text:
Q: autocomlete list with ajax function call show me an error in the jquery.min.js error "s.send(t.hasContent && t.data || null)" I try to make an autocomplete list and I'm getting data from a database using Ajax, I'm using CodeIgniter framework, but the problem is that I get an error in the console and it's related to the jequry.min.js,
this my code in 3 step mvc:
the controller :
 function get_autoComplete(){
    $keyword=$this->input->post('keyword');
    $data=$this->waiting_model->GetRow($keyword);
    echo json_encode($data);

}

model :
   public function GetRow($keyword) {
    $this->db->order_by('id', 'DESC');
    $this->db->like("firsName", $keyword);
    return $this->db->get('patients')->result_array();

}

and the view

            <input  type="text" id="patients_search" autocomplete="off" name="patients_search" class="form-control" placeholder="patient name">

intel that I show nothing in the console as an error for this function so I rewrite the ajax function in the console  I show this msg

I click jquery.min.js:2426


Newport Craft Brewing & Distilling Company announced in a press release today they have agreed to purchase the intellectual property assets of Braven Brewing Company, a recently shuttered brewery from Brooklyn, New York.
According to the release, the news of Braven's recent closing and availability for purchase sparked the interest of Newport Craft Brewing & Distilling Company. Brendan O'Donnell, CEO and Owner, along with Co-Owners, Derek Luke and Brent Ryan, recognized the many opportunities that acquiring Braven's brand would have.
Located in the historic Bushwick neighborhood of Brooklyn, Braven was co-founded by friends Marshall S. Thompson and Eric Feldman in 2013. The company began by brewing their beers in Saratoga Springs, New York in 2014, eventually opening its own brick-and-mortar brewery in Bushwick in 2018. Braven distributed its Bushwick PilsnerⓇ, White IPA, and other styles to more than 4,000 retailers during its four year existence under Thompson and Feldman's direction. Despite the brand's popularity, Braven faced a number of financial and personal challenges that led to its closure in March 2019.
As part of the transaction, Newport Craft will acquire the intellectual property, beer recipes, and customer lists, among other assets. However, Newport Craft will not continue Braven's overall operations or assume any of Braven's liabilities. Terms of the deal will not be released.


Union Public Service Commission will hold the Combined Medical Services Examination 2016 on 12th June, 2016. The examination will be held at various centres across the country.
Age: Candidates must not have attained the age of 32 years on 1st January, 2016. Relaxation in the upper age limit as per Govt. of India rules.
Educational Qualification : A candidate must have passed the written and practical parts of final MBBS Examination.
Fee: Rs. 200/- to be paid in any branch of SBI by cash or by using Net Banking facility of SBI, SBBJ, SBH, SBM, SBP SBT or using Visa/ Master Credit / Debit Card. SC/ ST/ PH/ Women candidates are exempted from payment of fees. OBC candidates are required to pay full fees.
How to Apply : Candidates must apply online at UPSC website at http://upsconline.nic.in/mainmenu2.php from 05/03/2016 to 01/04/2016.


Do you need a formal dining room for your Thanksgiving dinner for a small group? No! Over the years, I've had one long table, as well as multiple tables, and I'll tell you what is best: it's whatever works! And, if that means eating in the kitchen, so be it. Here are inspiring eat in kitchens good enough for today's main event, the Thanksgiving meal. Images from Desire To Inspire and the (departed) House & Garden. Enjoy!


text:
First Drive / Review
Audi Q2 India review, test drive
9th Oct 2020 7:13 pm
Part SUV, part large hatchback, Audi's Q2 is pure crossover. But what exactly does it bring to the table, and what's it like to drive?
Shapur Kotwal Author
Make : Audi
Model : Q2
Peppy performance
Agile handling
Interior quality
We Don't Like
Pricey for what you get
Dated infotainment system
The Audi Q2 is the latest, and smallest, addition to the brand's Q family of SUVs and crossovers in India. Audi has enjoyed considerable success with its luxury SUVs in India. First came the massive, gen-one Q7, followed by the 'right-sized' Q5, and then the compact, but superbly packaged Q3. While all three have done rather well here, and the Q7 and Q5 are well into their second generation, the Q2 is different kettle of fish.
Audi Q2 Exterior Design
Yes, it is an SUV, and clearly an Audi too, but unlike the rest of the family, it doesn't quite follow the same laws of arithmetic, or for that matter, geometric progression. So, while the Q7, Q5 and Q3 use the same basic formula, adapted for size and spec, the Q2 is designed to appeal to a different sort of customer. One who is younger in approach, more appreciative of an edgy design and, importantly, someone who sees real merit in compactness.
It's quite a forward-looking design by Audi's conservative standards, but it isn't really loud or in your face either.
While the Q2 is designed to be different, it is still an Audi, and therefore isn't a design that shouts out loud or draws attention to itself. That ice-cool, understated Audi demeanour is there in spades and the more you look at the Q2, the more you appreciate its chiselled features. While the Q2 may only be 4.2 metres long, it has a strong shoulder line, gently bulging wheel arches and fast roofline. And then there's the wide rectangular nose with its multiple trapezoidal details.
'T' shaped motif in headlight also seen in the tail-light.
Audi designers have also provided plenty of interplay between light and shade. The flanks of the Q2 look like they've been worked on with a giant chisel, and these scooped out sections look great, especially with their sharp cuts and ridges catching the light. The 'T' shaped pattern in the headlights and tail-lights also work well, and although the Q2 is neither an SUV nor a hatch, it is comfortable in its own skin and not awkward looking; a malady suffered by a good many crossovers.
Audi Q2 Interior and Features
The Q2 may be small on the outside and only has as much space on the inside as your average mid-size family SUV, but given the shadow it casts on the ground, packaging is still good. There's more than sufficient place for those sat up front, both front seats are extremely supportive and comfortable, and finding the right driving position is also easy. The India-spec car, however, has no powered seats – a shocking omission, and there's no touchscreen either; probably in an effort to keep the price in check.
Visibility out is excellent; seat is placed at a height, and the A pillar is slim.
The slightly elevated driving position, however, feels just right. Visibility up front is superb and the dash doesn't seem to be too in your face either. It's aesthetically pleasing on the inside too. Quality levels are almost as good as they are in an A4, and the dash even has some of the TT's vibe, with its rotary click-to-shut vents. The quality of the soft-top dash and the fit and finish of the various chrome elements make this a cabin you want to touch and feel all the time, and this is particularly true of the nicely put-together flat-bottom steering wheel that feels great to hold. The knurled metallic knobs and dials on the centre console are also a delight to use, and though the LED inlay in the dash feels a bit gimmicky, the snakeskin-like effect is quite interesting.
Medium-size sunroof brightens up the otherwise dingy and dark all-black cabin.
In addition, you get Audi's Virtual Cockpit – the configurable, fully digital instrument panel, a wireless charging tray in the elbow box and, though it isn't quite panoramic, there is a big sunroof too. Rear visibility, with those massive C pillars and the slot-like rear windscreen is poor, though.
Legroom is decent but backrest is upright and the seat isn't too comfortable.
Now clearly, you wouldn't buy a Q2 to be chauffeur driven in. That said, there's sufficient space in the rear to be reasonably comfortable. Headroom is decent, there's some surplus knee room, and since you can put your feet under the front seats, you can even stretch out a bit. The seat base is short, however, and the backrest is a bit upright, so no long drives with people sat in the rear. And it also feels a bit a bit dark and dingy in the back. Also, while there are USB sockets in the rear, both are Type C units, so you will need adapters.
405-litre boot is nicely shaped; and you get a space-saver spare tyre too.
While the boot is well shaped and easy to load, and you get a decent 405 litres, it is only large enough to carry a pair of big bags in the back. Should you need more space, you can fold the seatback down. So that you don't get stranded with a puncture, there is also a space-saver spare tyre placed below the boot floor.
Audi Q2 Engine and Performance
With all its power and the taut chassis, the Q2 conjures up plenty of thrills for the driver. A lot of this is down to its 190hp power output –strong enough for an SUV that tips the scales at just 1,430kg. The 2.0-litre engine is a familiar one. Powering all manner of Audi, VW and Skoda products, from the A6 to the Tiguan AllSpace to the Octavia RS and everything in between, this is one of the best four-cylinder engines around.
The Q2 is a joy to drive.
Idle is vibration free and refined, there is almost no lag, and responses are so immediate, it doesn't feel turbocharged. There's no sudden spike in power when the boost comes in, probably down to the BS6 tune, but there's still is a clear ramp up after 2,000rpm, and then performance rises steeply all the way to the conservative 6,200rpm redline. What adds in to the experience is that the engine just loves to rev. There's an enthusiasm here, as it winds up the powerband, that you'd normally find only in a naturally aspirated engine, and though the acceleration doesn't quite throw you back in your seat, performance is still very urgent.
A quick test we carried out revealed a 0-100kph time of 8.08 seconds, but with a stronger launch, the Q2 has the potential to be much closer to its claimed time of 6.5 seconds. The Q2 also, expectedly, comes with a dual-clutch gearbox, which is quick on the draw during regular driving, and allows you to snap through the gears smartly. What takes away from the driving experience, however, is that at times it can be reluctant to shift down. This happens just when you need it most; when you need to brake, downshift and get into a corner, and this can be quite frustrating.
Audi Q2 performance (as tested)*
0-20kph 1.04s
0-100kph 8.08s
0-120kph 11.79s
20-80kph (in kickdown) 5.00s
40-100kph (in kickdown) 6.79s
*Performance figures not to Autocar India test standards
Audi Q2 Comfort and Handling
What impresses is that the suspension rides silently over several of the bad patches we encounter. While it's fairly absorbent, there is some amount of stiffness here, and this shows up over some of the sharper-edged bumps. Also, while there is a bit of roll in corners, body control in general is very good. In fact, once you get stuck in, it feels more like a sporting hatch than an SUV. The long-travel suspension is kept on a tight leash and the Q2 settles down quickly after that initial bit of roll. And since there's plenty of grip, you can put the power down quite nicely too.
Body control in general is very good but there is some stiffness felt.
The steering could do with a bit more feedback though, and the Q2 begins to understeer if you push harder. Still, the driving experience in general is so pleasing, this is a car you just love to drive quickly, be it in a straight line or around a series of nicely meandering corners.
The Q2 will have a niche appeal for those looking for a bit of fun.
It's important to note, however, that while the badge on the grille says Quattro, this isn't one of Audi's typical, full-time all-wheel-drive systems. Unlike the arrangement used on its larger cars, which use a longitudinally mounted engine and a Torsen-based Quattro system, the Q2 is based on the VW Group's MQB platform, and so comes with a transverse engine, a front-biased setup, and an on-demand AWD system based on a multi-plate clutch.
Audi Q2 Price and Verdict
At an expected starting price of Rs 35 lakh, ex-showroom, Audi's imported Q2 isn't for those looking for value for money. Unlikely to be a rational choice, it is targeted more at self-driven owners who want a compact luxury crossover with good performance, easy everyday usability and a high-quality cabin. For your money, you get a high-riding car, an added dose of excitement thanks to the peppy petrol and relatively light weight (which, frankly, is central to its appeal), and it feels agile from behind the wheel too.
It should have been better equipped at this price, and there will be a mild facelift out internationally at around the same time the Q2 goes on sale in India, which customers are unlikely to be happy about. But if compactness is a virtue, and at times it truly is, the Q2 could be the compact luxury crossover you are looking for.
Audi Q2 India video review
Audi Q2 India launch on October 16, 2020
Audi Q2 bookings open; booking amount set at Rs 2 lakh
2020 Audi Q2 facelift gets subtle design updates
Petrol AT
Diesel AT
Fuel Type / Propulsion Petrol
Type 4 cyl, turbo-petrol
Cubic Capacity (cc) 1984cc
Max Power (hp @ rpm) 190hp at 4200-6000rpm
Max Torque (Nm @ rpm) 320Nm at 1500-4100rpm
Drive Layout All-wheel drive
Gearbox Type Dual-clutch automatic
No of Gears 7
0 - 20 kph (sec) 1.04s*
0 - 80 kph (sec) 5.6s*
0 - 100 kph (sec) 8.08s*
0 - 120 kph (sec) 11.79s*
20-80kph (sec) 5.0s (in kickdown)
40-100kph (sec) 6.79s (in kickdown)
Length (mm) 4318mm
Width (mm) 1805mm
Height (mm) 154mm
Wheelbase (mm) 2595mm
Boot Capacity (Lts) 405-1050 litres
/car-reviews/audi-q2-india-review-test-drive-418863 Audi Q2 India review, test drive - Autocar India Part SUV part large hatchback Audi's Q2 is pure crossover. But what exactly does it bring to the table and what's it like to drive? https://cdni.autocarindia.com/ExtraImages/20201009064101_2020-Audi-Q2-review-front-action-exclusive.jpg

