it is
non-empty and, for all closed subsets $C_1$ and $C_2$ of $X$, if
$C \subseteq C_1 \cup C_2$ then $C \subseteq C_1$ or
$C \subseteq C_2$.  The closures $\dc x$ of points are always
irreducible closed.  A \emph{sober space} is any $T_0$ space in which
the only irreducible closed subsets are closures of points.  $\creal$
is sober in its Scott topology.  Every quasi-continuous dcpo is sober
in its Scott topology (by our definition), every Hausdorff space is
sober; also, $\Val_\wk X$ is sober for every space $X$
\cite[Proposition~5.1]{heckmann96}.

The sober subspaces $Y$ of a sober space $X$ are exactly those that
are closed in the so-called \emph{Skula}, or \emph{strong} topology on
$X$ \cite[Corollary~3.5]{KL:dtopo}.  That topology is the coarsest one
that contains both the original open and the original closed sets as
open sets.  We note that $\Val_{\leq 1,\wk} X$ is closed in
$\Val_\wk X$, being the complement of $[X > 1]$.  Every closed set is
Skula-closed, so $\Val_{\leq 1, \wk} X$ is also a sober space.  Also,
$\Val_{1,\wk} X$ is the intersection of the closed set
$\Val_{\leq 1,\wk} X$ with the open sets $[X > 1-\epsilon]$,
$\epsilon > 0$, hence is also Skula-closed and therefore sober as
well.

The forgetful functor from the category of sober spaces and continuous
maps to the category of topological spaces has a left adjoint called
\emph{sobrification}.  Explicitly, this means that every topological
space $X$ has a sobrification $X^s$, which is a sober topological
space; there is a continuous map $\eta_X \colon X \to X^s$, called the
\emph{unit}; and every continuous map $f \colon X \to Y$ where $Y$ is
sober extends to a unique continuous map $\hat f \colon X^s \to Y$, in
the sense that $\hat f \circ \eta_X = f$.  Concretely, $X^s$ can be
realized as the space of all irreducible closed subsets of $X$, with a
suitable topology, and $\eta_X (x) \eqdef \dc x$.  By Proposition~3.4
of \cite{KL:dtopo}, given any subspace $Y$ of a sober space $X$, the
Skula-closure $cl_s (Y)$ of $Y$ in $X$ is also a sobrification of $Y$,
with $\eta_Y$ defined as the inclusion map.  In general, for a $T_0$
space $Y$, and a sober space $X$, together with a continuous map
$f \colon Y \to X$, $X$ is a sobrification of $Y$ with unit $f$ if and
only if $f$ is a topological embedding, with Skula-dense image
\cite[Proposition~3.2]{KL:dtopo}.

\section{\bf Simple and point-continuous valuations}
\label{sec:simple-valuations}

Among all the continuous valuations that exist on a space $X$, the
\emph{simple valuations} are those of the form $\sum_{x \in A} a_x
\delta_x$, where $A$ is a finite subset of $X$, $a_x \in \Rp$, and
$\delta_x$ is the Dirac mass, defined by $\delta_x (U) \eqdef 1$ if $x
\in U$, $0$ otherwise.  We let $\Val_{\ast,\fin} X$ be the subspace of
$\Val_{\ast,\wk} X$ that consists of its simple valuations.

Heckmann characterized the sobrification of $\Val_\fin X$ as being the
space $\Val_\pw X$ of so-called \emph{point-continuous valuations} on
$X$ \cite[Theorem~5.5]{heckmann96}, together with inclusion as unit.
Those are the valuations $\nu$ on $X$ that are continuous from
$\Open_\pw X$ to $\creal$.  $\Open_\pw X$ is the lattice of open
subsets of $X$ with the point topology, namely the coarsest topology
that makes $\{U \in \Open X \mid x \in U\}$ open for every point
$x \in X$.  We write $\Val_{\ast,\pw} X$ for the usual variants.
\begin{lemma}
  \label{lemma:VpX:dense}
  Let $X$ be a topological space.  $\Val_\fin X$ is Skula-dense in
  $\Val_\pw X$.
\end{lemma}
\begin{proof}
  By Proposition~3.2 of \cite{KL:dtopo}, cited earlier: since
  $\Val_\pw X$ is a sobrification of $\Val_\fin X$, with unit given by
  the inclusion map $i$, the image of $i$ must be Skula-dense.
\end{proof}

\begin{lemma}
  \label{lemma:mu:simple:approx:1}
  Let $X$ be a topological space and $\mathcal U$ be an open subset of
  $\Val_\pw X$.  For every $\nu \in \mathcal U$, there is a simple
  valuation $\nu'$ in $\mathcal U$ such that $\nu' \leq \nu$.
\end{lemma}
\begin{proof}
  $\mathcal U \cap \dc \nu$ is open in the Skula topology of
  $\Val_\wk X$, and is non-empty, since it contains $\nu$.  Using
  Lemma~\ref{lemma:VpX:dense}, it must contain an element $\nu'$ of
  $\Val_\fin X$.
\end{proof}

Heckmann also showed that, when $X$ is locally finitary compact,
\emph{all} continuous valuations are point-continuous, hence
$\Val_\wk X = \Val_\pw X$ \cite[Theorem~4.1]{heckmann96}.  Using that
information, we obtain the following.
\begin{lemma}
  \label{lemma:mu:simple:approx}
  Let $X$ be a locally finitary compact space, $\mathcal U$ be an open
  subset of $\Val_{\ast,\wk} X$, where $\ast$ is nothing or ``$\leq
  1$''.  For every $\nu \in \mathcal U$, there is a simple valuation
  $\nu'$ in $\mathcal U$ such that $\nu' \leq \nu$.
\end{lemma}
\begin{proof}
  When $\ast$ is nothing, this is
  Lemma~\ref{lemma:mu:simple:approx:1}, together with the fact that
  $\Val_\wk X = \Val_\pw X$.

  When $\ast$ is ``$\leq 1$'', we use the definition of the weak
  topology: $\nu$ is in some finite intersection
  $\bigcap_{i=1}^m [U_i > r_i]_{\leq 1}$ of subbasic open sets
  included in $\mathcal U$.  Then $\nu$ is also in the corresponding
  finite intersection $\bigcap_{i=1}^m [U_i > r_i]$ of subbasic open
  sets of $\Val_\wk X$.  We have just seen that there is a simple
  valuation $\nu' \leq \nu$ in $\bigcap_{i=1}^m [U_i > r_i]$.  Since
  $\nu' \leq \nu$, $\nu'$ is a subprobability valuation, so $\nu'$ is in
  $\bigcap_{i=1}^m [U_i > r_i]_{\leq 1}$, hence in $\mathcal U$.
\end{proof}

\section{\bf Capacities}
\label{sec:capacities}

Capacities are a generalization of valuations (or measures) introduced
by Choquet \cite{Choquet:capacities}, where modularity is abandoned in
favor of weaker properties.  We will need the following kind.

Given a subset $B$ of a topological space, the \emph{unanimity game}
$\ugame B \colon \Open X \to \creal$ maps every open set $U$ to $1$ if
$B \subseteq U$, to $0$ otherwise.  When $B = \{x\}$, $\ugame B$ is
simply the Dirac mass $\delta_x$, but in general $\ugame B$ is not
modular.

We will consider functions $\kappa$ of the form
$\sum_{x \in A} a_x \ugame {B_x}$, where $A$ is a finite subset of
$X$, and for each $x \in A$, $a_x$ is a number in $\Rp$ and $B_x$ is a
finite non-empty subset of $X$, which we call \emph{simple capacities}
here.  We compare capacities, and in general all functions from $\Open
X$ to $\creal$, by $\kappa \leq \nu$ if and only if $\kappa (U) \leq
\nu (U)$ for every $U \in \Open X$, extending the stochastic ordering
from continuous valuations to all maps.

In this setting, an element $f$ of $\Sigma \eqdef \prod_{x \in A} B_x$
is a function that maps each point $x \in A$ to an element $f (x)$ in
$B_x$. One can think of such functions $f$ as \emph{strategies} for
picking an element of $B_x$ for each $x \in A$.
We let $\Delta_\Sigma$ be the set of all families $\vec\beta \eqdef
{(\beta_f)}_{f \in \Sigma}$ of non-negative real numbers such that
$\sum_{f \in \Sigma} \beta_f = 1$.  $\Delta_\Sigma$ is simply the
standard $n$-simplex $\Delta_n \eqdef \{(\beta_0, \beta_1, \cdots,
\beta_n) \in \Rp^{n+1} \mid \sum_{i=0}^n \beta_i=1\}$, where $n$ is
the cardinality of $\Sigma$ minus $1$.

In order to show the following lemma, we will need to introduce the
Choquet integral $\int_{x \in X} h (x) d\nu$ of a lower semicontinuous
map $h \colon X \to \creal$ with respect to a set function
$\nu \colon \Open X \to \creal$.  By definition, this is equal to the
Riemann integral $\int_0^\infty \nu (h^{-1} (]t, \infty])) dt$.  Note
that this makes sense, because $h^{-1} (]t, \infty])$ is open for
every $t \in \Rp$, and because every non-increasing map is
Riemann-integrable.  In our setting, this form of the Choquet integral
was introduced by Tix \cite{Tix:bewertung}, and differs only slightly
from Choquet's original definition
\cite[Section~48]{Choquet:capacities}.  Tix proved that, when $\nu$ is
a continuous valuation, $\int_{y \in X} h (y) d\nu$ is linear and
Scott-continuous in $h$ \cite[Lemma~4.2]{Tix:bewertung}.  It is an
easy exercise to verify that
$\int_{y \in X} \chi_U (y) d\nu = \nu (U)$ for every open subset $U$
of $X$, where $\chi_U$ is the characteristic map of $U$.  It follows
that, when $h$ is of the form $\sum_{j=0}^m \alpha_j \chi_{U_j}$,
$\int_{y \in X} h (y) d\nu = \sum_{j=0}^m \alpha_j \nu (U_j)$.

For a simple capacity $\kappa \eqdef \sum_{x \in A} a_x \ugame {B_x}$,
we compute $\int_{y \in X} h (y) d\kappa$ as follows.  For each
$x \in A$,
$\int_{y \in X} h (y) d\ugame {B_x} = \int_0^\infty \ugame {B_x}
(h^{-1} (]t, \infty])) dt$ by the Choquet formula.  But
$\ugame {B_x} (h^{-1} (]t, \infty])) = 1$ if and only if
$B_x \subseteq h^{-1} (]t, \infty])$, if and only if
$\min_{y \in B_x} h (y) > t$.  It follows that
$\int_{y \in X} h (y) d\ugame {B_x} = \min_{y \in B_x} h (y)$.  Hence
$\int_{y \in X} h (y) d\kappa = \sum_{x \in A} a_x \min_{y \in B_x} h
(y)$.

\begin{lemma}
  \label{lemma:mincred}
  Let $X$ be a topological space, and $\kappa \eqdef \sum_{x \in A}
  a_x \ugame {B_x}$ be a simple capacity on $X$.

  Let $\nu$ be any bounded continuous valuation on $X$.  If
  $\kappa \leq \nu$, then, for some $\vec\beta \in \Delta_\Sigma$,
  $\sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f (x)} \leq \nu$.
\end{lemma}
\begin{proof}
  This is a consequence of von Neumann's original minimax theorem
  \cite{vN:minimax}, which says that given any $n \times m$ matrix $M$
  with real entries,
  \begin{equation}
    \label{eq:minimax}
    \min_{\vec\alpha \in \Delta_m} \max_{\vec\beta \in \Delta_n}
    {\vec\beta}^\intercal M \vec\alpha
    = \max_{\vec\beta \in \Delta_n} \min_{\vec\alpha \in \Delta_m}
    {\vec\beta}^\intercal
    M \vec\alpha.
  \end{equation}
  In particular: $(\dagger)$ if for every $\vec\alpha \in \Delta_m$,
  there is a $\vec\beta \in \Delta_n$ such that
  ${\vec\beta}^\intercal M \vec\alpha \geq 0$ (namely, if the
  left-hand side of (\ref{eq:minimax}) is non-negative), then there is
  a $\vec\beta \in \Delta_n$ such that, for every
  $\vec\alpha \in \Delta_m$,
  ${\vec\beta}^\intercal M \vec\alpha \geq 0$.

  We first show that: $(*)$ given finitely many open subsets $U_0$, $U_1$,
  \ldots, $U_m$ of $X$, we can find a $\vec\beta \in \Delta_\Sigma$
  such that, for every $j$, $0\leq j \leq m$,
  $\sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f (x)} (U_j) \leq
  \nu (U_j)$.

  Let $\kappa$ denote $\sum_{x \in A} a_x \ugame {B_x}$.  Since
  $\kappa \leq \nu$, for every lower semicontinuous map $h$,
  $\int_{y \in X} h (y) d\kappa = \int_0^\infty \kappa (h^{-1} (]t,
  \infty])) dt \leq \int_0^\infty \nu (h^{-1} (]t, \infty])) dt =
  \int_{y \in X} h (y) d\nu$.  In other words,
  $\sum_{x \in A} a_x \min_{y \in B_x} h (y) \leq \int_{y \in X} h (y)
  d\nu$.

  For every $\vec\alpha \in \Delta_m$, we consider
  $h_{\vec\alpha} \eqdef \sum_{j=0}^m \alpha_j \chi_{U_j}$ for $h$.
  The inequality we have just shown can be rewritten as
  $\sum_{x \in A} a_x \min_{y \in B_x} h_{\vec\alpha} (y) \leq
  \sum_{j=0}^m \alpha_j \nu (U_j)$.  For each $x \in A$, there is an
  element $y \in B_x$ that makes $h_{\vec\alpha} (y)$ minimal, and we
  call it $f_{\vec\alpha} (x)$.  Therefore
  $\sum_{x \in A} a_x h_{\vec\alpha} (f_{\vec\alpha} (x)) \leq
  \sum_{j=0}^m \alpha_j \nu (U_j)$.  By definition of
  $h_{\vec\alpha}$, and since
  $\chi_{U_j} (f_{\vec\alpha} (x)) = \delta_{f_{\vec\alpha} (x)}
  (U_j)$, this can be written equivalently as
  $\sum_{x \in A} \sum_{j=0}^m \alpha_j a_x \delta_{f_{\vec\alpha}
    (x)} (U_j) \leq \sum_{j=0}^m \alpha_j \nu (U_j)$.  It follows that
  there is a vector $\vec\beta$ in $\Delta_\Sigma$ such that, for
  every $j$, $0 \leq j\leq m$,
  $\sum_{j=0}^m \alpha_j \nu (U_j) - \sum_{j=0}^m \alpha_j \sum_{f \in
    \Sigma, x \in A} \beta_f a_x \delta_{f (x)} (U_j) \geq 0$: namely,
  $\beta_f \eqdef 1$ if $f=f_{\vec\alpha}$, and $\beta_f \eqdef 0$
  otherwise.