redpajama_set_name: RedPajamaCommonCrawl


text:
Kerala CPM Minister Sudhakaran Denigrates Hindu Sanyasis, Praises Christian and Muslim Priests
TOPICS:CPMHinduphobiaKeralaLeftist HypocrisyMarx
BY: HinduPost Desk June 7, 2016
It seems that even though Marxists in Bharat espouse communist atheism, the Abrahamic faiths (Chrsitianity and Islam) hold a special position in their hearts. CPM MLA from Ambalappuzha constituency, G Sudhakaran, who is also the minister for public works in the Government of Kerala, recently said that Hindu sanyasis (ascetic) do not wear underwear, while Christian & Muslim priests are 'decent'.
#Kerala CPM minister says Hindu Sanyasis do not wear underwear, while Christian, Muslim priests are decent. pic.twitter.com/F7ZWVzvamz
— Jayakrishnan (@savarkar5200) June 6, 2016
Sudhakaran was in the news recently for breaking the election code by peeping into the polling booth when veteran CPM leader VS Achutanandhan was casting his vote. Were Suhakaran's remarks alluding to sexual licentiousness? If so, he has conveniently forgotten the various sex scandals involving Muslim and Christian clergy that have hit Kerala in recent years.
Such contempt and denigration of Hindu beliefs and practices is common for Marxists. And it is not surprising, given the views on Hindu Dharma of their ideological fountainhead Karl Marx (1818 – 1883) :
1] "That man, the sovereign of nature, fell down on his knees in adoration of Hanuman, the monkey, and Sabbala, the cow" (The British Rule of India, New York Daily Tribune, June 25, 1853). Marx believed that British rule despite its iniquities was a progressive force for Bharat.
2] "England has to fulfill a double mission in India: one destructive, the other regenerating – the annihilation of the old Asiatic society, and the laying of the material foundation of Western society in Asia" (The Future Results of British Rule in India, New York Daily Tribune, August 8, 1853).
So while communists of Kerala and elsewhere indulge in Hindu-bashing, they suck up to Islamists and Christian missionaries as the support of either of these groups is vital for them to come to power in Kerala. This 2007 article in DNA shows how this vote-bank game works in Kerala.
And believe it or not, CPM's G Sudhakaran has also written poems in praise of that Muslim 'icon' Osama bin-Laden!
#Kerala : Same CPM minister Sudhakaran writes a poem about Osama Bin Laden. pic.twitter.com/5RKKw8Zf1H
Such repeated assaults on Hindu identity and heritage, while molly-coddling the most fanatical elements in Abrahamic society, has resulted in the sorry state of affairs in the country today; all this in the name of secularism.


text:
\section{Introduction}
 Let    $R= \Bbbk[x_1, x_2, x_3]$ and  $S = \Bbbk[x]$ be a polynomial rings over a field $\Bbbk$ of characteristic zero.
 Let $q$ and $m$ be positive integers, $d=2q+1$ and   $\gcd (d,m)=1$.  Consider the  homomorphism  $\phi: R {\longrightarrow} S$  defined by $\phi(x_i) = x^{ d + (i-1)m}$, where   $1 \leq i \leq 3$. Throughout this paper   $\mathfrak p:=\mathfrak p_{{\mathcal C}(d, d+m, d+2m)}= \operatorname{ker}(\phi)$. For $q=1$, the resurgence $\rho(\mathfrak p)$, the Waldschmidt constant $\gamma(\mathfrak p)$ and the Castelnuovo-Mumford regularity of the symbolic powers of $\mathfrak p$  have been computed in \cite{clare-shreedevi}. In this paper we generalise these results for all $q \geq 1$.
We also verify that certain conjectures posed in   \cite{harb-huneke} hold true for $\mathfrak p$.
Before we describe our main results we will  give some background on these quantities.

 For any ideal $I$ in a Noetherian ring  $A$ of positive dimension with no embedded components, the $n$-th symbolic power of $I$ is defined by
 $I^{(n)}:= \cap_{\mathfrak p \in Ass(R/I)} I^n A_{\mathfrak p} \cap A$. In general, the generators of $I^{(n)}$ are hard to describe.
Hence,  in order  to have a more precise   relation between symbolic powers and ordinary powers of ideals, Harbourne
posed the following conjecture: Let $I \subseteq \Bbbk[x_1, \ldots, x_t]$ be an homogeneous ideal. Then
$I^{(m)}\subseteq  I^r$ if $m \geq  r(t-1) - (t-2)$ \cite[Conjecture~8.4.2]{brauer}. In the same paper, the authors give  evidence  to show that  this conjecture is true if $\operatorname{char} \Bbbk >0$.
  Later, Bocci
and Harbourne  introduced an asymptotic quantity called resurgence which is defined as $\rho(I) :=\sup \{  m/r |
I^{(m)}  \not \subset I^r\}$ \cite{bocci-harbourne}.
This
supremum exists and in fact $1 \leq \rho(I) \leq t-1$  \cite[Lemma~2.3.2]{bocci-harbourne}.
Since resurgence in general is hard to compute, in  \cite{bocci-harbourne}  the authors  define another invariant which they call the Waldschmidt
 constant.  The Waldschmidt constant was first introduced by Waldschmidt  in \cite{waldschmidt}. We use the definition as in \cite{bocci-harbourne}.
Let $\alpha(I):= \min \{ n | I_n \not = 0 \}$. The Waldschmidt   constant is defined as
$
\gamma(I) = \limm~\f{\alpha( I^{(n)})}{n}.
$
Bocci and Harbourne showed that if $I$ is a homogenous ideal, then  $\alpha(I) / \gamma(I) \leq \rho(I)$,  and in addition if $I$ is a  zero
dimensional subscheme  in a projective space, then
$\alpha(I) / \gamma(I) \leq \rho(I) \leq \operatorname{reg} (I) / \gamma(I)$, where $\operatorname{reg}(I)$ id the Castelnuovo-Mumford
regularity of $I$ \cite[Theorem~1.2.1]{bocci-harbourne} .

The resurgence  and the Waldschmidt constant has been studied in a few cases: for certain general points in $
\mathbb P^2$  \cite{bocci-harbourne-2},  smooth subschemes
\cite{guardo}, fat linear subspaces  \cite{fatabbi},  special point configurations \cite{duminicki} and monomial ideals
\cite{bocco-waldschmidt}.

  If we put   weights on the variables $wt(x_i) = d  + (i-1)m$ for  $i=1, 2, 3$, then   from \cite[Theorem~6.8]{clare} it follows that  $\mathfrak p^{(n)}$ is  a   weighted homogenous  ideal of
height $2$.   Hence, we can define the the Waldschmidt constant $\gamma(\mathfrak p)$ in the same way as in  \cite{bocci-harbourne}.
From  \cite[Theorem~1.1]{cut-kurano}  it follows that $\limm  \operatorname{reg}(    (\mathfrak p^{n})^{sat}   ) /n $ exists and can even be irrational \cite{cutkosky}. Moreover,
 $\operatorname{reg}(\mathfrak p^{(n)})$  is eventually periodic  \cite[Corollary~4.9]{cut-kurano}.
In our case $ (\mathfrak p^{n})^{sat} = \mathfrak p^{(n)}$. We compute $ \operatorname{reg}(    (\mathfrak p^{n})^{sat} )$ (Theorem~\ref{final regularity}). It is clear from our result that the regularity depends on $q$ and $m$.
 In this paper we compute the exact formula for the resurgence of $\mathfrak p$  (Theorem~\ref{theorem resurgence}).

We briefly summarise the contents of this paper. In Section~2 we prove some preliminary results. In Section~3 we compute the resurgence of $\mathfrak p$. We  verify that   Conjecture~2.1 and Conjecture~4.1.5 in \cite{harb-huneke} hold true for $\mathfrak p$ (Corollary~\ref{conjecture 2.1}, Corollary~\ref{conjecture 4.1.5}). In Section~4 we compute the Waldschmidt constant.
We  verify that Chudnovsky's conjecture (Proposition~\ref{chudnovsky}) holds true in our case.  In section~5 we compute the Castelnuovo-Mumford regularity of $\mathfrak p^{(n)}$ for all $n \geq 1$ (Theorem~\ref{final regularity}).

We end this paper by observing that   Theorem~1.2.1 of \cite{bocci-harbourne} holds true for $\mathfrak p$, i.e.,  $\alpha(\mathfrak p) / \gamma(\mathfrak p) \leq \rho(\mathfrak p) \leq \operatorname{reg} (\mathfrak p) / \gamma(\mathfrak p)$.

\section{Preliminaries}
In this section we prove some results which may be well known.
\begin{lemma}
\label{f_2}
For all $q \geq 1$ and $m \geq 1$,
\been
\item
\label{generators of p}
$\mathfrak p = (g_1, g_2, g_3)$ where
\beq
\label{def of delta}
 g_1 &:=& x_1^{m+q}x_2 - x_3^{q+1}, \hspace{.2in}
  g_2  := x_1^{m+q+1} - x_2 x_3^q, \hspace{.2in}
 g_3 := x_2^2 - x_1 x_3.
\eeq
\item
\label{definition of f_i}
Let
 $
f := - x_1^{2(m+q)+1}- x_1^{m+q-1} x_2^3 x_3^{q-1}+3 x_1^{m+q} x_2 x_3^{q}-x_3^{2q+1}.$
\been
\item
\label{x_if_2}
For all  $i=1,2,3$, $x_i f \in \mathfrak p^2$.
\item
\label{f_2^i}
For all $j=1, \ldots, q+1$, $f^j \in \mathfrak p^{2j-1}$.
\item
 \label{Eqn:f_2}
  $\mathfrak p^{(2)} = \mathfrak p^2 + (f)$ and  for all $k \geq 1$,
\beq
        \mathfrak p^{(2k)}
&=&  (\mathfrak p^{(2)})^k        \hspace{.2in} \mbox{and}
          \hspace{.2in}
           \mathfrak p^{(2k + 1)} =\mathfrak p \mathfrak p^{( 2k)}\hspace{.1in}.
                    \eeq
\eeen
\eeen
\end{lemma}
\begin{proof} \eqref{generators of p} is well known.

(\ref{x_if_2}) Since
\beqn
x_1 f = -g_2^2                                        -    x_3^{q-1}g_1 g_3, \hspace{.2in}
x_2 f = - x_1^{m+q-1} x_3^{q-1} g_3^2 -  g_1g_2,  \hspace{.2in}
x_3 f =   - g_1^2                                     + x_1^{m+q-1} g_2 g_3
\eeqn
and $g_j \in \mathfrak p$ for all $j=1,2,3$,  we get $x_i f \in \mathfrak p^2$  for all $i=1,2,3$.

\eqref{f_2^i}  Let $1\leq j \leq q+1$. As $f = x_3^q g_1 - x_1^{m+q} g_2+x_1^{m+q-1}x_2x_3^{q-1} g_3$,
 \beqn
            f^{j}
   & =& (x_3^q g_1 - x_1^{m+q} g_2+x_1^{m+q-1}x_2x_3^{q-1} g_3)f^{j-1}\\
   & =& (x_3 f)^{j-1} x_3^{q-j+1}  g_1  -  (x_1 f)^{j - 1} x_1^{m+q-j+1}  g_2   + ( x_1f)^{j-1}   x_1^{m+q-j}x_2  g_3  \\
   & \in & \mathfrak p^{2(j-1)} \mathfrak p     \hspace{4.2in} \mbox{[from \eqref{x_if_2}]}  \\
   &=&  \mathfrak p^{2j-1}.
  \eeqn

(\ref{Eqn:f_2}) follows from  \cite[Theorem~5.9]{clare} and   by induction on $k$.
\end{proof}

\section{Computation of resurgence}
In this section we compute the resurgence $\rho(\mathfrak p)$. The resurgence can be computed in the following way.
Let $\rho_n(\mathfrak p):=\min\{r: \mathfrak p^{(n)} \nsubseteq \mathfrak p^r\}.$ Then
 $$
 {\displaystyle
 \rho(\mathfrak p):=\sup\left\{\frac{n}{\rho_n(\mathfrak p)}:  n \geq 1\right\}.}
 $$
We state Conjecture~4.1.1 in \cite{harb-huneke} in our context:
 Does $\mathfrak p^{(2n-1)} \subseteq \mathfrak p^n$ hold true for all $n$? The following proposition proves a stronger statement.

\begin{proposition}
\label{Prop:Contain}
 Let  $k \geq 0$.  Then
 \beqn
 \rho_{k (2q+2)  + j}(\mathfrak p)
 =\begin{cases}
 k(2q+1)  +j + 1& \mbox{ if } k \geq 1 \mbox{ and } j=0,1\\
   k(2q+1) +j  & \mbox{ if } k \geq 0 \mbox{ and }  j=2, \ldots, 2q+1
 \end{cases}.
 \eeqn
\end{proposition}
\begin{proof}
We first show that
\beq
\label{equation-contain}
 \mathfrak p^{(k (2q+2)  + j)}
 \subseteq
 \begin{cases}
 \mathfrak p^{k(2q+1)  +j} & \mbox{ if } k \geq 1 \mbox{ and } j=0,1\\
  \mathfrak p^{ k(2q+1) +j-1}  & \mbox{ if } k \geq 0 \mbox{ and }  j=2, \ldots, 2q+1
 \end{cases}.
 \eeq
Applying  Lemma~\ref{f_2},  (\ref{f_2^i}) and   (\ref{Eqn:f_2})  we get
\begin{align}
 \label{containment of even symb powers}
           \mathfrak p^{(2 j^{\prime})}
&=  (\mathfrak p^2+(f))^{ j^{\prime}}
=      (\sum_{i=0}^{ j^{\prime}} f^i \mathfrak p^{2( j^{\prime}-i)} )
         \subseteq  \mathfrak p^{2 j^{\prime}} + \sum_{i=1}^{ j^{\prime}} \mathfrak p^{2i - 1} \mathfrak p^{2( j^{\prime}-i)}
        \subseteq \mathfrak p^{2 j^{\prime} -1} ,&& j^{\prime}  = 1, \ldots, q+1
         \\
       \label{containment of  odd symb powers}
         \mathfrak p^{(2 j^{\prime}   + 1)}
&= \mathfrak p  \mathfrak p^{(2 j^{\prime} )} \subseteq \mathfrak p \mathfrak p^{2 j^{\prime}-1} =  \mathfrak p^{(2 j^{\prime})}
       \hspace{2.0in}   \mbox{[by   (\ref{containment of even symb powers})]},  &&j^{\prime}  = 1, \ldots, q.
 \end{align}
 Hence  (\ref{equation-contain}) is true for $k=0$ and $j = 2, \ldots, 2q+1$.

Let $k \geq 1$ and $j=0,1$. Then from Lemma~\ref{f_2}(\ref{Eqn:f_2}) and  (\ref{containment of even symb powers})  we get
\beq
\label{containment of 2q+2}
 \mathfrak p^{(k (2q+2) + j)}
= ( \mathfrak p^{( 2 q+2)})^k  \mathfrak p^{j}
\subseteq    \mathfrak p^{  k(2q+1) + j   }  .
\eeq
Let $k \geq 1$ and  $j = 2, \ldots, 2q+1$.  Then from  Lemma~\ref{f_2}(\ref{Eqn:f_2}), (\ref{containment of even symb powers}), (\ref{containment of  odd symb powers}) and  (\ref{containment of 2q+2})
 we get
 \beqn
 \mathfrak p^{(k (2q+2) + j)}
 =  \mathfrak p^{(k (2q+2))} \mathfrak p^{(j)}
 \subseteq  \mathfrak p^{ k (2q+1)}  \mathfrak p^{j-1}
=\mathfrak p^{k (2q+1)   + j-1}  .
\eeqn
To complete the proof of the lemma it remains to show that
\beq
\label{Eqn:non-Contain}
{\mathfrak p}^{k (2q+2)  + j}
 \not \subseteq \begin{cases}
 \mathfrak p^{k(2q+1)  +j + 1}& \mbox{ if } k \geq 1 \mbox{ and } j=0,1\\
  \mathfrak p^{ k(2q+1) +j } & \mbox{ if } k \geq 0 \mbox{ and }  j=2, \ldots, 2q+1
 \end{cases}.
 \eeq
By  Lemma~\ref{f_2},   (\ref{generators of p}) and (\ref{Eqn:f_2}), $g_1 \in \mathfrak p$ and  $f \in \mathfrak p^{(2)}$. Hence
\begin{align*}
          f^{k(q+1)}
&\in \mathfrak p^{(k(2q+2))},  & k &\geq 1 \\
         g_1 f^{k(q+1)}
&\in  \mathfrak p \mathfrak p^{(k(2q+2))}  = \mathfrak p^{(k(2q+2) + 1)}, & k &\geq 1 \\
          f^{k(q+1)  +j^{\prime}}
&\in \mathfrak p^{(2k(q+1) + 2j^{\prime} )},  & k &\geq 0,  j^{\prime}=1, \ldots, q\\
            g_1 f^{k(q+1)  +j^{\prime}}
&\in \mathfrak p \mathfrak p^{(2k(q+1) + 2j^{\prime} )} =  \mathfrak p^{(2k(q+1) + 2j^{\prime}  +1)},& k &\geq 0,  j^{\prime}=1, \ldots, q.
\end{align*}
From  (\ref{def of delta}) and Lemma~\ref{f_2}(\ref{definition of f_i}),
\begin{align}
 \label{equation-cong-f}
 f &\equiv x_3^{2q+1} \Mod {x_1}\\
  \label{equation-cong-g}
g_1 &\equiv x_3^{q+1}\Mod{x_1} \\
 \label{equation-cong-p}
\mathfrak p &\equiv  (x_2^2, x_2x_3^q, x_3^{q+1})\Mod{x_1}.
\end{align}
By (\ref{equation-cong-f}),
(\ref{equation-cong-g}) and (\ref{equation-cong-p}) we get
\begin{align*}
                 f^{k(q+1)}
&\equiv  ( x_3^{q+1})^{k (2q+1) }
\not \in      \mathfrak p^{ k(2q+1) + 1} \Mod{x_1}, &  k \geq 1\\
                 g_1 f^{k(q+1)}
&\equiv  (x_3^{q+1})^{(k (2q+1) +1)  }
\not \in  \mathfrak p^{ k(2q+1) + 2}\Mod{x_1}, & k \geq 1.
\end{align*}
As $   (2q+1)  (k (q+1)  + j^{\prime}  ) - (q+1) (k(2q+1) + 2j^{\prime} )= -j^{\prime} <0$, by (\ref{equation-cong-f}),
(\ref{equation-cong-g}) and (\ref{equation-cong-p}) we get
\begin{align*}
               f^{k(q+1)  +j^{\prime}}
&\equiv  x_3^{   (2q+1)  (k (q+1) + j^{\prime} )}
 \not \in  \mathfrak p^{ k(2q+1) + 2j^{\prime}}  \Mod{x_1},
&            k \geq 0, j^{\prime}=1, \ldots, q\\
               g_1 f^{k(q+1)  +j^{\prime}}
&\equiv  x_3^{   (2q+1)  (k (q+1) + j^{\prime} ) + (q+1) }
 \not  \in  \mathfrak p^{2k(q+1) + 2j^{\prime}  +1} \Mod{x_1},
&            k \geq 0, j^{\prime}=1, \ldots, q.
\end{align*}
\end{proof}

We are now ready to compute the resurgence.
\begin{theorem}
\label{theorem resurgence}
For all $q \geq 1$,
 $\rho(\mathfrak p)= \frac{2q+2}{2q+1} .$
\end{theorem}
\begin{proof}
By Proposition \ref{Prop:Contain}
\[
 \rho(\mathfrak p)=\sup_k\left\{   \frac{k(2q+2) }{k(2q+1) +1},     \frac{k(2q+ 2)  +1}{k(2q+1) +2},   \frac{k(2q+2) + j }{k(2q+1) +j}  : j=2, \ldots, 2q+1\right\}
 =\frac{2q+2}{2q+1}
 .
\]
\end{proof}

The following conjecture was stated for ideal of fat points  \cite[Conjecture~2.1]{harb-huneke}. We verify that the conjecture holds true for $\mathfrak p$.

\begin{corollary}
\label{conjecture 2.1} For all $n \geq 1$,
${\displaystyle
 \mathfrak p^{(2n)} \subseteq
 \begin{cases}
 \mathfrak m^n \mathfrak p^n & \mbox{ if } q=1 \\
 \mathfrak m^{2n} \mathfrak p^n  & \mbox { if } q>1
 \end{cases}
}$
and
${\displaystyle
 \mathfrak p^{(2n-1)} \subseteq
 \begin{cases}
 \mathfrak m^n \mathfrak p^n & \mbox{ if } q=1 \\
 \mathfrak m^{2n} \mathfrak p^n  & \mbox { if } q>1
 \end{cases}.
}$
\end{corollary}
\begin{proof} By Lemma~\ref{f_2}(\ref{Eqn:f_2}),  $\mathfrak p^{(2n)} = (\mathfrak p^{(2)})^n$. Hence    it is enough to prove the lemma for $n=1$. If $n=1$, then by Lemma~\ref{f_2}(\ref{Eqn:f_2}),
 \beqn
 \mathfrak p^{(2)} = \mathfrak p^2 + (f) = (g_1, g_2, g_3)\mathfrak p + (x_3^q g_1 - x_1^{m+q} g_2+x_1^{m+q-1}x_2x_3^{q-1} g_3)
 \subseteq
 \begin{cases}
 \mathfrak m \mathfrak p & \mbox { if } q=1\\
\mathfrak m^2 \mathfrak p &\mbox { if }q>1
 \end{cases}.
 \eeqn
 Hence the corollary is true for even powers.
By Lemma~\ref{f_2}(\ref{Eqn:f_2}),  $\mathfrak p^{(2n-1)} = \mathfrak p \mathfrak p^{(2(n-1))}$. Hence  the corollary is true for odd powers.
\end{proof}

 We rephrase Conjecture~4.1.5   of \cite{harb-huneke} in our context:
  \begin{corollary}
 \label{conjecture 4.1.5}
 For all $n \geq 1$, $\mathfrak p^{(2n-1)} \subseteq \mathfrak m^{n-1} \mathfrak p^{n}$.
 \end{corollary}
 \begin{proof}  The proof follows from Corollary~\ref{conjecture 2.1}.
 \end{proof}

\section{Waldschmidt Constant}
Put weights $d_i= wt(x_i)$ where $d_1  = d := 2q+1$, $d_2 = d+m = 2q+1+m$ and $d_3 = d+2m = 2q+1  + 2m$. With these weights, $\mathfrak p^n$ and  $\mathfrak p^{(n)}$ are  weighted homogenous ideals \cite{clare}.
Hence we can define  $\alpha(\mathfrak p):= \min \{ n | \mathfrak p_n \not = 0 \}$. The Waldschmidt
constant can be defined as
$$
\gamma(\mathfrak p) = \limm~\f{\alpha( \mathfrak p^{(n)})}{n}.
$$
In this section we compute $\alpha(\mathfrak p)$ and  $\gamma(\mathfrak p)$.

\begin{theorem}
\label{theorem gamma}
For all $q \geq 1$ and $m \geq 1$,
\been
\item
\label{theorem gamma 1}
$\alpha(\mathfrak p) =  2 d_2$.

\item
\label{theorem gamma 2}
$
{\displaystyle
\gamma(\mathfrak p)
= \begin{cases}
15 / 2  & \mbox{ if } q=1 \mbox{ and } m=1\\
2 d_2& \mbox{ otherwise } .
\end{cases}
}
$
\eeen
\end{theorem}
\begin{proof}
By Lemma~\ref{f_2}, $\mathfrak p = (g_1, g_2, g_3)$, $\mathfrak p^{(2n)} = (\mathfrak p^{2} + f)^n$ and $\mathfrak p^{(2n+1)} = \mathfrak p \mathfrak p^{(2n)}$
 where $g_1$, $g_2$, $g_3$ are  defined in (\ref{def of delta}) and $f$ is defined in Lemma~\ref{f_2}(\ref{definition of f_i}).  Hence,
 $\operatorname{deg}(g_1) = (q+1) d_3 =(q+1)(d+2m)$,
 $\operatorname{deg}(g_2) = d(m+q+1)$ and $\operatorname{deg}(g_3) = 2 d_2 = 2(d+m)$.
This gives $\operatorname{deg}(g_3) \leq \operatorname{deg}(g_2) \leq \operatorname{deg}(g_1)$. Hence,
$\alpha(  \mathfrak p^{(2n)}) = \min \{   2\operatorname{deg} ( g_3) n,  \operatorname{deg}(f) n\}$.