  That can also be written as
  $\sum_{f \in \Sigma, 0 \leq j \leq m} \alpha_j \beta_f \left(\nu
    (U_j) - \sum_{x \in A} a_x \delta_{f (x)} (U_j)\right) \geq 0$,
  hence as ${\vec\beta}^\intercal M \vec\alpha \geq 0$ for some matrix
  $M$.  Using $(\dagger)$, there is a vector
  $\vec\beta \in \Delta_\Sigma$ such that, for every
  $\vec\alpha \in \Delta_m$,
  ${\vec\beta}^\intercal M \vec\alpha \geq 0$, in other words
  $\sum_{j=0}^m \alpha_j \nu (U_j) - \sum_{j=0}^m \alpha_j \sum_{f \in
    \Sigma, x \in A} \beta_f a_x \delta_{f (x)} (U_j) \geq 0$.  In
  particular, for each $j$, $0\leq j\leq m$, taking $\vec\alpha$ such
  that $\alpha_j \eqdef 1$ and all its other components are $0$,
  $\nu (U_j) \geq \sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f
    (x)} (U_j)$.  This proves $(*)$.

  For every finite family $\mathcal A$ of open subsets of $X$, let
  $C_{\mathcal A}$ be the set of vectors $\vec\beta \in \Delta_\Sigma$
  such that
  $\sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f (x)} (U) \leq
  \nu (U)$ for every $U \in \mathcal A$.  Claim $(*)$ above states
  that $C_{\mathcal A}$ is non-empty (when $\mathcal A$ is non-empty;
  when $\mathcal A$ is empty, this is vacuously true).  It is also a
  closed subset of $\Delta_\Sigma$.  The family
  ${(C_{\mathcal A})}_{\mathcal A \in \Pfin (\Open X)}$ then has the
  finite intersection property: given any finite collection of
  elements $\mathcal A_1$, \ldots, $\mathcal A_k$ in
  $\Pfin (\Open X)$,
  $\bigcap_{i=1}^k C_{\mathcal A_i} = C_{\bigcup_{i=1}^k \mathcal
    A_i}$ is non-empty.  Since $\Delta_\sigma$ is compact, the
  intersection
  $\bigcap_{\mathcal A \in \Pfin (\Open X)} C_{\mathcal A}$ is
  non-empty.  Let $\vec\beta$ be any vector in that intersection.  For
  every $U \in \Open X$, since $\vec\beta$ is in $C_{\{U\}}$, we have
  $\sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f (x)} (U) \leq
  \nu (U)$, and we conclude.
\end{proof}

\section{\bf The main theorem}
\label{sec:bf-main-theorem}

We come to our main theorem.  It applies in particular to every
quasi-continuous dcpo $X$, namely to every locally finitary compact
sober space, as we have announced; but sobriety is not needed.  We
spend the rest of the section proving it.
\begin{theorem}
  \label{thm:V:locfin}
  For every locally finitary compact space $X$,
  $\Val_\wk X=\Val_\pw X$ and
  $\Val_{\leq 1, \wk} X=\Val_{\leq 1, \pw} X$ are compact, locally
  finitary compact, sober spaces.  In particular, they are
  quasi-continuous dcpos and the weak topology coincides with the
  Scott topology.

  The sets of the form $\interior {\upc E}$, where $E$ ranges over the
  finite non-empty sets of simple (resp., simple subprobability)
  valuations form a base of the topology.
\end{theorem}

Let $\ast$ be nothing or ``$\leq 1$''.  We recall that the equality
$\Val_\wk X = \Val_\pw X$ holds for every locally finitary compact
space $X$, as shown by Heckmann \cite[Theorem~4.1]{heckmann96}.  The
equality $\Val_{\leq 1, \wk} X = \Val_{\leq 1, \pw} X$ immediately
follows from it.

We also recall that the quasi-continuous dcpos are exactly the locally
finitary compact sober spaces, and in particular that their topology
must be the Scott topology.  The fact that $\Val_{\ast,\wk} X$ is
compact follows from the fact that it has a least element in the
stochastic ordering, namely the zero valuation: every open cover
${(\mathcal U_i)}_{i \in I}$ of $\Val_{\ast,\wk}$ must be such that
some $\mathcal U_i$ contains the zero valuation, and therefore
coincide with the whole of $\Val_{\ast,\wk} X$, since open sets are
upwards-closed.

Finally, we recall that $\Val_{\ast, \wk} X$ is sober.

Therefore, it remains to show that $\Val_{\ast, \wk} X$ is locally
finitary compact.  In the rest of this section, we fix
$\nu \in \Val_{\ast,\wk} X$, and an open neighborhood $\mathcal U$ of
$\nu$ in the weak topology.  Then $\nu$ is in some finite intersection
$\bigcap_{i=1}^n [U_i > r_i]_\ast$ included in $\mathcal U$, where
each $U_i$ is open in $X$ and $r_i \in \Rp \diff \{0\}$.  We will find
a finite set $E$ of simple valuations and an open subset $\mathcal V$
of $\Val_{\ast, \wk} X$ such that
$\nu \in \mathcal V \subseteq \upc E \subseteq \mathcal U$.

Let us simplify the problem slightly.  By
Lemma~\ref{lemma:mu:simple:approx}, there is a simple valuation
$\nu' \leq \nu$ in $\bigcap_{i=1}^n [U_i > r_i]_\ast$. Hence, without loss
of generality, we may assume that $\nu$ itself is a simple valuation
$\sum_{x \in A} a_x \delta_x$, where $A$ is a finite subset of $X$,
and $a_x \in \Rp \diff \{0\}$ for every $x \in A$.

Since $\nu (U_i) > r_i$ for every $i$, $1\leq i\leq n$, there is a
number $a \in ]0, 1[$ such that $a . \nu (U_i) > r_i$ for every $i$.
There is also a positive number $s_i$ such that
$a . \nu (U_i) > s_i > r_i$.  We will need those numbers $a$ and $s_i$
only near the end of the proof.

Let us define a suitable open set $\mathcal V$.  For each point
$x \in A$, let $I_x \eqdef \{i \in I \mid x \in U_i\}$.  Then
$\bigcap_{i \in I_x} U_i \diff \dc (A \diff \upc x)$ is an open
neighborhood of $x$.  It is easy to see that $x$ is in
$\bigcap_{i \in I_x} U_i$, but perhaps a bit less easy to see that $x$
is not in $\dc (A \diff \upc x)$: otherwise there woud be an element
$y \in A \diff \upc x$ above $x$, and that is impossible.

Since $X$ is locally finitary compact, for each $x \in A$, one can
find a finite set $B_x$ such that
$x \in \interior {\upc B_x} \subseteq \upc B_x \subseteq \bigcap_{i
  \in I_x} U_i \diff \dc (A \diff \upc x)$.  We will require a bit
more, and we will make sure that $B_x$ is also included in
$\interior {\upc B_y}$ for every $y \in A$ such that $y \leq x$.  This
can be done by finding $B_x$ in stages, starting from the lowest
elements $x$ of $A$ and going up.  Formally, since $A$ is finite, we
define $B_x$ by course-of-values induction on the number of elements
$y \in A$ such that $y \leq x$, as follows: for each $x \in A$, we
simply find a finite set $B_x$ such that
$x \in \interior {\upc B_x} \subseteq \upc B_x \subseteq \bigcap_{i
  \in I_x} U_i \diff \dc (A \diff \upc x) \cap \bigcap_{y \in A, y<x}
\interior {\upc B_y}$, where the last term is available, and an open
neighborhood of $y \in A$ hence of $x$, by induction hypothesis.

We also define $V_x \eqdef \interior {\upc B_x}$.  We note the
following three facts.
\begin{lemma}
  \label{lemma:z}
  For all $x, y \in A$ with $x \leq y$, $\upc B_y \subseteq V_x
  \subseteq \upc B_x$.  \qed
\end{lemma}

\begin{lemma}
  \label{lemma:a}
  For every $x \in A$, for every $i \in I$, if $x \in U_i$ , then
  $B_x \subseteq U_i$.
\end{lemma}
\begin{proof}
  If $x \in U_i$, then $i \in I_x$.  Since
  $B_x \subseteq \bigcap_{i \in I_x} U_i$, the claim follows.
\end{proof}

\begin{lemma}
  \label{lemma:b}
  For all $x, y \in A$, $x \in V_y$ if and only if $y \leq x$.
\end{lemma}
\begin{proof}
  If $y \leq x$, and since $V_y$ is an open neighborhood of $y$, and
  is in particular upwards-closed, $x$ is also in $V_y$. If
  $y \not\leq x$, then $x$ is in $A \diff \upc y$, hence in
  $\dc (A \diff \upc y)$. It follows that $x$ cannot be in
  $\bigcap_{i \in I_y} U_i \diff \dc (A \diff \upc y)$, hence cannot
  be in the smaller set $V_y$.
\end{proof}

\begin{definition}[$\mathcal V$]
  \label{defn:V}
  Let $\pow_{\upc} A$ denote the (finite) family of upwards-closed
  subsets of $A$.  For each $B \in \pow_{\upc} A$, let
  $V_B \eqdef \bigcup_{x \in B} V_x$.  Let also
  $s_B \eqdef a . \sum_{x \in B} a_x$. The open set $\mathcal V$ is
  $\bigcap_{B \in \pow_{\upc} A} [s_B \ll V_B]$.
\end{definition}
Recall that $\mu \in [s_B \ll V_B]$ if and only if $s_B \ll \mu
(V_B)$, if and only if $s_B=0$ or $s_B < \mu (V_B)$.

\begin{lemma}
  \label{lemma:nu:V}
  $\nu \in \mathcal V$.
\end{lemma}
\begin{proof}
  For every $B \in \pow_{\upc} A$, we claim that $A \cap V_B = B$. For
  every $x \in B$, $V_x$ is included in $V_B$, and since $V_x$ is an
  open neighborhood of $x$, it follows that $x$ is in $V_B$; $x$ is
  also in $A$, since $B \subseteq A$.  Conversely, if
  $x \in A \cap V_B$, then $x$ is in $V_y$ for some $y \in B$.  Both
  $x$ and $y$ are in $A$, so by Lemma~\ref{lemma:b}, we obtain that
  $y \leq x$.  Since $B$ is upwards-closed, $x$ is in $B$.

  Let us verify that $\nu$ is in $\mathcal V$, namely that, for every
  $B \in \pow_{\upc} A$, $s_B \ll \nu (V_B)$.  Indeed,
  $\nu (V_B) = \sum_{x \in A \cap V_B} a_x = \sum_{x \in B} a_x$,
  since $A \cap V_B = B$.  Now, since $a < 1$,
  $a . \sum_{x \in B} a_x \ll \sum_{x \in B} a_x$.  In other words,
  $s_B \ll \nu (V_B)$, as desired.
\end{proof}

Finding the finite set $E$ is more difficult.  As a first step in that
direction, let $\kappa \eqdef a . \sum_{x \in A} a_x \ugame {B_x}$,
and let us consider the set $\mathcal Q$ of all the continuous
valuation $\mu \in \Val_\ast X$ such that $\kappa \leq \mu$.
\begin{lemma}
  \label{lemma:V:Q}
  $\mathcal V \subseteq \mathcal Q$.
\end{lemma}
\begin{proof}
  Let $\mu$ be any element of $\mathcal V$.  We must show that, for
  every open subset $U$ of $X$,
  $a . \sum_{x \in A, B_x \subseteq U} a_x \leq \mu (U)$.

  Let $B \eqdef \{x \in A \mid B_x \subseteq U\}$.  For every
  $x \in B$ and every $y \in A$ with $x \leq y$, we have
  $B_y \subseteq \upc B_x \subseteq U$ by Lemma~\ref{lemma:z}, so $y$
  is in $B$.  Hence $B$ is upwards-closed in $A$.

  Then the left-hand side $a . \sum_{x \in A, B_x \subseteq U}
  a_x$ is just $s_B$.  Since $\mu \in \mathcal V$, $s_B \ll \mu
  (V_B)$. We recall that $V_B = \bigcup_{x \in B} V_x$, that
  $V_x$ is included in $\upc B_x$ for each
  $x$, and that (by the definition of $B$), $\upc
  B_x$ is included in $U$ for every $x \in B$. Therefore $V_B
  \subseteq U$, and hence $\mu (V_B) \leq \mu
  (U)$, which concludes the proof.
\end{proof}

Let $\Sigma \eqdef \prod_{x \in A} B_x$, and $\Delta_\Sigma$ be the
associated standard simplex.  Lemma~\ref{lemma:V:Q}, together with
Lemma~\ref{lemma:mincred}, immediately implies the following.
\begin{lemma}
  \label{lemma:V:beta}
  Every element $\mu$ of $\mathcal V$ is above a simple valuation of
  the form
  $a. \sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f (x)}$, for
  some $\vec\beta \in \Delta_\Sigma$.
\end{lemma}

Let $E_0$ be the set of simple valuations obtained this way, namely
the set of simple valuations
$a . \sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f (x)}$, where
$\vec\beta \in \Delta_\Sigma$.  We have just shown that every element
$\mu$ of $\mathcal V$ is above some element of $E_0$.