As $\operatorname{deg}(f) = d( d+2m)$, we get  $\operatorname{deg}(f) - 2 \operatorname{deg}(g_3) = d(d+2m) - 4(d+m) = d(d-4) + 2m (d-2)$,
we get,
 $deg(f) \leq 2  \operatorname{deg}(g_3)$ if and only if $q=1$ and $m=1$. Hence
\beq
\label{waldschmidt even}
\alpha( \mathfrak p^{(2n)} )&= &
\begin{cases}
 \operatorname{deg}(f) n  = 15n  &   \hspace{.8in}\mbox{ if } q=1 \mbox{ and } m=1\\
2 \operatorname{deg}( g_3) n =  2(2n) d_2 & \hspace{.8in}\mbox { otherwise }
\end{cases} \\
\alpha( \mathfrak p^{(2n + 1)}) \label{waldschmidt odd}
&= &
\begin{cases}
 n \operatorname{deg}(f) + \operatorname{deg}(g_3)    = 15n + 8&   \mbox{ if }  q=1 \mbox{ and } m=1\\
(2n+1)  \operatorname{deg}( g_3) =    2(2n+1)d_2       & \mbox { otherwise }
\end{cases}.
\eeq
This implies that
$
{\displaystyle
\gamma(\mathfrak p) =  \limm~\f{\alpha( \mathfrak p^{(n)})}{n}
= \begin{cases}
15 / 2  &   \mbox{ if }  q=1 \mbox{ and } m=1\\
 2 d_2& \mbox { otherwise } .
\end{cases}
}$
 \end{proof}

We verify Chudnovsky's conjecture (see \cite[Remark~3.4]{harb-huneke}).
\begin{proposition}
\label{chudnovsky}
$
{\displaystyle
\f{\alpha  ( \mathfrak p^{(n)})}{n} \geq \f{\alpha( \mathfrak p) + 1}{2}.}
$
\end{proposition}
\begin{proof}
If $q=1$ and $m=1$, then by Theorem~\ref{theorem gamma}(\ref{theorem gamma 1}), $\f{\alpha(\mathfrak p)  + 1}{2}= (8+1)/2  = 9/2$ and
\beqn
\f{\alpha  ( \mathfrak p^{(n)})}{n}
= \begin{cases}
\f{(15/2)2r}{2r}  = \f{15}{2} \geq \f{9}{2}& \mbox{ if $n=2r$ }\\
\f{ (15(2r+1)  + 1)/2} {2r+1}  = \f{15}{2} + \f{1}{2(2r+1)} \geq \f{9}{2}& \mbox { if $n=2r+1$ }
\end{cases}.
\eeqn
If either $q \not =1$ or $m \not = 1$, then by (\ref{waldschmidt even}), (\ref{waldschmidt odd}) and Theorem~\ref{theorem gamma}(\ref{theorem gamma 1}), for all $n \geq 1$
\beqn
\f{\alpha  ( \mathfrak p^{(n)})}{n}
= 2d_2 \geq \f{ 2 d_2 + 1}{2} = \f{\alpha(\mathfrak p) + 1}{2}.
\eeqn
\end{proof}

\section{Regularity}
Recall $d=d_1  = 2q+1$, $d_2 = d+m$ and
$d_3 = 2q+1 + 2m$.
We begin with some basic results comparing $\mathfrak p^{(n)}$ and $I_nT$ where $T = \Bbbk[x_2, x_3] \cong R / (x_1)$.
Let
\beq
\label{definition of Ji}
J_1 &:= &   \{x_2^2, x_2x_3^q, x_3^{q+1}\}, \hspace{.2in}
J_2 := \{x_3^{d}\}.
\eeq.
\begin{notation}
If $A_1, \ldots, A_n$  are $n$ sets of monomials we define the set $A_1 \cdots A_n$ by
$A_1\cdots A_n := \{a_1 \cdots a_n : a_i \in A_i \}$.
\end{notation}
With the above notation
\beq
\label{equation of In}
I_n
&:=&
\sum_{a_1 + 2 a_2 = n} J_{1}^{a_1} J_{2}^{a_{2} }.
\eeq
As an immediate consequence of Theorem~5.9 in \cite{clare} we have:
\begin{lemma}
\label{ideals in tprime}
For all $n \geq 1$,  $\mathfrak p^{(n)}R + (x_1) = I_nR + (x_1)$,  $\mathfrak p^{(n)} T = I_n T$,  $I_{2n} T = (I_2T)^n $ and $I_{2n+1} T = (I_2T) (I_{2n} T)$.
\end{lemma}

 \begin{lemma}
 \label{reg comparision}
For all $n \geq 1$,  $\operatorname{reg}( R/ \mathfrak p^{(n)}   )  =  \operatorname{reg}(T/ I_nT) $.
  \end{lemma}
 \begin{proof} As $x_1$ is a nonzerodivisor on $R/ \mathfrak p^{(n)}$ and $T / I_n T$,
  \beqn
  \operatorname{reg} \left(  \f{R}  { \mathfrak p^{(n)}} \right)
&=& \operatorname{reg} \left(  \f{R  }{ \mathfrak p^{(n)} + (x_1)}  \right) - (d_1-1)  \hspace{.6in}   \mbox{\cite[Remark~4.1]{chardin}}\\
&=& \operatorname{reg}    \left(  \f{R }  { I_{n}R + (x_1) }  \right)  - (d_1-1)   \hspace{.6in} \mbox{[Lemma~\ref{ideals in tprime}]}\\
&=& \operatorname{reg}  \left(  \f{R } { I_{n}    R }   \right) +  (d_1-1) -  (d_1-1)     \hspace{.4in} \mbox{\cite[Remark~4.1]{chardin}} \\
&=& \operatorname{reg}  \left(  \f{T} { I_{n}   T  }   \right)  .
  \eeqn
\end{proof}

From Lemma~\ref{reg comparision} it follows that  we need to compute $\operatorname{reg}(T/ I_nT)$.
\begin{corollary}
\label{regularity modulo x_2^2}
Let  $n \geq 1$.  Then
\beqn
\operatorname{reg}\left( \f{T}{I_{n}T + ( x_2^2) } \right)
=
\begin{cases}
\f{ d   d_3}{2}n + 2 d_2 -2& \mbox{ if } n = 2r,\\
\f{d  d_3}{2} n + d_2 + \left(\f{  -d}{2} + q+1\right) d_3 -2& \mbox{ if }  n = 2r-1    .
\end{cases}
\eeqn
\end{corollary}
\begin{proof}
If  $n=2r$,   then by Lemma~\ref{ideals in tprime}, $I_{2r}T + (x_2^2)= ( x_2^2, x_3^{  dr })$ and  hence
\beqn
\operatorname{reg} \left( \f{T }{ I_{2r}T + (x_2^2)} \right) = 2    d_2 + d d_3 r-2 = \f{ d  d_3}{2}n + 2 d_2 -2.
\eeqn
If $n= 2r-1$,  then by Lemma~\ref{ideals in tprime},
$I_{2r-1}T + (x_2^2)
=  (I_{2(r-1)  }T)(I_1T) + (x_2^2) =  (x_3^{d(r-1)}) (x_2 x_3^q, x_3^{q+1}) + (x_2^2)
= (x_2^2, x_2x_3^{ d(r-1) + q },   x_3^{  d(r-1) + q +1} )$.
 By Hilbert-Burch theorem the minimal free resolution  of $I_{2r-1} T+ (x_2^2)$ is of the form
$$
\small
\xymatrix@C=10pt{
0\ar[r]  &  {\begin{array}{c}  T[ - 2d_2 - (d(r-1) + q)d_3 ] \\ \oplus  \\T[ -d_2  - (d( r-1) + q+ 1) d_3]\end{array}}
\ar[rrrrr]^(0.5){  \left(
\begin{array}{cc}
x_3^{  d(r-1) + q } & 0\\
-x_2           & -x_3\\
0                & x_2
\end{array} \right)}
&&&&& { \begin{array}{c} T[-(2d_2)]  \\ \oplus \\ T[ - (d_2 + ( d(r-1) + q)d_3) ]  \\  \oplus \\ T[- ((d(r-1) + q+ 1) d_3)]\end{array}}
\ar[r] & T
\ar[r] &  {\displaystyle \f{T}{I_{2r-1} T+ (x_2^2)}}
\ar[r] & 0.
}
$$
\normalsize
Hence
\beqn
\operatorname{reg} \left(\f{T}{I_{2r-1} T+ (x_2^2)} \right)
=  d_2 + ( d(r-1) + q+1)d_3   -2
=  \f{d  d_3}{2} n + d_2 + \left(\f{  -d}{2} + q+1\right) d_3 -2 .
\eeqn
\end{proof}

\begin{lemma}
\label{lemma mod x_3}
For all $n \geq 1$,
\beqn
\operatorname{reg}\left( \f{T}{I_{2n} T+ ( x_3^d) } \right)
= 2d_2(2n) -2d_2+ d  d_3-2.
\eeqn
\end{lemma}
\begin{proof}
By  Lemma~\ref{ideals in tprime} we get
\beqn
    I_{2n} T+ (x_3^d)
= I_2^n T+ (x_3^d)
= (x_2^4, x_2^3x_3^q, x_2^2 x_3^{q+1}, x_3^d)^{n} + (x_3^d)
= (x_2^{4n}, x_2^{4n-1}x_3^{q}, x_2^{4n-2} x_3^{q+1}, x_3^d).
\eeqn
By Hilbert-Burch theorem the minimal free resolution of $ I_{2n}T + (x_3^d)$ is
\small
$$
\xymatrix@C=12pt{
0\ar[r]  &  {\begin{array}{c}
T[ - (4n-1 )d_2 -(q+1) d_3  ] \\ \oplus  \\T[ -4n d_2 - qd_3]
\\ \oplus \\
T[-(4n-2) d_2 - d  d_3]
\end{array}}
\ar[rrrrr]^(0.5){  \left(
\begin{array}{ccc}
0 & x_3^q      &0\\
x_3          & -x_2 & 0\\
 -x_2 & 0 &- x_3^q\\
 0& 0& x_2^{4n-2}
\end{array} \right)}
&&&&& { \begin{array}{c} T[-4n d_2]  \\ \oplus \\ T[ -(4n-1) d_2  - qd_3]  \\  \oplus \\ T[- (4n-2)  d_2 - (q+1) d_3]\\
\oplus \\T[ -d  d_3]
\end{array}}
\ar[r] & T
\ar[r] &  {\displaystyle \f{T}{I_{2n}T + (x_3^d)}}
\ar[r] & 0.
}
$$
\normalsize
This gives $\operatorname{reg}( T/ I_{2n} T+ (x_3^d)) =  (4n-2) d_2 + d d_3 - 2 = 2d_2(2n) -2d_2+ d  d_3-2
$.
\end{proof}