Note that the elements $\varpi$ of $E_0$ can all be written as
$\sum_{z \in Z} c_z \delta_z$, where $Z \eqdef \bigcup_{x \in A} B_x$,
and $c_z \in \Rp$.  For each such $\varpi$, let $\overline\varpi$ be
$\sum_{z \in Z} \frac 1 N \lfloor N c_z \rfloor$, where $N$ is a
fixed, large enough (in particular, non-zero) natural number that we
will determine shortly.

\begin{definition}[$E$]
  \label{defn:E}
  $E$ is the set of all simple valuations $\overline\varpi$, where
  $\varpi$ ranges over $E_0$.
\end{definition}

\begin{lemma}
  \label{lemma:E:fin}
  $E$ is a finite set.
\end{lemma}
\begin{proof}
  $Z$ is finite and the coefficients $\frac 1 N \lfloor N c_z \rfloor$
  are integer multiples of $\frac 1 N$ between $0$ and
  $\sum_{x \in A} a_x$.
\end{proof}

\begin{lemma}
  \label{lemma:overline:varpi}
  $\mathcal V \subseteq \upc E$.
\end{lemma}
\begin{proof}
  For every $z \in Z$, and every $c_z \in \Rp$,
  $\frac 1 N \lfloor N c_z \rfloor \leq c_z$.  It follows that
  $\overline\varpi \leq \varpi$ for every $\varpi \in E_0$.  Since
  every element of $\mathcal V$ is above some element $\varpi$ of
  $E_0$ by Lemma~\ref{lemma:V:beta}, it is also above the
  corresponding element $\overline\varpi$ of $E$.
\end{proof}

\begin{lemma}
  \label{lemma:upE:U}
  $\upc E \subseteq \mathcal U$.
\end{lemma}
\begin{proof}
  We show that $E$ is included in $\bigcap_{i=1}^n [U_i >
  r_i]_\ast$. For every $x \in A$, for every $y \in B_x$, we have
  $\delta_y \geq \ugame {B_x}$, simply because every open neighborhood
  of $B_x$ must contain $x$. Hence, for every $\varpi \in E_0$, say
  $\varpi = a . \sum_{f \in \Sigma, x \in A} \beta_f a_x \delta_{f
    (x)}$, where $\vec\beta \in \Delta_\Sigma$, we have
  $\varpi \geq a . \sum_{f \in \Sigma, x \in A} \beta_f a_x \ugame
  {B_x} = a. \sum_{x \in A} (\sum_{f \in \Sigma} \beta_f) a_x \ugame
  {B_x} = a . \sum_{x \in A} a_x \ugame {B_x} = \kappa$.  For every
  $i$, $1\leq i\leq n$, Lemma~\ref{lemma:a} states that for every
  $x \in A$, if $x \in U_i$ then $B_x$ is included in $U_i$.
  Therefore
  $\varpi (U_i) = a. \sum_{x \in A, B_x \subseteq U_i} a_x \geq
  a. \sum_{x \in A \cap U_i} a_x = a. \nu (U_i)$.  We now remember
  that $a . \nu (U_i) > s_i > r_i$.  In particular,
  $\varpi (U_i) > s_i$.

  It is time we fixed the value of $N$.  The values of $c_z$ and of
  $\frac 1 N \lfloor N c_z \rfloor$ differ by $\frac 1 N$ at most, so
  for any open set $U$, the values $\varpi (U)$ and
  $\overline\varpi (U)$ differ by $\frac 1 N |Z|$ at most, where $|Z|$
  is the cardinality of $Z$.  It follows that
  $\overline\varpi (U_i) > s_i - \frac 1 N |Z|$.  By picking any
  non-zero natural number $N$ larger than or equal to
  $\frac {|Z|} {s_i-r_i}$ for every $i$, $1\leq i\leq n$, we therefore
  ensure that $\overline\varpi (U_i) > r_i$ for every $i$, hence that
  $\overline\varpi$ is in $\mathcal U$.  Since that holds for every
  $\varpi \in E_0$, $E$ is included in $\mathcal U$, hence also
  $\upc E$.
\end{proof}

Hence, as promised, $\nu \in \mathcal V$ (Lemma~\ref{lemma:nu:V})
$\subseteq \upc E$ (Lemma~\ref{lemma:overline:varpi})
$\subseteq \mathcal U$ (Lemma~\ref{lemma:upE:U}), where $\mathcal V$
is open (Definition~\ref{defn:V}) and $E$ is finite
(Lemma~\ref{lemma:E:fin}).  This concludes the proof of
Theorem~\ref{thm:V:locfin}.  \qed

\section{The case of probability continuous valuations}
\label{sec:case-prob-cont}

We now apply the previous results to the space $\Val_{1,\wk} X$ of
probability continuous valuations.  A space $X$ is \emph{pointed} if
and only if it has a least element $\bot$ in its specialization
preordering.  We are not assuming $X$ to be $T_0$, so $\dc \bot$ is a
closed subset that may be different from $\{\bot\}$.  The open subsets
of $X \diff \dc \bot$ are just the proper open subsets of $X$.

The following is \emph{Edalat's lifting trick}, which was introduced
in \cite[Section~3]{edalat95a} for dcpos, and in
\cite[Section~7.4]{alvarez00} for stably locally compact spaces.
Every continuous valuation $\nu$ on $X$ gives rise to a continuous
valuation $\nu^-$ on $X \diff \dc \bot$ by $\nu^- (U) \eqdef \nu (U)$
for every $U \in \Open (X \diff \dc \bot)$.  If $\nu \in \Val_1 X$,
then $\nu^-$ is in $\Val_{\leq 1} X$, and we have much more, as we now
show.
\begin{lemma}
  \label{lemma:VwX:pointed}
  Let $X$ be a pointed topological space, with least element $\bot$.
  The map $\nu \mapsto \nu^-$ is a homeomorphism of $\Val_{1,\wk} X$
  onto $\Val_{\leq 1,\wk} (X \diff \dc\bot)$.  Its inverse maps every
  subprobability continuous valuation $\mu$ on $X \diff \dc \bot$ to
  $\mu^+$, defined by
  $\mu^+ (U) \eqdef \mu (U \diff \dc \bot) + (1-\mu (X \diff \dc
  \bot)) \delta_\bot$, for every $U \in \Open (X \diff \dc \bot)$.
\end{lemma}
\begin{proof}
  Let $\nu \in \Val_1 X$.  For every $U \in \Open X$,
  ${(\nu^-)}^+ (U) = \nu^- (U \diff \dc \bot) + (1-\nu^- (X \diff \dc
  \bot)) \delta_\bot (U)$.  If $U$ is a proper open subset of $X$,
  then $U$ does not contain $\bot$, so $U \diff \dc \bot = U$, and
  $\delta_\bot (U)=0$, so ${(\nu^-)}^+ (U) = \nu^- (U) = \nu (U)$.  If
  $U=X$, then
  ${(\nu^-)}^+ (U) = \nu^- (X \diff \dc \bot) + (1-\nu^- (X \diff \dc
  \bot)) = 1$, and this is equal to $\nu (U)$ since $U=X$ and
  $\nu \in \Val_1 X$.

  For every $U \in \Open (X \diff \dc \bot)$,
  ${(\mu^+)}^- (U) = \mu^+ (U) = \mu (U)$, since
  $U \diff \dc \bot = U$, and $\bot$ is not in $U$.

  Hence the two maps $\nu \mapsto \nu^-$ and $\mu \mapsto \mu^+$ are
  inverse of each other.

  For every open subset $U$ of $X$ and every $r \in \Rp \diff \{0\}$,
  the inverse image of $[U > r]_1$ by $\mu \mapsto \mu^+$ is equal to
  one of the following sets.  If $U=X$ and $r<1$, this is the whole of
  $\Val_{\leq 1,\wk} X$.  If $U=X$ and $r\geq 1$, this is empty.
  Finally, if $U$ is a proper subset of $X$, hence does not contain
  $\bot$, then this is the set of all
  $\mu \in \Val_{\leq 1} (X \diff \dc \bot)$ such that $\mu^+ (U) >
  r$, where $\mu^+ (U) = \mu (U)$: hence this is $[U > r]_{\leq 1}$.
  In any case, that inverse image is open, so $\mu \mapsto \mu^+$ is
  continuous.

  For every open subset $U$ of $X \diff \dc\bot$, for every
  $r \in \Rp \diff \{0\}$, the inverse image of $[U > r]_{\leq 1}$ by
  $\nu \mapsto \nu^-$ is $[U > r]_1$.  Therefore $\nu \mapsto \nu^-$
  is continuous.
\end{proof}

Lemma~\ref{lemma:VwX:pointed} allows us to obtain the following
corollary to Theorem~\ref{thm:V:locfin}.
\begin{corollary}
  \label{corl:V1:fin}
  For every locally finitary compact, pointed space $X$,
  $\Val_{1, \wk} X$ is compact, locally finitary compact, and sober.
  In particular, it is a quasi-continuous dcpo, and the weak topology
  coincides with the Scott topology.

  The sets of the form $\interior {\upc E}$, where $E$ ranges over the
  finite non-empty sets of simple probability
  valuations form a base of the topology.  \qed
\end{corollary}

\section{The Scott and weak topologies may differ}
\label{sec:scott-weak-topol}

The Scott and weak topologies on $\Val_\ast X$ seem to agree in many
situations, and Alvarez-Manilla, Jung and Keimel asked whether they
agree on $\Val_{\leq 1} X$ for every stably compact space $X$
\cite[Section~5, second open problem]{AMJK:scs:prob}.  We show that
this is not the case.

Let $\alpha (\nat)$ be the one-point compactification of the discrete
space $\nat$.  Its elements are the natural numbers, plus a fresh
element $\infty$.  Its open subsets are the subsets of $\nat$ (not
containing $\infty$), plus all the subsets $\alpha (\nat) \diff E$,
where $E$ ranges over the finite subsets of $\nat$.  A \emph{discrete
  valuation} on $\alpha (\nat)$ is any valuation of the form
$\sum_{n \in \nat} a_n \delta_n + a_\infty \delta_\infty$, where each
$a_n$ and $a_\infty$ are in $\creal$.  They are all continuous
valuations.
\begin{lemma}
  \label{lemma:aN:discrete}
  Letting $\ast$ be ``$\leq 1$'' or ``$1$''.
  \begin{enumerate}[label=(\roman*)]
  \item The continuous valuations $\nu$ on $\alpha (\nat)$ are
    exactly the discrete valuations.
  \item The function
    $f \colon \Val_\ast (\alpha (\nat)) \to Y_\ast$ that maps
    $\sum_{n \in \nat} a_n \delta_n + a_\infty \delta_\infty$ to
    ${(a_x)}_{x \in \alpha (\nat)}$ is an order-isomorphism onto the
    poset $Y_\ast$ of families of non-negative real numbers whose sum
    is at most $1$ (if $\ast$ is ``$\leq 1$'') or exactly $1$ (if
    $\ast$ is ``$1$''), ordered pointwise.
  \item The set $\mathcal V$ of families
    ${(a_x)}_{x \in \alpha (\nat)}$ of $Y_\ast$ such that
    $a_\infty > 0$ is Scott-open in $Y_\ast$, but
    $f^{-1} (\mathcal V)$ does not contain any basic open neighborhood
    $\bigcap_{i=1}^n [U_i > r_i]_\ast$ of $\delta_\infty$.
  \end{enumerate}
\end{lemma}
\begin{proof}
  (i) Let $\nu$ be any continuous valuation over $\alpha (\nat)$.  We
  recall that every continuous valuation on an LCS-complete space
  extends to a measure on the Borel $\sigma$-algebra
  \cite[Theorem~1.1]{dBGLJL:LCS}.  Every locally compact sober space
  is $G_\delta$ in itself, hence LCS-complete.  Since every Hausdorff
  space is sober, and clearly locally compact, $\alpha (\nat)$ is
  LCS-complete, and therefore $\nu$ extends to a measure $\tilde\nu$
  on the Borel $\sigma$-algebra of $\alpha (\nat)$.  It is easy to see
  that the latter $\sigma$-algebra is the whole of $\pow (\nat)$.  We
  define $a_n \eqdef \tilde\nu (\{n\}) = \nu (\{n\})$, and
  $a_\infty \eqdef \tilde\nu (\{\infty\})$.  By $\sigma$-additivity,
  for every (necessarily measurable) subset $E$ of $\alpha (\nat)$,
  $\tilde\nu (E) = \sum_{x \in E} a_x$.  In particular, for every open
  subset $U$ of $\alpha (\nat)$,
  $\nu (U) = \sum_{x \in U} a_x = (\sum_{n \in \nat} a_n \delta_n +
  a_\infty \delta_\infty) (U)$.

  (ii) Let $\nu$ be any element of $\Val_\ast (\alpha (\nat))$, and
  $\tilde\nu$ be a measure that extends $\nu$ to the Borel
  $\sigma$-algebra.  In a more precise way as in the statement of the
  lemma, we define $f (\nu)$ as ${(a_x)}_{x \in \alpha (\nat)}$, as
  given in item~(i), so that
  $\nu = \sum_{n \in \nat} a_n \delta_n + a_\infty \delta_\infty$.
  This defines a bijection $f$ of $\Val_\ast (\alpha (\nat))$ onto
  $Y_\ast$.

  Since $\{\infty\} = \bigcap_{n \in \nat} V_n$, where $V_n$ is the
  open set $\{n, n+1, \cdots, \infty\}$, and since $\tilde\nu$ is a
  bounded measure,
  $a_\infty = \tilde\nu (\{\infty\}) = \inf_{n \in \nat} \tilde\nu
  (V_n) = \inf_{n \in \nat} \nu (V_n)$.  This implies that $a_\infty$
  grows as $\nu$ grows.  It is clear that $a_n = \nu (\{n\})$ grows,
  too, as $\nu$ grows.  Therefore $f$ is monotonic, and its inverse is
  clearly monotonic as well.  (This discussion is superfluous when $\ast$ is
  ``$1$'', by the way, since in that case the ordering on
  $\Val_1 (\alpha (\nat))$ and on $Y_1$ is just equality.)

  (iii) $\mathcal V$ is clearly Scott-open in $Y_\ast$.  We now
  imagine that $f^{-1} (\mathcal V)$ contains a basic open
  neighborhood $\bigcap_{i=1}^n [U_i > r_i]_\ast$ of $\delta_\infty$,
  where each $U_i$ is open in $\alpha (\nat)$, and
  $r_i \in \Rp \diff \{0\}$.  Since
  $\delta_\infty \in [U_i > r_i]_\ast$, $U_i$ must contain $\infty$
  (and $r_i < 1$), so $U_i = \alpha (\nat) \diff E_i$ for some finite
  subset $E_i$ of $\nat$.  Let $n$ be a natural number that is not in
  any of the finite sets $E_i$, $1\leq i\leq n$.  Then
  $\delta_n (U_i) = 1 > r_i$, so $\delta_n$ is in
  $\bigcap_{i=1}^n [U_i > r_i]_\ast$, hence in $f^{-1} (\mathcal V)$.
  However, $f (\delta_n)$ is the family
  ${(a_x)}_{x \in \alpha (\nat)}$ such that $a_x=0$ for every
  $x \in \alpha (\nat)$ except for $a_n=1$; in particular,
  $a_\infty=0$, showing that $f (\delta_n)$ is not in $\mathcal V$:
  contradiction.
\end{proof}

\begin{theorem}
  Let $\ast$ be nothing, ``$\leq 1$'' or ``$1$''.  The Scott topology
  on $\Val_\ast (\alpha (\nat))$ is strictly finer than the weak topology.
\end{theorem}
\begin{proof}
  We recall that the Scott topology on any space of the form
  $\Val_\ast X$ is always finer than the weak topology.