\begin{proposition}
\label{regularity even}
Let  $n \geq 1$. Then
\beqn
  \operatorname{reg} \left( \f{T}{I_{2n}T }\right)
&=& \begin{cases}
(2d_2)(2n) - 2d_2 + d d_3 -2  & \mbox{ if } q=1 \mbox{ and } m=1,\\
 \f{d d_3}{2} (2n) + 2d_2-2& \mbox{ otherwise }    .
\end{cases}.
\eeqn
\end{proposition}
\begin{proof} For all $n \geq 1$,
\beq \nonumber
(I_{2n} T : x_3^d) \nonumber
&=& \sum_{a_1 + 2 a_2 = 2n} ((J_1T)^{a_1} ((J_2T)^{a_2} : x_3^d)\\ \nonumber
&=& \left( \sum_{a_2=1}^{n} ((J_1T)^{a_1} (J_2T)^{a_2} : x_3^d)\right) + (J_1T)^{2n} : x_3^d)\\ \nonumber
&\subseteq &  \left( \sum_{a_2=1}^{n} (  (J_1T)^{a_1} (J_2T)^{a_2-1} : x_3^d) \right) + J_1^{2n-2}
\hspace{.2in} \mbox{[\cite[(3.4)]{clare}]}\\
&\subseteq& I_{2n-2}T.
\eeq
The other inclusion follows from the fact that  $x_3^d I_{2n-2}T \subseteq (J_2T) (I_{2n-2}T) = I_{2n}T $.
Hence we have the exact sequence
$$
\xymatrix@C=20pt{
0 \ar[r] &{\displaystyle \f{T}{I_{2n-2}T } [-d d_3]}
\ar[r]^(0.7){.x_3^d}  & {\displaystyle \f{T}{I_{2n} T}}
\ar[r] & {\displaystyle \f{T}{I_{2n} T+ (x_3^d)}}
\ar[r] & 0\\
}
$$
This implies that
\beqn
 &&  \operatorname{reg}\left( \f{T}{I_{2n}T}\right)\\
&=& \max\left\{    \operatorname{reg}\left( \f{T}{I_{2n-2} T}\right) + d  d_3 ,
                            \operatorname{reg}\left( \f{T}{I_{2n} T  + (x_3^d) }\right)  \right\}\\
&=& \max\left\{    \operatorname{reg}\left( \f{T}{I_{2n-4} T}\right) + 2d d_3 ,
                           \operatorname{reg}\left( \f{T}{I_{2n-2}  T + (x_3^d) }  \right) + d   d_3 ,
                           \operatorname{reg}\left( \f{T}{I_{2n}T   + (x_3^d) }\right)   \right\}\\
&=& \vdots\\
&=& \max\left\{   \operatorname{reg}\left( \left. \f{T}{I_{2n-2i} T+ (x_3^d)}\right) + di d_3 \right|
                                                                                i=0, \ldots, n-1  \right\}\\
&=& \max\left\{ \left. 2d_2(2n-2i)   -2d_2 + d  d_3  -2         + di  d_3\right| i=0, \ldots, n-1  \right\}
        \hspace{1in}
        \mbox{[by Lemma~\ref{lemma mod x_3}]}\\
&=& \begin{cases}
(2d_2)(2n) - 2d_2 + d  d_3 -2  & \mbox{ if } q=1 \mbox{ and } m=1,\\
2d_2(2)   -2d_2 + d  d_3  -2         + d (n-1)  d_3 & \mbox{ otherwise}
\end{cases}\\
&=& \begin{cases}
(2d_2)(2n) - 2d_2 + d  d_3 -2  & \mbox{ if } q=1 \mbox{ and } m=1,\\
 \f{d d_3}{2} (2n) + 2d_2-2  & \mbox{ otherwise}
\end{cases}.
\eeqn
\end{proof}

\begin{proposition}
\label{regularity odd}
 Let  $n \geq 1$. Then
\beqn
  \operatorname{reg} \left( \f{T}{I_{2n+1} T}\right)
= \begin{cases}
  (2d_2) (2n+1)   -2 d_2 + d   d_3 -2 + 2d_2 & \mbox{ if }  q=1 \mbox{ and } m=1\\
\f{3d_3}{2} (2n+1) + 4d_2 -\f{d  d_3}{2} -2  & \mbox{ if } q=1 \mbox{ and }  m=2 \\
    \f{d  d_3}{2}  (2n+1) +d_2 + \left(\f{  -d}{2} + q+1\right) d_3-2     & \mbox{ if }   q=1  \mbox{ and }  m \geq 3 \mbox{ or } q  \geq 2
\end{cases}.
\eeqn
\end{proposition}
\begin{proof}
 For all $n \geq 1$, the sequence
\normalsize
$$
\xymatrix@C=20pt{
0 \ar[r] &{\displaystyle \f{T}{I_{2n}T}[-2d_2] }
\ar[r]^{.x_2^2}  & {\displaystyle \f{T}{I_{2n+1} T}}
\ar[r] & {\displaystyle \f{T}{I_{2n+1}T + (x_2^2)}}
\ar[r] & 0\\
}
$$
is exact by \cite[Theorem~3.1]{clare}. Hence
\beqn
\label{regularity odd and even}
 &&    \operatorname{reg}\left( \f{T}{I_{2n+1} T}\right) \\
&= &  \max\left\{    \operatorname{reg}\left( \f{T}{I_{2n} T}\right) + 2 d_2 ,
         \operatorname{reg}\left( \f{T}{I_{2n+1}   T+ (x_2^2) }\right)  \right\}\\
&=&  \begin{cases}
         \max\left\{  (2d_2) (2n)   -2 d_2 + d   d_3 -2 + 2 d_2,
         {d d_3} n +d_2 + ( q+1) d_3 -2    \right\}
&       \mbox{ if }  q=1 \mbox{ and } m=1\\
         \max\left\{ d d_3 (n) + 2d_2  -2 + 2 d_2,
         {d  d_3}  (n) +d_2 + ( q+1) d_3-2     \right\}
&       \mbox{ otherwise }
         \end{cases} \\
&&    \hspace{4in} \mbox{[Proposition~\ref{regularity even},  Corollary~\ref{regularity modulo x_2^2}]}\\
&=&  \begin{cases}
        (2d_2) (2n+1)   -2 d_2 + d d_3 -2
&     \mbox{ if }  q=1 \mbox{ and } m=1\\
       \f{d  d_3}{2} (2n+1) + 4d_2 -\f{d d_3}{2} -2
&     \mbox{ if } q=1 \mbox{ and }  m=2 \\
       \f{d d_3}{2}  (2n+1) +d_2 + \left(\f{  -d}{2} + q+1\right) d_3-2
&    \mbox{ if }   q=1  \mbox{ and }  m \geq 3 \mbox{  or } q  \geq 2
\end{cases}.
\eeqn
\end{proof}

\begin{theorem}
\label{final regularity}
\been
\item
\label{final regularity one}
 $\operatorname{reg} (R/ \mathfrak p)= d_2 + (q+1) d_3-2$.

\item
\label{final regularity two}
Let $n \geq 2$.
\been
\item If $q=1$ and $m=1$, then
$
{\displaystyle \operatorname{reg} (R/ \mathfrak p^{(n)})
= (2d_2)n -2d_2 + d  d_3-2}$.

\item
If $q=1$ and  $m=2$, then  ${
\displaystyle
  \operatorname{reg} \left( \f{R}{ \mathfrak p^{(n) } } \right)
= \begin{cases}
   \f{d  d_3}{2}n + 4d_2 -\f{d  d_3}{2} -2& \mbox{ if  $n$ is odd},\\
   \f{d  d_3}{2}n   + 2d_2-2  & \mbox{ if  $n$ is even}   .
\end{cases}
}
$

\item
If $q=1$ and $m \geq 3$ or $q \geq 2$, then  $${\displaystyle
  \operatorname{reg} \left( \f{R}{ \mathfrak p^{(n) } } \right)
= \begin{cases}
    \f{d  d_3}{2}  n +d_2 + \left(\f{  -d}{2} + q+1\right) d_3-2    & \mbox{ if  $n$ is odd}\\
  \f{d  d_3}{2}n   + 2d_2-2 & \mbox{ if  $n$ is even}  .
\end{cases}
}$$
\eeen
\eeen
\end{theorem}
\begin{proof}
By Lemma~\ref{reg comparision},   $\operatorname{reg} (R/ \mathfrak p^{(n)} )= \operatorname{reg} (T/  I_nT) $.
Hence  (\ref{final regularity one})  follows from Corollary~\ref{regularity modulo x_2^2} and
(\ref{final regularity two}) follows from Proposition~\ref{regularity even} and Proposition~\ref{regularity odd}.
\end{proof}

We end this paper with the following remark.
\begin{remark}
From our computations one can verify that for all $q \geq 1$ and $m \geq 1$,
\beqn
\f{\alpha(\mathfrak p)}{ \gamma(\mathfrak p)} \leq \rho(\mathfrak p) \leq \f{\operatorname{reg}(\mathfrak p)}{\gamma(\mathfrak p)}.
\eeqn
It follows that  Theorem~1.2.1 of \cite{bocci-harbourne} holds true for $\mathfrak p$.
\end{remark}

redpajama_set_name: RedPajamaArXiv


Dr K.S.Gangadharan has been venerated with many awards and recognitions. The paramount among these are ASHTANGARATHNA AWARD by the GOVT OF KERALA in the year 2013, NEHRU PEACE FOUNDATION AWARD for the best performance in the field of Ayurveda in the year 2003 -2004, other recognitions are felicitation by ARYA VAIDYA PHARMACY, GOLDEN JUBILY AWARD by Ayurveda Mandalam. Nangelil Excellence Award in 2010. Excellence award by Ayurveda Medical association Of India in 2006, recognition by MathaAmrithanandamay for his contribution in the field of Ayurveda etc.


text:
\section{Introduction}

Predicting the statistical properties of nonlinear random waves is at the core of research on turbulence~\cite{Nazarenko}. Generally speaking, nonlinear wave propagation in media with self-focusing nonlinearity tends
to produce heavy-tailed deviations from the initial Gaussian statistics,  observed in the probability density function (PDF) of the random wave field amplitude.  In recent years, the question of the emergence of heavy-tailed statistical distributions has been extensively studied in relation to the occurrence of extreme events such as  Rogue Waves (RW),  mainly in the physical contexts of fluid dynamics  \cite{kharif2009rogue, Onorato:04, Onorato:05, Onorato:13} and  optics \cite{Solli:07, Mussot:09, Walczak:15, Pierangeli:15,Suret:16, Narhi:16, Safari:17, Tikan:18}. Even though statistical
properties of nonlinear defocusing media have been less extensively examined, several experiments have shown that defocusing nonlinearities tend to produce low-tailed deviations from
the initial Gaussian statistical distribution \cite{Randoux:14,Bromberg:10}.

Among nonlinear random waves systems, those that are described by the  one-dimensional nonlinear Schr\"odinger Equation (1D-NLSE) play a unique and fundamental role. First of all, 1D-NLSE describes various physical systems at leading order in the amplitude and secondly, 1D-NLSE is integrable and exhibit many remarkable exact solutions, including solitons (both for the focusing and the defocusing cases)~\cite{Yang}. The salient features of focusing regime of 1D-NLSE are  modulational instability of plane waves and the existence of breather solutions localized both in space and in time and thus the focusing 1D-NLSE  plays a central role in the study of RWs ~\cite{Peregrine:83, Henderson:99,Akhmediev:09,Akhmediev:09b,Kibler:10,Chabchoub:11, Kibler:11}. The study of the statistics of nonlinear wave systems described by 1D-NLSE  enters within the framework of {\it integrable turbulence} proposed by Zakharov~\cite{Zakharov:09, Zakharov:13, Agafontsev:15}. Generally, integrable turbulence arises in nonlinear random wave systems described  by integrable equations such as the 1D-NLSE, the Korteweg de Vries (KdV) equation and other physically important equations. Given the absence of resonances in integrable systems, the mechanisms underlying integrable turbulence~\cite{Walczak:15, Agafontsev:15, Suret:16, Randoux:16, SotoCrespo:16, AkhmedievIntegrableTurbulence:16, randoux2016livre, Suret:17, Tikan:18} are of profoundly different nature than those found in the standard {\it wave turbulence}~\cite{Zakharov,Nazarenko, Picozzi:14}. Since many nonlinear wave systems can be described by partial differential equations having an integrable core part, integrable turbulence has become an active field of research both from the theoretical and experimental perspective ~\cite{Zakharov:09, Zakharov:13, Agafontsev:15,Walczak:15,  Suret:16, Randoux:16, SotoCrespo:16, AkhmedievIntegrableTurbulence:16, randoux2016livre, Suret:17, Tikan:18}.

Random wave fields composed of a linear superposition of a large number of {\it independent} Fourier modes having random phases and/or amplitudes play  a fundamental role in wave turbulence~\cite{Zakharov, Nazarenko} and in integrable turbulence  ~\cite{Randoux:16, randoux2016livre}. As an immediate consequence of the  central limit theorem, these nonlinear random waves are characterized by Gaussian statistics. Numerical simulations and optical fiber experiments show that the long-term evolution  of such Gaussian random wave fields in the focusing regime of 1D-NLSE is accompanied by the emergence of a heavy-tailed PDF ~\cite{Walczak:15,Suret:16,Tikan:18}. Contrastingly, in the defocusing regime of 1D-NLSE, the long-term statistics of  random waves with the same initial distributions, is characterized by a low-tailed PDF ~\cite{Randoux:14,Randoux:16, randoux2016livre}.  The theoretical description  of statistical changes that occur in the course of  nonlinear propagation of such wave fields in 1D-NLSE systems is a very active current field of research ~\cite{Randoux:16, randoux2016livre}.

The first statistical theory of wave systems described by the integrable
1D-NLSE has been established within the framework of wave turbulence
theory \cite{Janssen:03}. The wave turbulence theory fundamentally relies on a scale separation where
the linear dispersion time $T_L$ is supposed to be much shorter than the nonlinear
time $T_{NL}$. Wave turbulence theory is principally applicable
to sets of random dispersive waves that are engaged in a {\it weakly } nonlinear {\it resonant}  interaction~\cite{Nazarenko, Zakharov} (we note, howevere, that resonant energy exchanges are not possible in wave systems ruled by the integrable 1D-NLSE \cite{Janssen:03,Suret:11,Zakharov:09}).
A modified treatment of the wave turbulence theory based on the derivation
of quasi-kinetic equations for the lowest order moments of the wave field
has been proposed in ref. \cite{Janssen:03, Soh:10,Suret:11,Picozzi:14}.  By using this theoretical approach, the evolution of the kurtosis (the fourth standardized moment of the PDF) of partially coherent waves has been derived both in the focusing and defocusing regime of the 1D-NLSE~\cite{Janssen:03}. The kurtosis quantifies the ``heaviness'' of the tail in the PDF of the field amplitude distribution and this theory confirmed that the kurtosis increases in the focusing case and decreases in the defocusing case while the initial field is characterized by Gaussian statistics. However, we stress that this kind of  analytical treatment  is inherently {\it limited to the weakly nonlinear}  propagation regime.\\

The question of integrable turbulence (i.e. statistical theory of random nonlinear
wave systems governed by integrable partial differential equations) has been recently
approached from a completely different perspective. For a certain class of wave systems, one can take advantage of the mathematical framework of dispersive
hydrodynamics -- the semi-classical theory of nonlinear dispersive waves \cite{Biondini:16} --
in order to analyse the wave evolution asymptotically, by introducing two distinct spatio-temporal
scales: the long scale specified by initial conditions and the short scale by the internal coherence
length (i.e. the typical size of the coherent soliton-like structures).

The semi-classical, dispersive hydrodynamic approach describes the propagation regimes of a completely opposite nature compared to  the
regimes considered in the framework of wave turbulence theory.
 This approach can be applied  to the 1D-NLSE propagation if the initial scale of the fluctuations of the power of the complex field $|\psi|^2$ are much larger than the one corresponding to the balance between nonlinearity and  dispersion. In  most of the standard cases, this separation of scales correspond to situations where the nonlinear part of the energy is much greater that the linear (kinetic) part of the energy at the initial time.  As shown in ref. \cite{Randoux:17}, this scale separation permits one to split the development of integrable turbulence
into two distinct stages characterized by qualitatively different dynamical and statistical features.
At the initial (we shall call it ``pre-breaking'') stage of the evolution nonlinear effects dominate linear
dispersion and the wave fronts of the random initial field
experience gradual steepening leading to the formation
of gradient catastrophes that are subsequently regularized through the
generation of dispersive shock waves in the defocusing regime \cite{El:2016} and of Peregrine-like
breather sequences in the focusing regime \cite{Tikan:17}. As shown in ref. \cite{Randoux:17}, the dynamical and
statistical features that occur at the pre-breaking stage of the defocusing 1D-NLSE can be interpreted in
terms of the evolution of random Riemann waves.\\

In this paper, we extend the analysis of the previous works based on the semi-classical
treatment of the 1D-NLSE with random initial data  by calculating the short-time evolution of the
the normalized fourth moment $\kappa_4$ of the amplitude of the field. Similar to the standard kurtosis, the quantity $\kappa_4$ describes the degree of the deviation from the initial statistical distribution which is often assumed to be Gaussian~\cite{Onorato:16}.
Using the semi-classical Madelung transform and performing the zero dispersion limit, we derive a general  analytical expression for the short-time evolution
of  the fourth moment of the random 1D-NLSE wave field in terms of hydrodynamic
variables, and  show that this expression can be further simplified
for the wave field having  Gaussian statistics at initial time.  Our analytical asymptotic results are shown to be in excellent agreement with numerical simulations of the evolution of partially coherent initial data in 1D-NLSE.

This paper is organised as follows :

- Sec. II : using the semi-classical approximation, we identify the initial stage of the 1D-NLSE development of partially coherent waves with the nonlinearity dominated, dispersionless regime, and derive the general expression for the  short-time evolution of the fourth order moment $\kappa_4$ as a power series expansion in time $t$;

- Sec III : we apply the derived formula for $\kappa_4$ to the fundamental case of random waves characterized by Gaussian statistics at time $t=0$;

- Sec IV : we provide a comparison between our semi-classical  analytical results and numerical simulations of 1D-NLSE.

\section{The dispersionless limit of the 1D-NLSE and the time evolution of the fourth-order moment of a random wave field}\label{sec:main}

We consider the  1D-NLSE in the normalized form
 \begin{equation}
 i\varepsilon\frac{\partial\psi}{\partial t}+\frac{\varepsilon^2}{2}\frac{\partial^2\psi}{\partial x^2}+\sigma |\psi|^2\psi=0,
 \label{eq:NLS}
 \end{equation}
 where $\psi$ is a complex field,  $\varepsilon$ is the dispersion parameter, $\sigma=-1$ in the defocusing regime and  $\sigma=+1$ in the focusing regime.

The 1D-NLSE  \eqref{eq:NLS} is considered in a periodic box of size $L$,
 $\psi(x+L,t)=\psi(x,t) \, \forall t $. The field $\psi$ then can  be
represented as a Fourier series:
\begin{equation}\label{eq:psi}
\psi(x,t)=\sum_k\psi_k(t)e^{\frac{2i\pi}{L}kx} \quad \hbox{with} \quad k\in \mathbb{Z},
\end{equation}
where the Fourier coefficients are given by
\begin{equation}
\psi_k(t)=\frac{1}{L}\int_0^L\psi(x,t)e^{-2i\pi kx} dx.
\end{equation}

The ``density of particles'' $N$ and the
momentum $P$ represent  integrals of motion and are expressed in terms of Fourier coefficients:
\begin{equation}
 N=\frac{1}{L}\int_{0}^{L}|\psi|^2dx=\sum_k|\psi_k|^2 ,
\label{Npart}
\end{equation}
\begin{equation}
P=\frac{1}{L}\int_{0}^{L}\psi_x\psi^*dx=\sum_k \Big(\frac{2\pi ik}{L}\Big) |\psi_k|^2 .
\label{Momentum}
\end{equation}

The Hamiltonian, that we represent in the form
\begin{equation}
  H=\varepsilon^2H_{\rm L}+ H_{\rm NL}
  \label{eq:H}
\end{equation}
is also integral of motion, which is naturally split into two parts: the linear (kinetic energy) part
\begin{equation}
\varepsilon^2 H_{\rm L}(t)=\frac{\varepsilon^2}{2L}\int_{0}^{L}|\psi_x|^2dx=\frac{\varepsilon^2}{2}\sum_k \Big(\frac{2\pi k}{L}\Big)^2 |\psi_k|^2
\label{HL}
\end{equation}
and the nonlinear part
\begin{equation}\label{HNL}
H_{\rm NL}(t)=\frac{\sigma}{2L}\int_{0}^{L}|\psi|^4dx.
\end{equation}

 We now assume that the Fourier modes at initial time
$\psi_{0k}= \psi_{k}(t=0) =|{\psi_{0k}}|e^{i \phi_{0k}}$
are  complex random variables.  The complex  field \eqref{eq:psi} is then a  random periodic solution of the 1D-NLSE. No particular
hypothesis about  statistical properties of $\psi_{0k}$ needs to be
introduced  at this step
but we will show in Sec. \ref{sec:gauss_stat} that the main result  of our analysis can be simplified if the initial statistics
of the random wave field is assumed to be Gaussian. We consider random initial
conditions for which  $N$,  $H_{\rm L}(t=0)$ and $H_{\rm NL}(t=0)$ are all $\mathcal{O}(1)$.
This is typically achieved  by taking the initial power
spectrum $n_{0k}=|\psi_{0k}|^2$ with the characteristic width $\Delta k \simeq 1$, which
implies that the typical spatial size of the initial random fluctuations
is also of
the order of unity and much larger than the internal coherence length (that is $\epsilon$). Such random waves are  often called {\it partially coherent}, particularly in the statistical optics context \cite{Mandel_wolf}.

Given the 1D-NLSE evolution of individual realizations of the random field $\psi(x,t)$  the challenge is to determine the associated evolution of its statistical characteristics  such as the probability density function (PDF) of the amplitude $|\psi|$, the power spectrum $|\psi_k|^2$ etc.
The particular  objective of this paper is to determine the short-time
evolution of the normalized fourth moment $\kappa_4(t)$
 defined as
\begin{equation}\label{kappa4}
  \kappa_4(t)=\frac{\langle \frac{1}{L}\int_0^L |\psi(x,t)|^4 \, dx \rangle}
        {\langle  \frac{1}{L}\int_0^L |\psi(x,t)|^2 \, dx \rangle^2},
\end{equation}
where the brackets $\langle \dots \rangle$ denote  ensemble average that
is made over a large number of realizations of the random process $\psi(x,t)$.
The independence of Fourier modes composing the signal implies statistical homogeneity, hence the average over space  $\lim\limits_{L \to \infty} \frac{1}{L}\int_0^L f(\psi) \, dx$ of an arbitrary function $f(\psi)$ of the random complex field $\psi$ is equivalent to the ensemble
average  over a large number of realizations. Nevertheless, it is convenient for our purposes in this paper
to use the double (space and ensemble) averaging in the definition \eqref{kappa4} of $\kappa_4$.
We also note that the limit of a large box ($L \to \infty$)
corresponds to the situation when a large number of random Fourier components
are included inside the core part of the Fourier spectrum, thus
fulfilling the pre-requisites of the central limit theorem for the initial condition.

The fourth  moment \eqref{kappa4} is an important characteristic of the PDF of a random process that quantifies the ``heaviness`` of its tail. In particular,  it can be used to characterize the deviation from Gaussianity  in the course of evolution, when the initial statistics is  Gaussian, in which case   $\kappa_4$ is known to be equal $2$ \cite{Onorato:16,Koussaifi:18}. The determination of $\kappa_4$ is particularly relevant to the rogue wave studies as the formation of a ``heavy tail'' of the PDF is associated with the frequent appearance of large-amplitude events in the random process' realizations \cite{Onorato:13, Walczak:15, Onorato:16, Randoux:16}.

Fig. \ref{fig:dynamics} shows a typical initial evolution of a random wave in the
regime where the cubic (Kerr) nonlinearity dominates linear dispersive effects, which corresponds to the semiclassical regime described by Eq.~\eqref{eq:NLS} with $\varepsilon \ll 1$ (in the numerics we took $\varepsilon=0.1$).
As shown in Fig. \ref{fig:dynamics}, the self-focusing dynamics tends to produce bright peaks while the self-defocusing
dynamics leads to a decrease of the peak amplitudes but is accompanied
by steepening of  slopes in the random amplitude profile. While only the short-time
evolution of the wave system is shown in Fig. 1, a longer development leads to the formation of gradient catastrophes -- the explosion of the first derivatives of the wave's profile. These gradient catastrophes  have qualitatively different geometrical nature in the defocusing regime (the wave-breaking singularity \cite{Pomeau:2008}) and the focusing regime (the elliptic umbilic singularity \cite{dubrovin_universality_2008-1}).
 In both cases the gradient catastrophes are regularized by dispersive effects via
the generation of nonlinear short wavelength oscillations: breather structures in the focusing regime \cite{Bertola:13} and
dispersive shock waves  in the defocusing regime (see \cite{El:2016} and references therein).  For convenience, we shall call the initial nonlinear evolution preceding the formation of  gradient catastrophes, the ``pre-breaking stage'' in both defocusing and focusing regimes. The advantage of the semi-classical, dispersive-hydrodynamic  approach  employed in this paper is that it enables one to asymptotically separate the pre-breaking and post-breaking stages of the evolution, which exhibit qualitatively different behaviors and require very different  analytical methods for their descriptions.

\begin{figure}[h]
\includegraphics[width=9.cm]{dynamics.pdf}
\caption{Numerical simulations of Eq. (\ref{eq:NLS}) ($\varepsilon=0.1$) showing the time evolution
  of a random field having Gaussian statistics at initial time $t=0$ (green lines).
  (a) Focusing regime ($\sigma = +1$). At short evolution time ($t<0.56$ in the plot)  the self-focusing
  dynamics produces bright peaks
  having the amplitude that grows in time. (b) Defocusing regime ($\sigma = -1$). The
  self-defocusing dynamics induces the decrease in time
  of the amplitudes of  random
  peaks. After some time (not reached in the plot), the random wave
  develops  gradient catastrophes that
  are regularized by dispersive effects leading to the generation of breather structures
  in the focusing regime and of dispersive shock waves in the defocusing regime.
}
\label{fig:dynamics}
\end{figure}