  When $\ast$ is ``$\leq 1$'' or ``$1$'', this is
  Lemma~\ref{lemma:aN:discrete}, item~(iii): $f^{-1} (\mathcal V)$ is
  a Scott-open neighborhood of $\delta_\infty$ in
  $\Val_\ast (\alpha (\nat))$ that is not open in the weak topology.

  When $\ast$ is nothing, we notice that
  $\Val_{\leq 1} (\alpha (\nat))$ is Scott-closed in
  $\Val (\alpha (\nat))$.  This easily implies that the Scott topology
  on $\Val_{\leq 1} (\alpha (\nat))$ is the subspace topology induced
  by the Scott topology on $\Val (\alpha (\nat))$.  If the latter
  agreed with the weak topology, then the Scott topology on
  $\Val_{\leq 1} (\alpha (\nat))$ would be the subspace topology
  induced by the inclusion in $\Val_\wk (\alpha (\nat))$.  But the
  latter is just the weak topology on $\Val_{\leq 1} (\alpha (\nat))$,
  and we have just seen that it differs from the Scott topology.
\end{proof}
The gap between the Scott and weak topologies on
$\Val_1 (\alpha (\nat))$ is really enormous.  By Corollary~37 of
\cite{AMJK:scs:prob}, $\Val_{\leq 1} X$ and $\Val_1 X$ are stably
compact for any stably compact space $X$.  This applies to
$X \eqdef \alpha (\nat)$, since every compact Hausdorff space is
stably compact.  One checks easily (e.g., by using
Lemma~\ref{lemma:aN:discrete}, item~(ii)) that the stochastic ordering
on $\Val_1 (\alpha (\nat))$ is simply equality, hence that the Scott
topology is the discrete topology.  But the only discrete spaces that
are (stably) compact are finite, and $\Val_1 (\alpha (\nat))$ is far
from finite.

The coincidence of the Scott and weak topologies of
Theorem~\ref{thm:V:locfin}, and first observed by Kirch in the case
where $X$ is a continuous dcpo, is probably exceptional.  We leave
open the question of the exact characterization of those spaces $X$
for which the weak and Scott topologies agree on $\Val_\ast X$.

\section*{Acknowledgments}
\label{sec:acknowledgments}

My deepest thanks to Xiaodong Jia, who found a mistake in a previous
version of Lemma~\ref{lemma:mincred}, and another one in a previous
version of Lemma~\ref{lemma:V:Q}.

\bibliographystyle{plain}
\ifarxiv

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Discrete nonlinear equations are ubiquitous and play an important
role in diverse physical contexts~\cite{clark,scott}.  Some examples
of integrable discrete equations include Ablowitz-Ladik (AL)
lattice~\cite{AL} and Toda lattice~\cite{toda}. Certain non-Abelian
discrete integrable models are also known \cite{nonabel}.  There are
many non-integrable discrete nonlinear equations such as discrete
sine-Gordon (DSG)~\cite{DSG1,DSG2}, discrete nonlinear Schr\"odinger
(DNLS) equation~\cite{DNLS} and the Fermi-Pasta-Ulam (FPU)
problem~\cite{ford}. DSG is a physical realization of the dynamics
of dislocations in crystals where it is known as the Frenkel-Kontorova
model~\cite{nabarro}.  It also arises in the context of ferromagnets
with planar anisotropy~\cite{mikeska}, adsorption on a crystal lattice
and pinned charge-density waves~\cite{sacco}.  Similarly, DNLS plays
a role in the propagation of electromagnetic waves in doped glass
fibers~\cite{fibers} and other optical waveguides~\cite{OP}, and it
describes Bose-Einstein condensates in optical lattices~\cite{ST}.
FPU has served as a fertile paradigm for understanding solitons,
discrete breathers, intrinsic localized modes, chaos, anomalous
transport in low-dimensional systems and the fundamentals of
statistical mechanics~\cite{ford}.  A discrete double well or
discrete $\phi^4$ equation is a model for structural phase
transitions~\cite{dphi4}, and may be relevant for a better
understanding of the collisions of relativistic kinks~\cite{luis}.

Obtaining exact (soliton-like) solutions is always desirable,
particularly for discrete systems where the notion of a discreteness
(or Peierls-Nabarro) barrier~\cite{PN,peyrard} is an important one
in that its absence is a likely indication of integrability, e.g. in
the AL case~\cite{scott}.  In addition, exact solutions allow one
to calculate certain important physical quantities analytically as
well as they serve as diagnostics for simulations.  Recently derived
summation identities~\cite{khare} involving Jacobi elliptic
functions~\cite{stegun,byrd} led to exact periodic solutions of a
modified DNLS equation~\cite{dnls}.  In this paper we exploit similar
identities to obtain exact periodic solutions of the discrete
$\phi^4$ model.

The paper is organized as follows. In the next section we summarize
the exact solitary wave solutions of the continuum $\phi^4$ model.
In Sec.~\ref{sec:discrete}, by identifying the relevant elliptic
function identities, we derive a discrete $\phi^4$ model which
allows for exact solutions. We then obtain these solutions and study
their stability. In Sec.~\ref{sec:hdyn} we explicitly write down the
corresponding Hamiltonian dynamics using a modified Poisson bracket
algebra.  Sec.~\ref{sec:scat} contains numerical results for the
scattering of both kink- and pulse-like solitary waves.
Kink--anti-kink collisions appear to create a breather with some
radiation, and pulse--anti-pulse collisions lead to a flip and
little radiation. In Sec.~\ref{sec:PN} we compute the energy of the
solitary waves, and show that the Peierls-Nabarro
barrier~\cite{PN,peyrard} due to discreteness is zero not only for
the two localized solutions but even for the three periodic
solutions. Finally, we summarize our main findings in
Sec.~\ref{sec:concl}.

\section{Continuum solitary waves}
\label{sec:cont}

The double-well potential with the coupling parameter $\lambda$ and
the two minima at $\phi=\pm a$
\begin{equation} V= \frac{\lambda}{4} \ (\phi^2-a^2)^2
\end{equation}
leads to the following relativistic field equation
\begin{equation}
   - \frac{\partial^2\phi}{\partial t^2}
   + \frac{\partial^2\phi}{\partial x^2}
   - \lambda \phi(\phi^2-a^2) = 0 \>.
\end{equation}
For $\lambda>0$, if one assumes a moving periodic solution with velocity
$v$ in terms of the Jacobi elliptic function ${\rm sn(x,m)}$ with
modulus $m$
\begin{equation}
\phi(x,t) = Aa ~{\rm sn}\left[\beta(x+x_0-vt),m\right]
\end{equation}
one finds that
\begin{align}
   - \frac{\partial^2\phi(x,t)}{\partial t^2}  &
   + \frac{\partial^2\phi(x,t)}{\partial x^2}
   \\ \notag &
   = - \beta^2 a~ (1-v^2)~[(1+m)~{\rm sn} -2m~{\rm sn}^3]
   \>,
\end{align}
where we have suppressed the argument of ${\rm sn}$. Matching terms
proportional to ${\rm sn}$ and ${\rm sn}^3$ we find we have a moving
periodic solution with
\begin{align}
   \beta & = \left[ \frac{\lambda a^2}{(1-v^2) (1+m)} \right]^{1/2} ,
   \nonumber \\
   A & = \sqrt \frac{2m}{1+m} .
\end{align}
The usual kink solitary wave is obtained in the limit
$m~\rightarrow~1$. Since \begin{equation}
   {\rm sn}(x,1) = \tanh(x) \>,
\end{equation}
we obtain
\begin{equation}
    \phi(x,t) = a \tanh \Biggl \{  \left[ \frac{\lambda
    a^2}{2(1-v^2)} \right]^{1/2}(x+x_0-vt) \Biggr \} \>.
\end{equation}
If instead $\lambda
<0$, then there are $\mathrm{dn}$ Jacobi elliptic function solutions to
the field equations. Assuming \begin{equation}
   \phi(x,t) = Aa ~{{\rm dn}}\left[\beta(x+x_0-vt),m\right] \>,
\end{equation}
and matching terms proportional to ${\rm dn}$ and ${\rm dn}^3$,
we find a moving periodic pulse solution with
\begin{align}
   \beta & = \left[ \frac{-\lambda a^2}{(1-v^2) (2-m)} \right]^{1/2} ,
   \nonumber \\
   A & = \sqrt \frac{2}{2-m}
   \>.
\end{align}

In fact, there is another pulse solution in terms of the Jacobi elliptic
function of ${\rm cn}$ type.
Assuming \begin{equation} \phi(x,t) = Aa ~{{\rm cn}}\left[\beta(x+x_0-vt),m\right]\>. \end{equation}
Matching terms proportional to ${\rm cn}$ and ${\rm cn}^3$ we find
we have a moving periodic solution with
\begin{align}
   \beta & = \left[ \frac{-\lambda a^2}{(1-v^2) (2m-1)} \right]^{1/2} ,
   \nonumber \\
   A & = \sqrt \frac{2m}{2m-1} .
\end{align}
Note that this solution is valid only if $m > 1/2$.

The usual pulse solitary wave is obtained in the limit $m
\rightarrow 1$. Since \begin{equation}
   {\rm dn}(x,1) = {\rm cn}(x,1) = {\rm sech}(x) \>,
\end{equation} we obtain \begin{equation}
   \phi(x,t) = a \sqrt{2} {\rm sech}
   \left[  \left( \frac{-\lambda a^2}{1-v^2} \right)^{1/2}(x+x_0-vt) \right] \>.
\end{equation}
Note that similar solutions and their stability were considered by
Aubry in the context of structural phase transitions~\cite{aubry}.
In the rest of this paper we will refer to the $\mathrm{sn}$ solutions
as kink-like, and to the $\mathrm{dn}$ and $\mathrm{cn}$ solutions as
pulse-like.

\section{Discretization of $\lambda \phi^4$ field theory}
\label{sec:discrete}

A naive discretization of the field equation above is
\begin{equation}
   - \ddot \phi_n +
   \frac{1}{\epsilon^2} (\phi_{n+1}+\phi_{n-1}-2\phi_n)
   + \lambda \phi_n (a^2 -\phi_n^2)=0 \>,
\end{equation}
where $\epsilon$ is the lattice parameter and the overdots represent
time derivatives.  However, this equation does not admit solutions
of the form
\begin{equation}
   \phi_n(t)  =Aa ~ {{\rm sn}} (\beta [(n+c)\epsilon-vt],m),
\end{equation}
where $c$ is an arbitrary constant.  To find solutions, one has to
modify the naive discretization.  The key for understanding how to
modify the equation comes from the following identities~\cite{khare}
of the Jacobi elliptic functions

\noindent ${\rm sn}$:
\begin{align}
   m~{\rm sn}(x,m)^2 & \left[ {\rm sn}(x+\beta,m) + {\rm sn}(x-\beta,m) \right]
   \\ \notag
   = \ & {\rm ns}^2(\beta,m) \bigl [ {\rm sn}(x+\beta,m) + {\rm sn}(x-\beta,m) \bigr]
   \\ \notag
     & - 2 {\rm cs}(\beta,m) {\rm ds}(\beta,m) {\rm sn}(x,m) \>,
\end{align}
${\rm cn}$:
\begin{align}
   m~{\rm cn}(x,m)^2 & \left[ {\rm cn}(x+\beta,m) + {\rm cn}(x-\beta,m) \right]
   \\ \notag
   = \ & -{\rm ds}^2(\beta,m) \bigl [ {\rm cn}(x+\beta,m) + {\rm cn}(x-\beta,m) \bigr]
   \\ \notag
     & + 2 {\rm cs}(\beta,m) {\rm ns}(\beta,m) {\rm cn}(x,m) \>,
\end{align}
and ${\rm dn}$:
\begin{align}
   {\rm dn}(x,m)^2 & \left[ {\rm dn}(x+\beta,m) + {\rm dn}(x-\beta,m) \right]
   \\ \notag
   = \ & - {\rm cs}^2(\beta,m) \bigl [ {\rm dn}(x+\beta,m) + {\rm dn}(x-\beta,m) \bigr ]
   \\ \notag
     & + 2 {\rm ds}(\beta,m) {\rm ns}(\beta,m) {\rm dn}(x,m) \>,
\end{align}
where
\begin{align}
   ns=\frac{1}{{\rm sn}(x,m)} \>, \quad
   cs=\frac{{\rm cn}(x,m)}{{\rm sn}(x,m)} \>, \quad
   ds=\frac{{\rm dn}(x,m)}{{\rm sn}(x,m)} \>.
\end{align}
It will be useful later to rewrite these identities in the
form:
\begin{align}
\label{eq:sn_id}
   {\rm sn}(x+\beta,m) & + {\rm sn}(x-\beta,m)
   \\ \notag
   = & \
   2 \ {\rm cs}(\beta,m) \ {\rm ds}(\beta,m) \ {\rm sn}(x,m)
   \\ \notag & \times
   [ {\rm ns}^2(\beta,m) -  m \ {\rm sn}(x,m)^2] ^{-1} ,
\end{align}
\begin{align}
\label{eq:cn_id}
   {\rm cn}(x+\beta,m) & + {\rm cn}(x-\beta,m)
   \\ \notag
   = & \
   2 \ {\rm cs}(\beta,m) \ {\rm ns}(\beta,m) \ {\rm cn}(x,m)
   \\ \notag & \times
   [ {\rm ds}^2(\beta,m) +  m \ {\rm cn}(x,m)^2] ^{-1} ,
\end{align}
and
\begin{align}
\label{eq:dn_id}
   {\rm dn}(x+\beta,m) & + {\rm dn}(x-\beta,m)
   \\ \notag
   = & \
   2 \ {\rm ds}(\beta,m) \ {\rm ns}(\beta,m) \ {\rm dn}(x,m)
   \\ \notag & \times
   [  {\rm dn}(x,m)^2  + {\rm cs}^2(\beta,m)  ] ^{-1} .
\end{align}
For the sake of brevity, in what follows we will suppress the
modulus $m$ in the argument of the Jacobi elliptic functions, except
when needed for added clarity.