The starting point of our analysis is the evolution of the nonlinear part $H_{\rm NL}$
of the Hamiltonian. Differentiating \eqref{HNL} we obtain
\begin{equation}
\frac{d H_{\rm NL}}{dt}=\frac{\sigma}{L}\int_0^L |\psi|^2 \, \Big [\psi \frac{\partial \psi^*}{\partial t} + \psi^*\frac{\partial \psi}{\partial t} \Big ] \; dx. \\
\end{equation}
Using Eq. (\ref{eq:NLS}) and integrating by parts, one readily finds
\begin{equation}
\frac{dH_{\rm NL}}{dt}=\frac{\sigma\varepsilon}{L}\int_{0}^{L}Im[(\psi_x\psi^*)^2]dx.
\label{eq:dHnldt}
\end{equation}
Now, using the   Madelung transformation
\begin{equation}
\psi=\sqrt{\rho}e^{i\frac{\phi}{\varepsilon}} , \qquad  u=\frac{\partial\phi}{\partial x},
\label{Madelung}
\end{equation}
Eq. (\ref{eq:dHnldt}) can be rewritten as
\begin{equation}
\frac{dH_{\rm NL}}{dt}=\frac{\sigma}{L}\int_{0}^{L}\rho u\rho_xdx.
\end{equation}

Noticing from \eqref{kappa4}, \eqref{Npart}, \eqref{HNL} that
\begin{equation}\label{kappa4t}
\kappa_4(t)=\frac{2\langle H_{\rm NL}\rangle}{\sigma \langle N \rangle ^2}
\end{equation}
  one obtains
\begin{equation}
\frac{d \kappa_{4}}{dt}=\frac{2}{\sigma \langle N \rangle^2}\frac{d\langle H_{\rm NL}\rangle}{dt}=\frac{2}{\langle N \rangle ^2L}\int_{0}^{L}\langle\rho u\rho_x\rangle dx.
\label{eq:dk4dt}
\end{equation}

We now derive an analytical expression for  $\kappa_4(t)$ for short evolution times, $t\ll1$. If the dispersion parameter is small, $\varepsilon \ll 1$, the initial dynamics are dominated by nonlinearity.
To describe these dynamics analytically, we  consider the semi-classical limit of the 1D-NLSE \eqref{eq:NLS}  which is found by applying the Madelung transform \eqref{Madelung} and letting $\varepsilon \to 0$. Assuming smooth evolution of $\rho(x,t)$ and $u(x,t)$ in the pre-breaking regime,  we obtain in the limit $\varepsilon \to 0$
the following well-known set of nonlinear geometric optics equations
\cite{forest_exact_2009,Wabnitz:13b,Fatome:14,Kodama1995, Randoux:17} \begin{equation}\label{shallow}
\begin{cases}
\rho_t + (\rho u)_x =0, \\
 u_t+uu_x - \sigma\rho_x=0.
\end{cases}
\end{equation}
If $\sigma=-1$, equations (\ref{shallow}) are identical to the shallow-water equations
for an incompressible fluid with $\rho>0$ and $u$ interpreted as the fluid depth and
the depth-averaged horizontal fluid velocity, respectively. In the nonlinear fiber optics context,
$\rho$ represents the instantaneous optical power and $u$ represents the instantaneous frequency
(or chirp) \cite{Wetzel:16}.

Rigorous proofs of the point-wise convergence, as $\varepsilon \to 0$, of solutions of the 1D-NLSE \eqref{eq:NLS} to the solutions of the dispersionless system \eqref{shallow} with the same initial data, prior to the formation of gradient catastrophe, can be found for certain classes of initial data in \cite{jin:1999} (defocussing) and in \cite{Kamvissis, Tovbis:04} (focussing). Some important particular exact solutions of system \eqref{shallow} for the focusing case have been found as early as in 1960-70s  (see \cite{talanov_1965, akhmanov_self-focusing_1966, gurevich_exact_1970}). A detailed mathematical analysis of the pre-breaking dynamics in the defocusing case can be found in  \cite{forest_exact_2009} (see also \cite{Moro:14} for the special case of the wave breaking into vacuum).

It follows from the above consideration that, to study  the pre-breaking dynamics of partially coherent waves in 1-D NLSE \eqref{eq:NLS} we need to be able to describe  {\it random} solutions of system \eqref{shallow} obtained by evolving initial data  $\rho(x,0)$, $u(x,0)$ with given statistics (e.g. corresponding to the Gaussian statistics of $\psi$).  The study of such solutions has been recently initiated   for the defocusing case in the context of the interaction of random Riemann waves in fiber optics \cite{Randoux:17} (see also \cite{Suret:17}). In this connection one must stress that the term ``pre-breaking dynamics'' is understood here in the probabilistic sense, as for random initial data there is always a non-zero probability of having gradient catastrophe at any, arbitrarily small, moment of time. However, due to initial data having typical size $\Delta x = \mathcal{O}(1)$, we assume that for small $\varepsilon$ the contribution of such early gradient catastrophes to the statistics  is negligibly small.

To this end, with the short-time, pre-breaking evolution in mind,  we look for the solutions of Eqs. (\ref{shallow}) in the form of the  time power series
expansions for the realizations $\rho(x,t)$ and $u(x,t)$:
\begin{equation}\label{eq:series}
\begin{aligned} & \rho(x,t)=\rho_0(x)+\rho_1(x)t+\rho_2(x)t^2+\rho_3(x)t^3+\mathcal{O}(t^4) \\ & u_0(x,t)=u_1(x)t+u_2(x)t^2+u_3(x)t^3+\mathcal{O}(t^4) \, .
\end{aligned}
\end{equation}

 In \eqref{eq:series} we assumed   that initially,  $u_0 =0$, which agrees with  typical physical condition $u_0\ll\rho_0$  satisfied in standard realistic experimental conditions. Indeed timescales of amplitude and phase in partially coherent waves are generally similar ($\mathcal{O}(1)$ here). Considering the normalizations given by the Eq. (\ref{Madelung}), this means that the derivative of the phase $\frac{\partial(\phi_0/\epsilon)}{\partial x}=\mathcal{O}(1)$ and thus $u_0=\frac{\partial\phi_0}{\partial x}=\mathcal{O}(\epsilon)$ whereas $\rho_0 =\mathcal{O}(1)$. This assumption is for example satisfied in the experiments on the propagation of partially coherent light through optical fibers, see \cite{Randoux:17}.\\

Substituting Eqs. (\ref{eq:series}) into Eqs. (\ref{shallow}) we obtain
\begin{equation}\label{eq:rho_u_time}
\begin{aligned} & \rho(x,t)=\rho_0-\frac{1}{4}\sigma[\rho_0^2]_{xx}t^2+\mathcal{O}(t^4), \\ &u(x,t)=\sigma\rho_{0x} t-(\frac{1}{12}[\rho_0^2]_{xxx}+\frac{1}{3}\rho_{0x}\rho_{0xx}) t^3+\mathcal{O}(t^4).
\end{aligned}
\end{equation}
Next, substituting Eqs.~\eqref{eq:rho_u_time} into Eq. (\ref{eq:dk4dt}) and
integrating in time, we obtain the following expression for the time evolution of the
normalized fourth moment of the field amplitude:
\begin{equation}
\begin{aligned}
 \kappa_4(t)-\kappa_4(0) = & \frac{\sigma t^2}{\langle N\rangle^2L}\int_{0}^{L}\langle \rho_0\rho_{0x}^2\rangle dx\\ &-\frac{t^4}{2\langle N\rangle^2L}\int_{0}^{L}\langle\frac{2}{3}\rho_0^2\rho_{0x}\rho_{0xxx} \\ & + \frac{17}{6}\rho_0\rho_{0x}^2\rho_{0xx}+\frac{1}{2}\rho_{0x}^4\rangle dx +\mathcal{ O}(t^6).
\end{aligned}
\label{eq:k4_t}
\end{equation}

Eq. (\ref{eq:k4_t}) is our main general result. One can make now two important observations. The first one is that Eq.~\eqref{eq:k4_t} shows that the normalized fourth-order
moment $\kappa_4(t)$ of the field evolves quadratically with time at leading order for $t \ll 1$. The second observation is that Eq. (\ref{eq:k4_t})
 shows that  the increasing or decreasing nature of the time evolution of $\kappa_4(t)$ is determined by the value taken by $\sigma$.
In the focusing regime ($\sigma=+1$), $\kappa_4(t)$ is an increasing function of
time which means that the nonlinear evolution of the wave field is characterized
by PDFs that exhibit heavy tailed deviations from the initial statistical distribution.
On the other hand, in the defocusing regime ($\sigma=-1$) $\kappa_4(t)$ becomes a decreasing function
of time which implies  low-tailed deviations from the initial statistics occurring
in this regime. The statistical features described by Eq. (\ref{eq:k4_t}) are in full qualitative agreement
with the results that have been recently obtained in numerical and experimental investigations of  integrable turbulence \cite{Randoux:16,Randoux:14,Walczak:15,randoux2016livre,Suret:16,Suret:17}.

Let us emphasize that the decreasing or increasing nature of the time evolution of $\kappa_4$ has
also been shown to be determined by the defocusing or focusing nature of the propagation regime
for {\it weakly} nonlinear dispersive random waves that are described by the 1D-NLSE \cite{Janssen:03}.
Theoretical approaches that have been used in the weakly nonlinear regime are based on the
wave turbulence theory and they consist in deriving quasi-kinetic equations for the lowest order moments
of the wave field \cite{Suret:11,Janssen:03}. Dispersion plays crucial role in that consideration. Our work is based on a completely different, dispersive-hydrodynamic
approach, where dispersive effects  are initially not of dominant but of perturbative nature.

\section{Initial conditions with Gaussian statistics}\label{sec:gauss_stat}

Eq. (\ref{eq:k4_t}) represents a general result that is derived with the only assumption
that $\varepsilon \ll 1$. As we already stressed, it is valid before the typical time of the gradient catastrophe occurrence, i.e. for $t \ll 1$
(for random initial conditions with typical scales for $\rho$- and $x$-variations at $\mathcal{O}(1)$).
Importantly, Eq. (\ref{eq:k4_t}) is derived without any assumption on the nature of
the initial statistics of the random wave field. In this section we show that Eq. (\ref{eq:k4_t})
can be further simplified if the random wave field taken as initial condition has  Gaussian
statistics. To this end, we assume that the random initial field $\psi(x,0)$  is composed of a
linear superposition of a large number of {\it independent} random Fourier
modes $\psi_{k}(t=0)=\psi_{0k}=|{\psi_{0k}}|e^{i \phi_{0k}}$,   so that by the central limit theorem $\psi(x,0)$ is a Gaussian
random field  ~\cite{Nazarenko}.

In the random phase and amplitude (RPA)
model, $|\psi_{0k}|$ and $\phi_{0k}$ are both taken as randomly-distributed
variables \cite{Nazarenko}.
Here, we will mainly use the so-called random phase (RP) model
in which only the phases $\phi_{0k}$ of the Fourier modes are considered as
being random \cite{Nazarenko}. In this model, the phase of each Fourier
mode is randomly and uniformly distributed between $-\pi$ and $\pi$.
Moreover, the phases of separate Fourier modes are assumed to be uncorrelated so that
$\langle e^{i\phi_{0k}}e^{i\phi_{0k'}}\rangle= \delta_{k}^{k'}$. In the above expression,
the brackets, as usual,  represent the averaging  over  an ensemble
of many realizations of the random process; $\delta_{k}^{k'}$ is the
Kronecker symbol defined by $\delta_{k}^{k'}=1$ if $k=k'$ and $\delta_{k}^{k'}=0$
if $k\neq k'$. With the assumptions of the RP model described above,
the statistics of the initial field is homogeneous, which means that all statistical
moments of the initial complex field $\psi(x,t=0)=\psi_0(x)$ do not depend
on $x$ \cite{Picozzi:07,Picozzi:14}. This RP description of the initial
random field has been shown to describe in a satisfatory way many experiments
performed in the field of integrable turbulence \cite{Randoux:16,Randoux:14,Walczak:15,randoux2016livre,Suret:16,Suret:17,Onorato:04,Onorato:05,Onorato:01,onorato2005modulational}.