\subsection{Static lattice solutions}

Consider the naive lattice equation:
\begin{equation}
   \frac{1}{\epsilon^2}
   \bigl ( \phi_{n+1}+\phi_{n-1}-2\phi_n \bigr )
   + \lambda \phi_n (a^2 - \phi_n^2) = 0 \>.
\end{equation}
Here $\phi_n \equiv \phi[ \beta\epsilon (n+c), m]$.
A general ansatz to modify the solution is to multiply the second difference
operator by the factor $(1-\alpha \phi_n^2)$, with $\alpha$ chosen so that
we get a consistent set of equations.  That is, we consider instead the equation:
\begin{equation}
   \frac{1- \alpha \phi_n^2}{\epsilon^2}
   \bigl ( \phi_{n+1}+\phi_{n-1}-2\phi_n \bigr )
   +\lambda \phi_n (a^2 - \phi_n^2) =0.  \label{eq:static}
\end{equation}
Using $\alpha$ to eliminate the $\phi_n^3$ term leads to the result
\begin{equation}
\alpha = \lambda \epsilon^2/2.
\end{equation}
This implies, in the static case, that the lattice equation is just a smeared
discretization of the $\phi_n^3$ term.  Namely, one merely needs to study
the lattice equation:
\begin{equation}
   \frac{1}{\epsilon^2}
   \bigl ( \phi_{n+1}+\phi_{n-1}-2\phi_n \bigr )
   = \frac{\lambda \phi_n^2}{2}  \bigl ( \phi_{n+1}+\phi_{n-1} \bigr ) - \lambda \phi_n  a^2 .
\end{equation}
Assume a solution of the form (for $\lambda>0$)
\begin{equation}
\phi_n = Aa~ {\rm sn}_n\,,
\end{equation}
where ${\rm sn}_n$ denotes ${\rm sn}[\beta\epsilon (n+c),m]$ with $c$ being an
arbitrary constant. Note that we only need to consider $c$ between 0
and 1/2 (half the lattice spacing).
On matching the coefficients of ${\rm sn}_n$ and ${\rm sn}_{n+1}+{\rm sn}_{n-1}$ we obtain
\begin{equation}
A^2 a^2 = \frac{(1+m) ~{\rm sn}^2(\beta \epsilon)}{\lambda \epsilon^2} ,
   \label{eq:sn1}
\end{equation}
and
\begin{equation}
\lambda a^2 = \frac{2}{\epsilon^2} \left[ 1- {\rm dn} (\beta \epsilon) {\rm cn} (\beta \epsilon)\right] \>.
   \label{eq:sn2}
\end{equation}
In the continuum limit ($\epsilon \rightarrow 0$), we get
\begin{equation}
\beta^2 = \frac{\lambda a^2}{1+m};  ~~~ A^2= \frac{2m}{1+m} \>,
\end{equation}
which agrees with our continuum result (2.5) with $v=0$.
Also as $m \rightarrow 1$ we recover the usual kink solution
\begin{equation}
\phi_n =  a \ \tanh [\beta\epsilon (n+c)] \,.
\end{equation}

If instead, we have $\lambda < 0$ and we assume
\begin{equation}
\phi_n = Aa~ {\rm dn}_n,
\end{equation}
we obtain:
\begin{equation}
A^2 a^2  = - \frac{2m}{\lambda \epsilon^2} \frac{{\rm sn}^2(\beta \epsilon)}{{\rm cn}^2(\beta \epsilon)} \>,
   \label{eq:dn1}
\end{equation}
and
\begin{equation}
\lambda a^2 \frac{2}{\epsilon^2} \left[ 1-  \frac{{\rm dn}(\beta \epsilon)}{ {\rm cn}^2(\beta \epsilon)} \right] \>.
   \label{eq:dn2}
\end{equation}
In the limit when the lattice spacing $\epsilon$ goes to zero, we obtain
\begin{equation}
\beta^2= - \frac{\lambda a^2}{2-m}; ~~~   A^2 = \frac {2}{2-m},
\end{equation}
same as the continuum case (2.9) with $v=0$.

For $\lambda < 0$ we also have a possible solution of the form
\begin{equation}
\phi_n = Aa~ {\rm cn}_n,
\end{equation}
we obtain:
\begin{equation}
A^2 a^2  = - \frac{2m}{\lambda \epsilon^2} \frac{{\rm sn}^2(\beta \epsilon)}{{\rm dn}^2(\beta \epsilon)} ,
   \label{eq:cn1}
\end{equation}
and
\begin{equation}
\lambda a^2 = \frac{2}{\epsilon^2} \left[ 1-  \frac{{\rm cn}(\beta \epsilon)}{ {\rm dn}^2(\beta \epsilon)} \right].
   \label{eq:cn2}
\end{equation}
Again taking the lattice spacing to zero, we obtain
\begin{equation}
  \beta^2= - \frac{\lambda a^2}{2m-1}; ~~~   A^2 = \frac {2m}{2m-1},
\end{equation}
same as the continuum case (2.11) with $v=0$.  Note that as before this
solution is only valid if $m > 1/2$.  For both these solutions, as
$m \rightarrow 1 $, we get our previous continuum solution:
\begin{equation}
\phi_n =  \sqrt{2} a \ {\rm sech}[\beta\epsilon(n+c)] \>.
\end{equation}

\subsection{Stability of solutions}

\begin{figure}[t!]
   \includegraphics[width=0.47\textwidth]{Fig1}
   \caption{\label{fig:stab_sn}
   (Color online) Kink-like ($\lambda>0$) case: N-dependence of the lowest eigenvalue, $\omega_1^2$, for $\lambda=1,\ a=1$, and various values of the
elliptic modulus $m$.
   }
\end{figure}

\begin{figure}[!]
   \includegraphics[width=0.47\textwidth]{Fig2}
   \caption{\label{fig:eps_sn}
   (Color online) Kink-like ($\lambda>0$) case: N-dependence of the lattice spacing, $\epsilon$,
   for $\lambda=1,\ a=1$, and various values of $m$.
   }
\end{figure}

\begin{figure}[t!]
   \includegraphics[width=0.47\textwidth]{Fig3}
   \caption{\label{fig:stab_dn}
   (Color online) Pulse-like ($\lambda<0$) ${\rm dn}$ case: N-dependence of
the lowest eigenvalue, $\omega_1^2$, for $\lambda=-1,\ a=1.5$, and
various values of $m$.
}
\end{figure}

\begin{figure}[!]
   \includegraphics[width=0.47\textwidth]{Fig4}
   \caption{\label{fig:eps_dn}
   (Color online) Pulse-like ($\lambda<0$) ${\rm dn}$ case: N-dependence of the lattice spacing, $\epsilon$,
   for $\lambda=-1,\ a=1.5$, and various values of $m$.
   }
\end{figure}

Let us now discuss the stability of these static solutions. To that purpose,
let us expand
$$\phi_n = \phi_n^{(s)} + \psi_n \exp(-i\omega t) , $$
where $\phi_n^{(s)}$ is the known solitary wave exact solution and
$\psi_n$ is a small perturbation.  Then we find that to the lowest order
in $\psi_n$ the stability equation is
\begin{eqnarray}  &&\omega^2\psi_n +\frac{1}{\epsilon^2}(\psi_{n+1}
+\psi_{n-1}-2\psi_n)
\nonumber \\ && - \lambda  \ \psi_n \phi_n^{(s)}[\phi^{(s)}_{n+1}+\phi^{(s)}_{n-1}] \nonumber \\
&& +\lambda \psi_n  a^2 - \frac{\lambda \phi_n^2}{2}
(\psi_{n+1}+\psi_{n-1})=0\>. \end{eqnarray} Using the
identities~\eqref{eq:sn_id}, (3.8) and~\eqref{eq:dn_id}, the combination
$[\phi^{(s)}_{n+1}+\phi^{(s)}_{n-1}]$ can be written as a function
of $\phi_n$ alone.  Thus schematically we have
\begin{equation} \bigl ( \omega^2 - f_n \bigr ) \psi_n
\ + \ g_n \bigl ( \psi_{n+1}+\psi_{n-1} \bigr ) \ = \ 0 \>,
\end{equation}
or
\begin{equation}
\psi_{n+1}+\psi_{n-1} + h_n \psi_n =0 \>,
\end{equation}
with
\begin{equation} h_n = \frac{\omega^2 - f_n}{g_n} \>,
\end{equation}
where $f_n$ and $g_n$ are well defined functions.  We also have
that the solutions are periodic on the lattice with $N$ sites ($\psi_{n+N}
= \psi_{n}$). If we write the system of $N$ equations in matrix form, as
\begin{equation}
\mathbf{A} \ [\psi] \ = \ \omega^2 \ [\psi] ,
\end{equation}
then the condition for nontrivial solutions is that
\begin{equation}
  \mathrm{Det}\  | \mathbf{A} - \omega^2 \mathbf{1}| = 0.
\end{equation}

In our simulations we require that the solution has exactly one
period in the model space. Therefore, the equations for $\beta$ and
$A$ are supplemented by the equations
   \begin{equation}
      N \beta \epsilon \ = \ 4 \ K(m) \>,
   \label{eq:sn3}
   \end{equation}
for solutions, $\phi_n=\mathrm{sn}(\beta n \epsilon)$ or
$\phi_n=\mathrm{cn} (\beta n \epsilon)$, and
   \begin{equation}
      N \beta \epsilon \ = \ 2 \ K(m) \>,
   \label{eq:dn3}
   \end{equation}
when $\phi_n=\mathrm{dn}(\beta n \epsilon)$.  Here $K(m)$ is the
complete elliptic integral of the first kind \cite{stegun,byrd}.
Hence, once the parameters $m$, $\lambda$ and $a$ are
specified, we need to solve a system of equations for
$\beta$, $\epsilon$, and~$A$, i.e. we have to solve
Eqs.~(\ref{eq:sn1}, \ref{eq:sn2}, \ref{eq:sn3}) or
Eqs.~(\ref{eq:dn1}, \ref{eq:dn2}, \ref{eq:dn3}), or Eqs. (\ref{eq:cn1}, \ref{eq:cn2}, \ref{eq:sn3}),
for the solutions $\mathrm{sn}$, $\mathrm{dn}$, and $\mathrm{cn}$, respectively, for
$\beta$, $\epsilon$, and $A$ as a function of~N.

Typical results are depicted in Figs.~\ref{fig:stab_sn} and
\ref{fig:eps_sn} for kink-like ${\rm sn}$ solutions, and in
Figs.~\ref{fig:stab_dn} and \ref{fig:eps_dn}, for pulse-like ${\rm dn}$
solutions, respectively. Similar results can also be obtained in the
case of the ${\rm cn}$-type pulse solution. For the case of the kink-like
solutions and our choice of parameters, we find from
Fig.~\ref{fig:stab_sn}, that stability requires $N>7$ lattice sites.
In the pulse-like case, we find stability for arbitrary values of
$N$. The magnitude of the lowest eigenvalues increases with $N$ in
the kink-like case, while it decreases with N for pulse-like
solutions. For fixed $m$, $\lambda$, and $A$, the lattice spacing,
$\epsilon$, is not an independent quantity, but it is a well-defined
function of~$N$. As seen from Figs.~\ref{fig:eps_sn} and~\ref{fig:eps_dn},
the lattice spacing, $\epsilon$, is always a decreasing function
of~$N$.

\section{Hamiltonian dynamics}
\label{sec:hdyn}
In this section we demonstrate that our discrete model is a Hamiltonian
system.  The equation which the static solutions obey,
Eq.~(\ref{eq:static}), can be written as:
\begin{equation}
   \frac{1}{\epsilon^2} \bigl ( \phi_{n+1}+\phi_{n-1}-2\phi_n \bigr )
+\lambda \frac{\phi_n (a^2 - \phi_n^2)}{1- \alpha \phi_n^2} =0.  \label{eq:static2}
\end{equation}
We recognize that this equation can be derived from the potential function
\begin{equation}
V_0[\phi_n]= \sum_n \frac{(\phi_{n+1}-\phi_n)^2}{2 \epsilon^2}
+ \lambda \sum_n \int d\phi_n \frac{ \phi_n (a^2 - \phi_n^2)}{1- \alpha \phi_n^2}.
\end{equation}
Performing the integral we obtain explicitly:
\begin{align}
   V_0[\phi_n]= & \sum_n \frac{(\phi_{n+1}-\phi_n)^2}{2 \epsilon^2}
   \label{eq:V0} \\ \notag &
   \ - \ \lambda  \frac{ \phi_n^2}{2\alpha}
   \ + \
   \lambda (a^2 \alpha-1) \ \frac{\ln(1- \alpha \phi_n^2)}{2 \alpha^2}.
\end{align}
One easily verifies that
\begin{align}
   - (1- \alpha \phi_n^2) \ \frac{\partial V_0}{\partial \phi_n}
   \ = \ &
   \frac{1- \alpha \phi_n^2}{\epsilon^2}
   \bigl ( \phi_{n+1} +\phi_{n-1}-2\phi_n \bigr )
   \notag \\ &
   \ + \ \lambda \phi_n (a^2 - \phi_n^2) .
\end{align}
The continuum limit $\alpha \rightarrow 0$ (or $\epsilon \rightarrow
0$) of the last two terms in Eq.~\eqref{eq:V0} is given by
\begin{align}
   \frac{\lambda}{4} (\phi_n^2 - a^2)^2 \ - \ \frac{\lambda}{4} a^4
   \>.
\end{align}

We will now show that if we want our class of solutions to be the
static limit of a Hamiltonian dynamical system, that we will be able
to have single solitary waves obey a simple equation, but general
solutions will obey a more complicated dynamics with terms
proportional to ${\dot \phi}^2$. For simplicity we will assume that
the Hamiltonian takes the form: \begin{equation} H = \sum_n \left( \frac{\pi^2_n}{2}
 \, g[\phi_n] \ + \ V[\phi_n] \right ) \label{eq:H}, \end{equation}
with $V$ given by $V_0$ in Eq.~(\ref{eq:V0}) plus possibly some
additional terms that vanish in the continuum limit.  Here $\pi_n$
is the conjugate momentum and $g[\phi_n]$ a weight function.  For
generality we will assume, as in the case of the discrete nonlinear
Schr\"odinger (DNLS) equation~\cite{dnls,das}, that an extended
Poisson bracket structure exists, namely: \begin{equation} \left\{ \phi_m, \pi_n
\right\} = \delta_{nm} f[\phi_n] , \end{equation} and
\begin{align}
{\dot \phi}_n = \left\{\phi_n, H \right\} &= \frac {\partial H}{\partial \pi_m} \left\{\phi_n, \pi_m\right\} = f[\phi_n] \frac {\partial H}{\partial \pi_n} ,
\nonumber \\
{\dot \pi}_n = \left\{\pi_n, H\right\} &= \frac {\partial H}{\partial \phi_m}\left\{\pi_n, \phi_m\right\}= -f[\phi_n] \frac {\partial H}{\partial \phi_n} .
\nonumber \\
\end{align}
From our ansatz, Eq.~(\ref{eq:H}), we obtain the first order equations:
\begin{equation}
{\dot \phi}_n = \pi_n f[\phi_n] g[\phi_n] \equiv \pi_n h[\phi_n] ,
\end{equation}
and
\begin{equation}
{\dot \pi}_n= - f[\phi_n] \left( \frac{\pi_n^2}{2} \frac{\partial g}{\partial \phi_n} + \frac{\partial V }{\partial \phi_n}   \right) .
\end{equation}
This leads to the following second order differential equation for $\phi_n$
\begin{equation}
{\ddot \phi}_n = - f[\phi_n] h[\phi_n] \frac{\partial V}{\partial \phi_n} +{\dot \phi}^2 \left(\frac{1}{h} \frac{\partial h}{\partial \phi_n} - \frac{f}{2h} \frac{\partial g}{\partial \phi_n} \right).
\end{equation}
For this equation to have the previously found static solitary waves, as well as the correct
continuum limit, we need only that
\begin{equation}
f[\phi_n] h[\phi_n] = f^2[\phi_n] g[\phi_n] = 1- \alpha \phi_n^2.
\end{equation}
The two simplest cases are: (a) $f=1$  (ordinary Poisson brackets) and
(b) $h=1$ (extended Poisson brackets).  First consider the case $ f=1$.
This requires
\begin{equation}
h[\phi_n] = g[\phi_n] = 1- \alpha \phi_n^2 .
\end{equation}
From this we get, if we choose $V=V_0$ with $V_0$ given by
Eq.~(\ref{eq:V0}), the equation:
\begin{align}
   \ddot \phi_n \ = \ &
   \frac{1- \alpha \phi_n^2}{\epsilon^2} \bigl ( \phi_{n+1}+\phi_{n-1}-2\phi_n \bigr )
   \\ \notag &
   \ + \ \lambda \phi_n (a^2 - \phi_n^2)
   \ - \ \frac{\alpha{\dot \phi_n}^2 \phi_n}{1-\alpha\phi_n^2} .
\end{align}
For $\alpha = \lambda {\epsilon^2}/{2}$ this equation has the previously
found static lattice solitary wave solutions as well as the correct
continuum limit.
For the case $h=1$, this leads to
\begin{equation}
f=  1- \alpha \phi_n^2\>, ~~~  g= (1- \alpha \phi_n^2)^{-1}\>.
\end{equation}
We find that for $ V = V_0$  the equation of motion again becomes
the {\it same} as Eq.~(4.14).

Unfortunately the presence of the ${\dot \phi}^2$ term does not allow
for single elliptic time dependent solitary wave solutions that are of
the form ${\rm sn}[\beta(x-ct)]$, ${\rm cn}[\beta(x-ct)]$
or ${\rm dn}[\beta(x-ct)]$ where $x = n \epsilon$.
This is because the second derivative of the ${\rm sn}$ and ${\rm dn}$ (or ${\rm cn}$)
functions contains both linear as well as cubic terms, and
the quantity ${\dot \phi}^2$ is a quartic polynomial in ${\rm sn}$ or
${\rm dn}$ (or ${\rm cn}$).  Thus the last term is equivalent to a non-polynomial
potential when applied to a single elliptic function solution.
Therefore in order to obtain a simple elliptic function solution in the
time dependent case, one must add a non-polynomial potential to $V_0$
which is chosen to exactly cancel the last term when evaluated for a
single elliptic solitary wave solution of the form  ${\rm sn}[\beta(x-ct)]$,
${\rm cn}[\beta(x-ct)]$ or
${\rm dn}[\beta(x-ct)]$. This means that to obtain an equation that is
derivable from a Lagrangian or a Hamiltonian we should add to the
static potential $V_0$ an additional contribution $\Delta V$ such
that
\begin{equation}\label{4.16}
(1-\alpha\phi_n^2)\ \frac{\partial\Delta
V}{\partial\phi_n}=-\frac{\alpha\phi_n {\dot
\phi_n}^2}{1-\alpha\phi_n^2} ,
\end{equation}
where $\phi_n$ is a single
solitary wave described by a time translated elliptic function. It
will turn out that $\Delta V$ needed to obtain a simple solution of
the elliptic kind will depend on the velocity squared of the
solitary wave as well as the type of solution (pulse- or kink-like).

In general we have that the Hamiltonian dynamics leads to
\begin{equation}
{\ddot \phi}_n=-fh\frac{\partial V_0 +\Delta V}{\partial\phi_n}+{\dot \phi_n}^2
\left(\frac{1}{h}\frac{\partial h}{\partial\phi_n}-\frac{f}{2h}\frac{\partial
g}{\partial\phi_n}\right) .
\end{equation}
Since $fh=1-\alpha\phi_n^2$ and $h=fg$, it follows that
\begin{equation}
\frac{1}{h}\frac{\partial h}{\partial\phi_n}-\frac{f}{2h}\frac{\partial
g}{\partial\phi_n} = -\frac{\alpha\phi_n}{1-\alpha\phi_n^2} ,
\end{equation}
from where the above condition for $\Delta V$ follows readily.  For the single solitary
wave solutions of the ${\rm sn}$, ${\rm cn}$ and ${\rm dn}$ type one  wants to choose
\begin{equation}\label{4.19}
\frac{\partial\Delta V}{\partial\phi_n}=\frac{a_1\phi_n^5+a_2\phi_n^3+a_3\phi_n}
{(1-\alpha\phi_n^2)^2} ,
\end{equation}
with $a_i$ dependent on the choice of elliptic function, in order to
cancel the effect of the ${\dot \phi}^2$ terms in the corresponding
equation of motion.

Integrating, we get for $\Delta V$
\begin{align}
   \Delta V
   \ = \ & \frac{a_1\phi^2}{2\alpha^2}+\frac{1}{2(1-\alpha\phi^2)}\left(\frac{a_1}
{\alpha^3}+\frac{a_2}{\alpha^2}+\frac{a_3}{\alpha}\right)
   \notag \\ &
\label{4.20}
   +\left(\frac{a_1}
   {\alpha^3}+\frac{a_2}{2\alpha^2}\right) {\rm ln}(1-\alpha\phi^2) .
\end{align}
If we assume a soliton solution of the form
\begin{equation} \phi_n = Aa ~{\rm sn} \Bigl \{ \beta [(n+c)\epsilon-vt] ,m \Bigr \} \>,
\end{equation}
where $c$ is an arbitrary constant, then
\begin{equation}
{\dot \phi_n}^2=\beta^2v^2(Aa)^2[1-(1+m)\ {\rm sn}^2+m\ {\rm sn}^4] ,
\end{equation}
which leads to
\begin{align}
   a_1 \ = \ & -\ m\alpha\left(\frac{\beta v}{Aa}\right)^2 ,
   \\
   a_2 \ = \ & \alpha\beta^2 v^2(1+m) ,
   \\
   a_3 \ = \ & -\ \alpha(\beta v Aa)^2 .
\end{align}
Explicitly,  for the ${\rm sn}$ type of solitary wave with a kink limit
when $m=1$  we need to choose the extra potential term to satisfy
\begin{equation} \label{4.23}
-\frac{\partial\Delta
V}{\partial\phi_n}=\frac{\alpha(\beta vAa)^2\phi_n}
{(1-\alpha\phi_n^2)^2}\left[1-\frac{(1+m)\phi_n^2}{(Aa)^2}+\frac{m\phi_n^4}
{(Aa)^4}\right] . \end{equation}

If instead, we assume a pulse soliton solution (for $\lambda~<~0$) of
the form
\begin{equation}
\phi_n=Aa ~{\rm dn} \Bigl \{ \beta [(n+c)\epsilon-vt] ,m \Bigr \} \>,
\end{equation}
then we have
\begin{equation}
{\dot \phi_n}^2=\beta^2v^2(Aa)^2[(m-1)+(2-m){\rm dn}^2-{\rm dn}^4] ,
\end{equation}
which leads to the (different set) of coefficients $\{ a_i \}$ for
the extra term needed to be added to the Hamiltonian
\begin{align}
   a_1 \ = \ & \alpha\left(\frac{\beta v}{Aa}\right)^2 \>,
   \\
   a_2 \ = \ & - \ \alpha\beta^2 v^2(2-m) \>,
   \\
   a_3 \ =\ & (1-m)\alpha({\beta v}{Aa})^2 \>.
\end{align}
Explicitly, for the  case of ${\rm dn}$ solitary waves we choose our additional potential term to satisfy:
\begin{align}
\label{4.29}
   - \frac{\partial\Delta V}{\partial\phi_n}
   = &
   \frac{\alpha(\beta vAa)^2\phi_n}{(1-\alpha\phi_n^2)^2}
   \\ \notag & \quad \times
   \left[(m-1)+\frac{(2-m)\phi_n^2}{(Aa)^2}-\frac{\phi_n^4} {(Aa)^4}\right] .
\end{align}

Finally, we assume a pulse solution (for $\lambda < 0$) of the form
\begin{equation}
\phi_n=Aa ~{\rm cn} \Bigl \{ \beta [(n+c) \epsilon-vt +c] ,m \Bigr \} \>,
\end{equation}
then we have
\begin{equation}
{\dot \phi_n}^2=\beta^2v^2(Aa)^2[(1-m)+(2m-1){\rm cn}^2-m \ {\rm cn}^4] ,
\end{equation}
which leads to the (different set) of coefficients $\{ a_i \}$ for
the extra term needed to be added to the Hamiltonian
\begin{align}
   a_1 \ = \ & m\alpha\left(\frac{\beta v}{Aa}\right)^2 \>,
   \\
   a_2 \ = \ & - \ \alpha\beta^2 v^2(2m-1) \>,
   \\
   a_3 \ =\ & -(1-m)\alpha({\beta v}{Aa})^2 \>.
\end{align}
Explicitly, for the  case of ${\rm cn}$ solitary waves we choose our
additional potential term to satisfy:
\begin{align}
\label{4.35}
   - \frac{\partial\Delta V}{\partial\phi_n}
   = &
   \frac{\alpha(\beta vAa)^2\phi_n}{(1-\alpha\phi_n^2)^2}
   \\ \notag & \quad \times
   \left[(1-m)+\frac{(2m-1)\phi_n^2}{(Aa)^2}-\frac{m\phi_n^4} {(Aa)^4}\right] .
\end{align}

Because of our ``fine tuning" of $\Delta V$, in all cases, the solitary
waves effectively obey the second order
differential difference equation:
\begin{align}
   - \ddot\phi_n &
   + \frac{1-\alpha \phi_n^2}{\epsilon^2} \bigl ( \phi_{n+1}+\phi_{n-1}-2\phi_n \bigr )
   \notag \\ &
   + \lambda \phi_n (a^2 -\phi_n^2)=0.
\end{align}
As long as $\alpha$  is proportional to $\epsilon^2$, we expect that this
equation has the correct continuum limit.  Next, we demonstrate this for
the three cases explicitly.

\subsection{Positive $\lambda$ and kink solutions}

If we assume a  solution of the form
\begin{equation} \phi_n = Aa ~{\rm sn} \Bigl \{ \beta[ (n+c) \epsilon-vt ] ,m \Bigr \} \>,
\end{equation}
where $c$ is an arbitrary constant we obtain matching terms proportional to
$\bigl [ {\rm sn}_{n+1}+{\rm sn}_{n-1} \bigr ]$ that
\begin{equation}
{\rm sn}^2(\beta \epsilon) = \alpha \frac{(Aa)^2}{m} \label{eq:plus1} .
\end{equation}
From the term linear in ${\rm sn}$ we then obtain
\begin{equation}
\lambda a^2 = \frac {2}{\epsilon^2} [1- {\rm dn}(\beta \epsilon) {\rm cn}(\beta \epsilon)] - (1+m) \beta^2 v^2 .
\end{equation}
When $v=0$, we get our previous static solution (3.16).  This equation
becomes in the continuum limit ($\epsilon \rightarrow 0$)
\begin{equation}
\beta^2 = \frac{\lambda a^2}{(1+m)(1-v^2)} .
\end{equation}
Finally, matching the cubic terms in ${\rm sn}$ yields an equation for $A$ namely
\begin{equation}
   \lambda A^2 a^2 =2 m \frac{ {\rm sn}^2(\beta \epsilon)}{\epsilon^2} - 2 m \beta^2 v^2 .
   \label{eq:cubic1}
\end{equation}
From this we deduce that
\begin{equation}
A^2 =  \frac{ m \left[ {\rm sn}^2(\beta \epsilon)-  \beta^2 v^2 \epsilon^2 \right]}
{ [1- {\rm dn}(\beta \epsilon) {\rm cn}(\beta \epsilon)] - (1+m) \epsilon^2  \beta^2 v^2/2} .
\end{equation}
In the continuum limit we get
\begin{equation}
A^2 = \frac {2m}{1+m} ,
\end{equation}
which agrees with our continuum result (3.17).
If we divide (\ref{eq:plus1}) by (\ref{eq:cubic1}) we obtain:
\begin{equation}
\frac{\alpha}{\lambda} = \frac {{\rm sn}^2(\beta \epsilon)} {2 [{{\rm sn}^2(\beta \epsilon)}/{\epsilon^2} - \beta^2 v^2 ]} .
\end{equation}
For small lattice spacing we get the result
\begin{equation}
\alpha \rightarrow \frac {\lambda \epsilon^2}{2(1-v^2)} .
\end{equation}
This shows that in order to have a solitary wave of velocity $v$, the
discretization depends on $v^2$.

When $v=0$ this reduces to the static case:
\begin{align*}
   \alpha a^2 A^2 & = m \ {\rm sn}^2(\beta \epsilon)\>,
   \\
   \lambda a^2 A^2 & = \frac{2 m \ {\rm sn}^2(\beta \epsilon)}{\epsilon^2}\>,
   \\
   \lambda a^2 & = \frac{2}{\epsilon^2} \bigl [ 1-{\rm cn}(\epsilon \beta){\rm dn}(\epsilon \beta) \bigr]
   \>,
\end{align*}
which implies that
\begin{align*}
   \alpha a^2 & =1-{\rm cn}(\beta \epsilon){\rm dn}(\beta \epsilon)\>,
   \\
   \alpha & = \lambda \epsilon^2 /2 \>,
   \\
   A^2 & = \frac{m \ {\rm sn}^2(\beta \epsilon)}{1-{\rm cn}(\beta \epsilon){\rm dn}(\beta \epsilon)}
   \>.
\end{align*}

\emph{Localized mode}: Let us now consider the $m \rightarrow 1$ limit
in which case we get a localized kink solution
\begin{equation}\label{4.71}
\phi_n = Aa \ \tanh[\beta(n+c)\epsilon-vt]\>,
\end{equation}
where
\begin{align}
\label{4.72}
   A=1\>, \quad \alpha a^2 = \tanh^2(\beta \epsilon)\>,
   \\ \notag
   \lambda a^2 =\frac{2}{\epsilon^2} \tanh^2(\beta\epsilon)-2\beta^2 v^2\,.
\end{align}

\subsection{Negative $\lambda$ and pulse solutions}

If we assume a  solution of the form
\begin{equation} \phi_n = Aa ~{\rm dn} \Bigl \{ \beta ([n+c] \epsilon-vt) ,m \Bigr \} \>,
\end{equation}
where $c$ is an arbitrary constant we obtain matching terms proportional to
$\bigl ( {\rm dn}_{n+1}+{\rm dn}_{n-1}\bigr )$ that
\begin{equation}
\frac {{\rm sn}^2(\beta \epsilon)}{{\rm cn}^2(\beta \epsilon)} = - \alpha (Aa)^2.
\label{eq:alpha2}
\end{equation}
From the term linear in ${\rm dn}$ we then obtain
\begin{equation}
-\lambda a^2 = \frac {2}{\epsilon^2}\left[ \frac{{\rm dn}(\beta \epsilon)}
{{\rm cn}^2(\beta \epsilon)}-1\right] - (2-m) \beta^2 v^2 .
\end{equation}
When $v=0$, we get our previous static solution (3.21). This equation
becomes in
the continuum limit ($\epsilon \rightarrow 0$)
\begin{equation}
\beta^2 =-  \frac{\lambda a^2}{(2-m)(1-v^2)} .
\end{equation}
Finally, matching the cubic terms in ${\rm dn}$ yields an equation for $A$ namely
\begin{equation}
-\lambda A^2 a^2 =  \frac{2} {\epsilon^2} \frac {{\rm sn}^2(\beta \epsilon)}
{{\rm cn}^2(\beta \epsilon)} - 2 \beta^2 v^2  \label{eq:lambda2} .
\end{equation}
From this we deduce that
\begin{equation}
A^2 =  \frac{ {\rm sn}^2(\beta \epsilon)-
\beta^2 v^2 \epsilon^2 {\rm cn}^2(\beta \epsilon)}
{ \Bigl [{\rm dn}(\beta \epsilon)-{\rm cn}^2(\beta \epsilon) \Bigr ]
- (2-m) \epsilon^2  \beta^2 v^2 {\rm cn}^2(\beta \epsilon) /2} \>.
\end{equation}
In the continuum limit we get
\begin{equation}
A^2 = \frac {2}{2-m} \>,
\end{equation}
which agrees with our  previous continuum result (3.22).

Dividing (\ref{eq:alpha2}) by (\ref{eq:lambda2}) we obtain
\begin{equation}
\frac{\alpha}{\lambda} = \frac{\epsilon^2 {\rm sn}^2(\beta \epsilon) }
{2  \Bigl[ {\rm sn}^2(\beta \epsilon) - \epsilon^2 \beta^2 v^2 {\rm cn}^2(\beta \epsilon) \Bigr]} \>.
\end{equation}
For small $\epsilon$ we have
\begin{equation}
\alpha =\frac{ \lambda \epsilon^2}{ 2 (1-v^2)} .
\end{equation}
When $v \rightarrow 0$, $ \alpha = \lambda \epsilon^2 /2$.  This exactly
cancels the $\lambda \phi_n^3$ term in the equation of motion and we get the
simple discretization for the time independent case.  In the time dependent
case we again  have the unusual result that the discretization needed is a
function of the velocity of the solitary wave.

If instead, we assume a  solution of the form
\begin{equation} \phi_n = Aa ~{\rm cn} \Bigl \{ \beta [(n+c)\epsilon-vt], m \Bigr \} \>,
\end{equation}
where $c$ is an arbitrary constant we obtain matching terms proportional to
$\bigl ( {\rm cn}_{n+1}+{\rm cn}_{n-1}\bigr )$ that
\begin{equation}
m \ \frac {{\rm sn}^2(\beta \epsilon)}{{\rm dn}^2(\beta \epsilon)} = - \alpha (Aa)^2.
\label{eq:alpha21}
\end{equation}
From the term linear in ${\rm cn}$ we then obtain
\begin{equation}
-\lambda a^2 = \frac {2}{\epsilon^2}\left[ \frac{{\rm cn}(\beta \epsilon)}
{{\rm dn}^2(\beta \epsilon)}-1\right] - (2m-1) \beta^2 v^2 .
\end{equation}
When $v=0$, we get our previous static solution (3.25). This equation
becomes in the continuum limit ($\epsilon \rightarrow 0$)
\begin{equation}
\beta^2 =-  \frac{\lambda a^2}{(2m-1)(1-v^2)} .
\end{equation}
Finally, matching the cubic terms in ${\rm cn}$ yields an equation for $A$ namely
\begin{equation}
-\lambda A^2 a^2 =  \frac{2m} {\epsilon^2} \frac {{\rm sn}^2(\beta \epsilon)}
{{\rm dn}^2(\beta \epsilon)} - 2 m \beta^2 v^2  \label{eq:lambda21} .
\end{equation}
From this we deduce that
\begin{equation}
A^2 =  \frac{ m \Bigl [ {\rm sn}^2(\beta \epsilon) - \beta^2 v^2 \epsilon^2 {\rm dn}^2(\beta \epsilon) \Bigr ]}
{ \Bigl [{\rm cn}(\beta \epsilon)-{\rm dn}^2(\beta \epsilon)\Bigr ]  - (2m-1) \epsilon^2  \beta^2 v^2 {\rm dn}^2(\beta \epsilon) /2}
\>.
\end{equation}
In the continuum limit we get
\begin{equation}
A^2 = \frac {2m}{2m-1} \>.
\end{equation}
which agrees with our  previous continuum result (3.26).
Dividing (\ref{eq:alpha21}) by (\ref{eq:lambda21}) we obtain
\begin{equation}
\frac{\alpha}{\lambda} = \frac{\epsilon^2 {\rm sn}^2(\beta \epsilon)}
{2  \Bigl [{\rm sn}^2(\beta \epsilon) - \epsilon^2 \beta^2 v^2 {\rm dn}^2(\beta \epsilon) \Bigr ]} \>.
\end{equation}
For small $\epsilon$ we have
\begin{equation}
\alpha =\frac{ \lambda \epsilon^2}{ 2 (1-v^2)} .
\end{equation}
When $v \rightarrow 0$, $ \alpha = \lambda \epsilon^2 /2$.  This exactly
cancels the $\lambda \phi_n^3$ term in the equation of motion and we get the
simple discretization for the time independent case.  In the time dependent
case we again  have the unusual result that the discretization needed is a
function of the velocity of the solitary wave.

\emph{Localized mode}: In the limit of $m \rightarrow 1$, both ${\rm cn}$
and ${\rm dn}$ solutions reduce to the localized pulse solution
\begin{equation}\label{4.73}
\phi_n = Aa \ {\rm sech} \Bigl \{ \beta[(n+c)\epsilon-vt] \Bigr \} \>,
\end{equation}
where
\begin{align}
\label{4.74}
-\alpha a^2 A^2 & = \sinh^2(\beta\epsilon)\>,
\\ \notag
-\lambda a^2 & =\frac{2}{\epsilon^2} [\cosh(\beta\epsilon)-1] -\beta^2 v^2
\>,
\\ \notag
-\lambda \ a^2 A^2 & \frac{2}{\epsilon^2}\sinh^2(\beta\epsilon)-2\beta^2 v^2\>.
\end{align}

\begin{figure}[t!]
   \includegraphics[height=0.8\textheight]{Fig5}
   \caption{\label{fig:Fig_sn}
   (Color online) Scattering of kink--anti-kink waves. For completeness, at $t=0$, we
   also depict the time-derivative of the initial wave function (in
   green).
   }
\end{figure}

\begin{figure}[t!]
   \includegraphics[height=0.8\textheight]{Fig6}
   \caption{\label{fig:Fig_pm}
   (Color online) Scattering of ${\rm dn}$ pulse--anti-pulse waves. For completeness, at $t=0$, we
   also depict the time-derivative of the initial wave function (in
   green).
   }
\end{figure}

\section{Scattering of solitary waves}
\label{sec:scat}

When we have two solitary waves colliding with opposite velocity,
then the equation of motion depends on whether we are considering
${\rm sn}$, ${\rm cn}$ or ${\rm dn}$ type solitary waves. In general we have
\begin{align}
   {\ddot \phi}_n  = &
   \frac{1- \alpha \phi_n^2}{\epsilon^2} \bigl [ \phi_{n+1}+\phi_{n-1}-2\phi_n \bigr ]
   + \lambda \phi_n (a^2 - \phi_n^2)
   \notag \\ &
   -\frac {\alpha} {1- \alpha \phi_n^2} {\dot \phi_n}^2 \phi_n
   -(1-\alpha \phi_n^2) \frac{\partial \Delta V}{\partial \phi_n}
   \>,
\end{align}
where the partial derivatives of $\Delta V$ are given by
Eqs.~\eqref{4.23}, \eqref{4.29}, and~\eqref{4.35},
for the three types of solutions, respectively. The
$\Delta V$ term given by Eq.~\eqref{4.20}
adds an extra term to the energy conservation
equation.
The conserved energy is given by
\begin{align}\label{5.5}
  E = \sum_n \biggl [ & \frac{\dot \phi_n^2}{2 (1- \alpha \phi_n^2)}
  + \frac{(\phi_{n+1}-\phi_n)^2}{2 \epsilon^2}
  - \frac{\lambda}{2\alpha} \, \phi_n^2
  \notag \\ &
  + \lambda (a^2 \alpha-1) \frac{\ln(1- \alpha \phi_n^2)}{2 \alpha^2} \biggr ]
  \ + \ \Delta V \>.
\end{align}

Typical scenarios are depicted in Figs.~\ref{fig:Fig_sn}
and~\ref{fig:Fig_pm}, for the scattering case of two kink--anti-kink
waves, and two pulse--anti-pulse waves, respectively. The solution
of the kink--anti-kink ``scattering'' waves appears to correspond to
a spatially localized, persistent time-periodic oscillatory
bound-state (or a breather~\cite{breather1,breather2}) with some
radiation (phonons) at late times.  Breathers are intrinsically
dynamic nonlinear excitations and can be viewed as a bound state
of phonons.  The scattering of pulse--anti-pulse is different: there
is a flip after collision and relatively less radiation.

\section{ Energy of solitary waves and the Peierls-Nabarro barrier}
\label{sec:PN}

In a discrete lattice there is an energy cost associated with moving
a localized mode by a half lattice constant, known as
Peierls-Nabarro (PN) barrier~\cite{PN,peyrard}.  In the $
m\rightarrow 1$ limit the elliptic functions become localized and
become either pulses, or kink-like solitary waves. We shall now show
that, rather remarkably, the PN barrier vanishes in this model for
both types of localized modes. In fact we prove an even stronger
result, that the PN barrier is zero for all three periodic
solutions (in terms of ${\rm sn},{\rm cn}$ and ${\rm dn}$). If the folklore of
nonzero PN barrier being indicative of non-integrability of the
discrete nonlinear system is correct, then \'a l\`a AL lattice, this
could also be an integrable model. It would be worthwhile if one can
prove this explicitly by finding the Lax pair for this model.

We start from the conserved energy expression given by Eq.~(\ref{5.5})
with $\Delta V$ of Eq.~(\ref{4.20}). First of
all, we shall show that in the time-dependent as well as in the static
cases, the conserved energy is given by
\begin{align}
\label{6.1}
  V[\phi_n]    \sum_n \left[
   - \frac{\phi_{n+1}\phi_n}{ \epsilon^2}
 +B\ln (1- \alpha \phi_n^2) +A
   \right]\>,
\end{align}
where the constants A and B vary for each case.
We first note that in view of Eqs. (\ref{4.16}) and (\ref{4.19}), we
have
\begin{align}
\label{6.2}
\frac{\dot \phi_n^2}{2(1-\alpha \phi_n^2)} \ = \ &
-\frac{1}{2\alpha(1-\alpha \phi_n^2)}
\Bigl ( a_1 \phi_n^4 + a_2 \phi_n^2 + a_3 \Bigr )
\>.
\end{align}
Combining this term with $\Delta V$ of Eq.~\eqref{4.20}
in the energy expression~(\ref{5.5}),
we then find that the conserved energy is given by
\begin{align}
\label{6.3}
  E = \sum_{n}
  \biggl [ &
     \left (
     \frac{1}{\epsilon^2}-\frac{\lambda}{2\alpha}
     + \frac{a_1}{\alpha^2}
     \right )\phi_n^2
     - \frac{\phi_{n+1}\phi_n}{\epsilon^2}
  \\ \notag &
  + \frac{1}{2\alpha^2} \Bigl ( a_2+\frac{a_1}{\alpha} \Bigr)
  \\ \notag &
  + \left (\frac{\lambda a^2}{2 \alpha}
           -\frac{\lambda}{2\alpha^2}
           +\frac{a_1}{\alpha^3}
           +\frac{a_2}{2\alpha^2}
  \right )
 \ln(1- \alpha \phi_n^2)
 \biggr ] \>.
\end{align}

Quite remarkably we find that in the case
of all the (i.e. ${\rm sn},{\rm cn}$ as well as ${\rm dn}$) solutions
\begin{equation}\label{6.4}
  \frac{1}{\epsilon^2}-\frac{\lambda}{2\alpha}
+\frac{a_1}{\alpha^2} = 0\>,
\end{equation}
where use has been made of relevant equations in Sec.~\ref{sec:hdyn}. As a result
the $\phi_n^2$ term vanishes. Thus the conserved energy takes a
rather simple form as given by Eq.~(\ref{6.1})
in the case of all of our solutions, and the constants $A$ and $B$ are
given by
\begin{align}
\label{6.4a}
  A \ = \ & \frac{1}{2\alpha^2} \ \Bigl ( a_2+\frac{a_1}{\alpha} \Bigr)
  \>,
  \\
\label{6.4b}
  B \ = \ & \frac{\lambda a^2}{2\alpha}
  -\frac{\lambda}{2\alpha^2}
  +\frac{a_1}{\alpha^3}+\frac{a_2}{2\alpha^2}
  \>.
\end{align}
Here $a_1,a_2,a_3$ have different values for the three cases and
are as given in Sec.~\ref{sec:hdyn}. In the special
case of the static solutions, there is a further simplification in
that the constant term also vanishes, since $a_{1,2,3}$ are all
zero in that case.

Let us now calculate the PN barrier in the case of the three
periodic solutions obtained in Sec.~\ref{sec:hdyn} and show that it
vanishes in all the three cases. Before we give the details, let us
explain the key argument. If we look at the conserved energy
expression as given by Eq.~(\ref{6.1}), we find that there are two
$c$-dependent sums involved here. We also observe that in these
expressions, time $t$ and the constant $c$ always come together in
the combination \begin{equation}\label{6.5} k_1=\beta(c\epsilon-vt)~. \end{equation}
Further, using the recently discovered identities for Jacobi
elliptic functions we explicitly show that for all the solutions the
first sum is $c$-independent. Since the total energy $E$ as given by
Eq.~(\ref{6.1}) is conserved, its value must be independent of time
$t$ and since time $t$ and the constant $c$ always come together in
the combination $k_1$ as given by Eq.~(\ref{6.5}), it then follows
that the second sum must also be $c$-independent and thus there is
no PN barrier for any of our periodic, and hence also the localized
pulse or kink, solutions.

In particular, the following three
cyclic identities~\cite{khare} will allow us to explicitly perform the
first sum in Eq.~(\ref{6.1}):
\begin{align}
   m \ {\rm sn}(x) \ {\rm sn}(x+a)    - \ & {\rm ns}(a)
   \\ \notag & \times \ \left[ Z(x+a) - Z(x) - Z(a) \right]
   \>,
\end{align}
\begin{align}
   m \ {\rm cn}(x) \ {\rm cn}(x+a) = &
   m \ {\rm cn}(a) \\ \notag
   & + {\rm ds}(a) \left[
   Z(x+a) - Z(x) - Z(a) \right]
   \>,
\end{align}
and
\begin{align}
   {\rm dn}(x) \ {\rm dn}(x+a) = & {\rm dn}(a)
   \\ \notag &
   + {\rm cs}(a) \left[
   Z(x+a) - Z(x) - Z(a) \right]
   \>.
\end{align}
Here $Z(x) \equiv Z(x,m)$ is the Jacobi zeta function \cite{stegun,byrd}.
In addition, we use the fact that
\begin{align}
   \sum_{n=1}^{N} \
   \Bigl \{ Z[\beta\epsilon (n+1)+k_1, m] - Z(n \beta\epsilon +k_1, m) \Bigr \}
   = 0 \>.
\end{align}

Let us now consider the sum as given by first term of Eq.~(\ref{6.1})
for the three cases. We first consider the kink-like solution
which can also be written as
\begin{equation}\label{6.6}
   \phi_n = Aa \ {\rm sn}(n \beta\epsilon+k_1,m) \>.
\end{equation}
Using the identities given above, we obtain
\begin{align}
\label{6.7}
   \sum_{n=1}^{N}  \left[
   -\frac{(\phi_{n+1}\phi_n)}{ \epsilon^2}\right] = - \frac{(A a)^2
   {\rm ns}(\beta \epsilon)}{m \epsilon^2} \ N \ Z(\beta \epsilon)
\>.
\end{align}
For the ${\rm dn}$ pulse-like case we have instead
\begin{equation}\label{6.8}
   \phi_n = Aa \ {\rm dn} (n \beta \epsilon+k_1,m)
   \>,
\end{equation}
and the first sum in Eq.~(\ref{6.1}) becomes
\begin{align}\label{6.9}
  &
  \sum_{n=1}^{N} \left[ -\frac{\phi_{n+1}\phi_n}{ \epsilon^2}\right]
  \\ \notag &
  = - \frac{N(A a)^2}{\epsilon^2}
  \left[ {\rm dn}(\beta \epsilon) - {\rm cs}(\beta \epsilon)Z(\beta \epsilon)
  \right] \>.
\end{align}
Finally, for the ${\rm cn}$ pulse-like case we have
\begin{equation}\label{6.10}
   \phi_n = Aa \ {\rm cn}(n \beta\epsilon+k_1,m)
   \>.
\end{equation}
In this case, the first sum in Eq.~(\ref{6.1}) becomes
\begin{align}\label{6.10a}
  &- \
  \sum_{n=1}^{N} \frac{\phi_{n+1}\phi_n}{ \epsilon^2}
  = - \frac{N(A a)^2}{\epsilon^2}
  \left[ \frac{1}{m} Z(\beta \epsilon) {\rm ds}(\beta \epsilon) - {\rm cn}(\beta \epsilon)
  \right] \>.
\end{align}
It is worth noting that all these sums are independent of the constant
$k_1$ and hence $c$. As argued above, since the total energy E as
given by Eq.~(\ref{6.1}) is conserved (and hence time-independent) and
since $t$ and $c$ always appear together in the combination $k_1$ in Eq.~(\ref{6.1})
therefore it follows that the second sum in Eq.~(\ref{6.1})
must also be independent of $k_1$ and hence $c$. We thus have shown
that the PN barrier is zero for the three periodic solutions and hence
also for the localized solutions (which are obtained from them in the
limit $m=1$).

In fact, it is easy to show that for all the three cases, the second
sum (apart from a trivial $k_1$-independent constant) in Eq. (6.1)
is given by
\begin{equation}\label{6.11}
\sum_{n=1}^{N} \ln
\left [1+\frac{{\rm sn}^2(\beta\epsilon)}{{\rm cn}^2(\beta\epsilon)}
{\rm dn}^2(n\beta\epsilon+k_1) \right ]~.
\end{equation}
In particular, for the solution (\ref{6.6}) the second sum is given by
\begin{equation}
B\sum_{n=1}^{N} \bigg \{ \ln [{\rm cn}^2(\beta\epsilon)] +
\ln \left [ 1+\frac{{\rm sn}^2(\beta\epsilon)}{{\rm cn}^2(\beta\epsilon)}
{\rm dn}^2(n\beta\epsilon+k_1) \right ] \bigg \}~.
\end{equation}
On the other hand, for the solution (\ref{6.8}), the second sum is
given by
\begin{equation}
B \sum_{n=1}^{N} \ln \Bigl [ 1+\frac{{\rm sn}^2(\beta\epsilon)}{{\rm cn}^2(\beta\epsilon)}
{\rm dn}^2(n\beta\epsilon+k_1) \Bigr ]~.
\end{equation}
Finally, for the solution (\ref{6.10}), the second sum is given by
\begin{equation}
B\sum_{n=1}^{N} \bigg \{ \ln \frac{{\rm cn}^2(\beta\epsilon)}{{\rm dn}^2(\beta\epsilon)} +
\ln \left [ 1+\frac{{\rm sn}^2(\beta\epsilon)}{{\rm cn}^2(\beta\epsilon)}
{\rm dn}^2(n\beta\epsilon+k_1)\right ] \bigg \} \>.
\end{equation}

It is worth remarking at this point that by following the above
arguments, it is easily shown that even in the AL model~\cite{AL}
the PN barrier is zero for the periodic ${\rm dn}$ and ${\rm cn}$ solutions.
In particular, since in  that case the energy is essentially given
by the first sum in Eq.~(\ref{6.1}), hence using Eqs. (\ref{6.9})
and (\ref{6.10a}) it follows that indeed for both ${\rm dn}$ and ${\rm cn}$
solutions \cite{scott} there is no PN barrier in the AL model.

In general, we do not know how to write the sum in Eq.~(\ref{6.11})
in a closed form. However, for $m=1$ and $N\rightarrow \infty$, the
sum  of
logarithms in Eq.~\eqref{6.3} can be carried out explicitly, and the
energy of the solitary
wave can be given in a closed form.

We proceed as follows: One can show that, for $N \rightarrow \infty$,
the following identity can be derived from the AL equation (see, for instance,
Ref.~\cite{cai}):
\begin{align}
   \sum_{n=-\infty}^{\infty}
   \ln \Bigl \{ 1 + \sinh^2(\rho) \ {\rm sech}^2[ \rho(n - x) ] \Bigr \}
   \ = \ \frac{2}{\rho}
   \>.
\end{align}
Then, for kink-like solutions and $m=1$, we have
\begin{align*}
   \ln (1 - & \alpha \phi_n^2)
      \ln [1 - \tanh^2(\beta \epsilon) \tanh^2(n\beta \epsilon + k_1) ]
   \\
   = \ &
   \ln {\rm sech}^2(\beta \epsilon) \ + \
   \ln [ 1 + \sinh^2(\beta \epsilon) \ {\rm sech}^2n\beta \epsilon + k_1) ]
   \>.
\end{align*}
Thus, in the limit when $N \rightarrow \infty$ and $m=1$, the sum of
logarithms in Eq.~\eqref{6.3} becomes
\begin{align}
   \frac{2}{\beta \epsilon}
   \ + \
   \sum_n \
   \ln {\rm sech}^2(\beta \epsilon)
   \>.
\end{align}
Similarly, in the case of the two pulse solutions,
${\rm dn}$ and ${\rm cn}$ are identical for $m = 1$, and we can write
\begin{align*}
   \ln (1 - & \alpha \phi_n^2)
      \ln [1 + \sinh^2(\beta \epsilon) \ {\rm sech}^2(n\beta \epsilon + k_1) ]
   \>.
\end{align*}
In the limit when $N \rightarrow \infty$, the sum of
logarithms in Eq.~\eqref{6.3} for the two
pulse solutions, ${\rm dn}$ and ${\rm cn}$, for $m~=~1$, is simply equal to
$2/(\beta \epsilon)$.

For the case of the kink solution, for $m=1$, $B$ as given by
Eq.~(\ref{6.4b}) simplifies to
\begin{equation}
B=-\frac{{\rm sech}^2 (\beta\epsilon)}{\alpha\epsilon^2}\>,
\end{equation}
whereas for the case of the pulse solutions, at $m=1$, $B$ simplifies
to
\begin{equation}
B=-\frac{\cosh (\beta\epsilon)}{\alpha\epsilon^2}\>.
\end{equation}

\section{Conclusions}
\label{sec:concl}

In this paper we have shown how to modify the naive discretization
of $\lambda \phi^4 $ field theory so that the discrete theory is a
Hamiltonian dynamical system containing both static and moving
solitary waves.  We have found three different periodic elliptic
solutions.  For time-dependent solutions we found the unusual result
that the discretization is dependent on the velocity.

In the static case, we have studied the stability of both kink-like
and pulse-like solutions, and have found different qualitative
behavior of the lowest eigenvalue of the stability matrix in the two
cases. For typical values of the model parameters, in the case of
kink-like solutions, we found that stability requires the number of
sites, $N$, to be larger than a minimum value, while for pulse-like
solutions stability is achieved for arbitrary values of $N$. The
magnitude of the lowest eigenvalues increases with $N$ in the
kink-like case, and decreases with $N$ for pulse-like solutions.
The lattice spacing, $\epsilon$, is not an independent parameter
and always decreases with $N$.

We also determined the energy of the solitary wave in the three
cases. Using the Hamiltonian structure, we were able to argue that
the PN barrier~\cite{PN,peyrard} for all solitary waves is zero.
This leads to the intriguing question whether this Hamiltonian
system is integrable.  As an additional result we explicitly showed
that for the two elliptic solutions (${\rm dn}$ and ${\rm cn}$) \cite{scott}
of the integrable Ablowitz-Ladik model \cite{AL} the PN barrier is
zero--as one would expect.

The single solitary wave solutions were found to be stable and when
we scattered two such single-kink solitary waves there were two
different behaviors. For pulses, the pulse--anti-pulse solution
leads to scattering with a flip and a little radiation (phonons). The
existence of phonons brings into question the integrability of this
model, in spite of the zero PN barrier. For kink-like solutions we
found a breather-like behavior~\cite{breather2} during the collision.
However, we have \emph{not} found \emph{exact} two-solitary wave or
breather solutions, which would help clarify the integrable nature
of this system.

The results presented here are useful for structural phase
transitions~\cite{dphi4,aubry} and possibly for certain field theoretic
contexts~\cite{luis}.  Our results also hold promise for appropriate
discretizations of other discrete nonlinear soliton-bearing
equations~\cite{clark,scott}.  Possible extension to discrete
integrable models in $2+1$ dimensions, e.g. Kadomtsev-Petviashvili
(KP) hierarchies \cite{KP}, would be especially desirable. Extension
to time-discrete integrable models \cite{capel} is another interesting
possibility.

\section{Acknowledgment} This work was supported in part by the U.S.
National Science Foundation and in part by the Department of Energy.
FC and BM would like to thank the Santa Fe Institute for hospitality.
AK acknowledges the hospitality of the Theoretical Division at LANL
where this work was initiated.


text:
Image from: news18.com
Current Club: Bournemouth.
Previous Clubs: Blackburn Rovers, Manchester United, Valerenga (U19).
Loan Clubs: Blackburn Rovers, Hull City, Borussia Moenchengladbach, Preston North End.
International: Norway 28 caps 7 goals.
Up until now his incredible pace, allied to flashes of ability, have been enough to see him get a place in a Premier League team. King has always been about pace, the kind of pace that leaves even quick defenders trailing in his wake, but with little thought to its use. Usually playing as a striker, his pace could be devastating, especially against a team utilising a high line, but he would often be too quick to make the best use of the ball when a chance came his way.
Last season, Eddie Howe at the Cherries used him a lot on the wing and behind a main striker, and he seemed to learn a lot about runs, when and where to run. King's game improved markedly in that respect over the season. However that was not the difference which turned him from a striker who fluffed his lines so often when chances arrived. Last season there was more thought behind his play, King did not just rely on his pace and try to do it all at a sprint. He learnt to just take his foot off the pedal once in and control his shot, which led, in large part, to his improvement.
Next season will be a big test of his improvement, as defenders will not be as willing to take a chance on giving him space, unlike in the past. Teams would often play a high line, risking giving him space, because he was such a poor finisher. That is unlikely to happen next season after his 16 goals in the 16/17 season.
Written by Ed001 June 26 2018 22:54:54
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