Given the delta-correlation of the random phases, the second moment of a field composed from the linear superposition of a large number
of {\it independent} Fourier components having Gaussian statistics is readily evaluated as
\begin{equation}\label{eq:2nd-order}
\big <\psi_{k}\psi_{k'}^*\big > = n_k \, \delta_k^{k'},
\end{equation}

and the  sixth moment can be factored into products of the second moments by using Wick's decomposition~\cite{Nazarenko}
\begin{equation}\label{eq:4th-order}
\begin{aligned}
  \langle\psi_{k_1}\psi_{k_2}\psi_{k_3}\psi_{k_4}^*\psi_{k_5}^*\psi_{k_6}^* \rangle=n_{k_1}n_{k_2}n_{k_3} && \\
             [\delta_{k_4}^{k_1}\delta_{k_5}^{k_2}\delta_{k_6}^{k_3}+
               \delta_{k_4}^{k_1}\delta_{k_5}^{k_3}\delta_{k_6}^{k_2}+
               \delta_{k_4}^{k_2}\delta_{k_5}^{k_1}\delta_{k_6}^{k_3}+&& \\
               \delta_{k_4}^{k_2}\delta_{k_5}^{k_3}\delta_{k_6}^{k_1}+
               \delta_{k_4}^{k_3}\delta_{k_5}^{k_1}\delta_{k_6}^{k_2}+
               \delta_{k_4}^{k_3}\delta_{k_5}^{k_2}\delta_{k_6}^{k_1}
             ]
\end{aligned}
\end{equation}
Now, using Eq. (\ref{eq:4th-order}), one can evaluate the coefficient for the  $\mathcal{O}(t^2)$ term in the expansion  (\ref{eq:k4_t}):
\begin{equation}\label{eq:t_square_term}
\begin{aligned}
 \frac{\sigma}{\langle N\rangle^2L}&\int_{0}^{L}\langle \rho_0\rho_{0x}^2\rangle dx  = \frac{\sigma}{\langle N\rangle^2}\sum_{k_1,...k_6} \Big(\frac{2 i \pi}{L}\Big)^2 \delta_{k_4+k_5+k_6}^{k_1+k_2+k_3} \\ &\times\langle\psi_{0k_1} \psi_{0k_2}\psi_{0k_3}\psi_{0k_4}^*\psi_{0k_5}^*\psi_{0k_6}^* \rangle (k_2-k_5)(k_3-k_6),
\end{aligned}
\end{equation}
where we have used the notation $\psi_{0k_i} = \psi_{k_i}(0)$ for the Fourier component at $t=0$. Using Eq. (\ref{eq:4th-order}), we obtain the following expression
for the short time evolution of the fourth-order moment of a random wave field that has Gaussian
statistics at initial time, i.e. $\kappa_4(0)=2$:
\begin{equation}\label{eq:k4_time_bis}
\begin{aligned}
\kappa_4(t)-\kappa_4(0)&=-\frac{\sigma}{\langle N\rangle^2}\sum_{k_1,k_2,k_3}n_{0k_1}n_{0k_2}n_{0k_3}\Big(\frac{2 \pi}{L}\Big)^2\\&\times[-2(k_2-k_3)^2]t^2+\mathcal{O}(t^4),
\end{aligned}
\end{equation}
where $n_{0k_i}=n_{k_i}(0)$ are the components of the  power spectrum at $t=0$. Using Eqs. (\ref{Momentum}), (\ref{Npart}), (\ref{HL}) and taking into account that $P=0$
for our random Gaussian  field,
we can finally rewrite Eq. (\ref{eq:k4_time_bis}) as
\begin{equation}\label{eq:k4_time_final_order2}
\kappa_4(t)-\kappa_4(0)=8\sigma \langle H_{\rm L}(0)\rangle t^2+\mathcal{O}(t^4).
\end{equation}
(Note that  $H_{\rm L} = \mathcal{O}(1)$ and $H_{\rm NL} = \mathcal{O}(1)$ whereas the linear part of the Hamiltonian (\ref{eq:H}) is $\mathcal{O}(\epsilon^2)$).

A similar, but somewhat lengthy, calculation permits one to obtain a more accurate expression
that includes $\mathcal{O}(t^4)$ correction (see Appendix \ref{appendixa}):
\begin{equation}
\begin{aligned}
\kappa_4(t)-&\kappa_4(0)=8\sigma \langle H_{\rm L}(0)\rangle t^2+\Bigg[\frac{208}{3} \langle H_{\rm L}(0)\rangle^2\\&+4\langle N\rangle\langle\sum_k\Big(\frac{2 \pi k}{L}\Big)^4|\psi_{0k}|^2\rangle\Bigg]t^4+\mathcal{O}(t^6).
\end{aligned}
\label{KHl}
\end{equation}

Eqs. (\ref{eq:k4_time_final_order2}) and (\ref{KHl}) show that the time evolution
of the fourth moment of the initially Gaussian random  wave field is determined by the linear part
$ H_{\rm L}(0)$ of the Hamiltonian computed for the initial condition.

Eq. (\ref{KHl}) can be further simplified if we assume that the shape of the Fourier power spectrum
of the initial random field is described by a Gaussian
\begin{equation}\label{eq:CI_gauss}
|\psi_{0k}|^2=n_0e^{-\frac{k^2}{(\Delta k)^2}}.
\end{equation}
The amplitude $n_0 \in \mathbb{R^+}$ has to be determined from the normalization
condition provided by Eq. (\ref{Npart}).
The linear energy density determined from Eq. (\ref{HL})
is $\langle H_{\rm L}\rangle=\frac{N(\Delta k)^2}{4}$, and we can finally rewrite
Eq.~\eqref{KHl} only in terms of the density of particles (or optical power) $N$
and of the width $\Delta k$ of the initial Fourier spectrum
\begin{equation}\label{eq:k4_gauss}
\kappa_4(t)-\kappa_4(0)=2\sigma N(\Delta k)^2t^2+\frac{22N^2(\Delta k)^4}{3} t^4 + \mathcal{O}(t^6).
\end{equation}

\section{Numerical simulations}

In this section, we use numerical simulations of Eq. (\ref{eq:NLS}) to
investigate the range and the degree of validity of the semi-classical
approach to the statistics of integrable turbulence presented in Secs. \ref{sec:main} and \ref{sec:gauss_stat}.
The initial condition used in our numerical simulations is a random complex field having Gaussian
statistics.  The amplitudes of the Fourier components are taken to be distributed
according to  Eq. (\ref{eq:CI_gauss}). In our numerical
simulations, the spectral phases $\phi_{0k}$ are random, statistically independent real numbers, uniformly
distributed between $-\pi$ and $+\pi$. The width $\Delta k$ of the
initial spectrum profile \eqref{eq:CI_gauss} is taken to be unity ($\Delta k=1$), and the value
of $\varepsilon$ in \eqref{eq:NLS}  is taken to be $0.1$. The numerical
simulations are performed by using a pseudo-spectral method with
the numerical box having  size $L=256$ that is discretized by using $2^{16}$ points.
\begin{figure}[h]
\includegraphics[width=9.cm]{focusing.pdf}
\caption{ Black line: numerical simulations of
  Eq. (\ref{eq:NLS}) with $\varepsilon=0.1$ in the focusing regime ($\sigma=+1$) showing the time evolution of the normalized fourth-order moment of the random field having at $t=0$ the Gaussian statistics and the Fourier spectrum defined by Eq. (\ref{eq:CI_gauss}) with
 $\Delta k=1$, $N=1$. Green line: analytical  result given by
  Eq. (\ref{eq:k4_gauss}) at leading order in $t^2$. Red line:  Analytical
  result given by Eq. (\ref{eq:k4_gauss}) including $t^2$ and $t^4$ evolution terms.
  The inset shows an enlarged view of the evolution of $\kappa_4(t)$ for $0<t<0.2$.
}
\label{fig:k4_focusing}
\end{figure}

Fig. \ref{fig:k4_focusing} shows the time evolution of the normalized
fourth moment $\kappa_4(t)$ of the random wave field in the focusing
regime ($\sigma=+1$). The curve plotted with
black line represents the result of the numerical simulation of Eq. (\ref{eq:NLS}).
At time $t=0$, the theoretical value taken by $\kappa_4(t)$ is $2$ due to the
Gaussian nature of the initial statistics.  As can be seen from Fig.~\ref{fig:k4_focusing}, in the numerical experiments the value taken by $\kappa_4(t)$ at $t=0$ slightly differs from $2$ (see the inset in Fig. \eqref{fig:k4_focusing}) because the  conditions of the central limit theorem are not perfectly fulfilled in our numerical simulations.  Indeed, because of the finite value of $L$, the number of Fourier modes in the spectrum  given by Eq. \ref{eq:CI_gauss} is finite, in particular, in the Full Width at Half Maximum (FWHM) we count $\frac{\sqrt{2\ln 2} L}{\pi}=95$ modes. Importantly, the deviation of the initial condition from  Gaussian statistics affects only the value of $\kappa_4(0)$ in Eq. \eqref{eq:4th-order} but not the evolution.
In the initial (before the formation of a gradient catastrophe) stage of the nonlinear evolution of the random wave, $\kappa_4(t)$
is at first an increasing function of time that later reaches a maximum around $t\sim 0.6$.
Then $\kappa_4(t)$ becomes a decaying function of time that reaches a stationary
value around $\sim 4$ at long evolution time. A similar evolution of $\kappa_4(t)$
has already been evidenced in numerical simulations presented in ref. \cite{Onorato:16}.

The curves plotted with green and red lines in Fig. \ref{fig:k4_focusing}
show monotonic evolutions of $\kappa_4(t)$ that are obtained from Eq. (\ref{eq:k4_gauss}).
In particular the curves plotted in the inset of Fig. \ref{fig:k4_focusing}
 clearly reveal a very good quantitative agreement
between numerical and theoretical results. In particular, a better
agreement between numerics and theory is obtained by including the
fourth-order correction term found in the
time expansion of the solution, see Eq. (\ref{eq:k4_gauss}).
A significant quantitative disagreement is found
between our theoretical results and the numerical simulation at evolution
times greater than $\sim 0.2$. This arises from the fact that our approach
is only valid at evolution times that are shorter that the typical wave breaking time
(the pre-breaking description). The significant occurrence of wave breakings
at evolution times greater than $\sim 0.2$ has strong influence on the wave evolution
and subsequently the wave statistics in a way that cannot be accounted for
by using our pre-breaking treatment.

\begin{figure}[h]
\includegraphics[width=9.cm]{defocusing.pdf}
\caption{Black line: numerical simulations of
  Eq. (\ref{eq:NLS}) with $\varepsilon=0.1$ in the defocusing regime ($\sigma=-1$) showing the time evolution of the normalized fourth-order moment of the random field having at $t=0$ the Gaussian statistics and the Fourier spectrum defined by Eq. (\ref{eq:CI_gauss}) with
 $\Delta k=1$, $N=1$. Green line: analytical  result given by
  Eq. (\ref{eq:k4_gauss}) at leading order in $t^2$. Red line:  Analytical
  result given by Eq. (\ref{eq:k4_gauss}) including $t^2$ and $t^4$ evolution terms.
  The inset shows an enlarged view  of $\kappa_4(t)$ plot for $0<t<0.2$.
}
\label{fig:k4_defocusing}
\end{figure}

Fig. \ref{fig:k4_defocusing} shows the comparison between the numerical simulation
of Eq. (\ref{eq:NLS}) and the theoretical result given by Eq. (\ref{eq:k4_gauss})
in the defocusing regime ($\sigma=-1$). In the defocusing regime, $\kappa_4(t)$ is
a monotonically decreasing function of time, as already evidenced
in ref. \cite{Onorato:16}. As for the focusing regime, a very good quantitative
agreement is obtained between numerics and the theory at short evolution time ($t<0.2$),
i.e. before the typical occurrence of gradient catastrophes.

\section{Conclusion}
In this paper we  have undertaken an analytical study of the problem of the  evolution of a random wave field in the 1D-NLSE for both focusing and defocusing regimes. This has been done from the perspective
of dispersive hydrodynamics, a semiclassical theory of
nonlinear dispersive waves exhibiting two distinct spatio-temporal scales:
the long scale specified by initial conditions and the short scale by the internal coherence length
(i.e., the typical size $\varepsilon$ of the coherent structures) \cite{Biondini:16}.
This scale separation enabled us to split the time evolution of the nonlinear
random wave system (integrable turbulence) into the initial,  ``pre-breaking'' stage, preceding the formation of  gradient catastrophes, when the evolution of the 1D NLSE wave field is almost everywhere smooth, and the ``post-breaking'' stage characterised by the generation of short-scale nonlinear oscillations (breathers or dispersive shock waves depending on the focusing vs. defocusing character of the 1D-NLSE).

Our work is concerned with the initial, pre-breaking stage of the semi-classical integrable turbulence, when the dynamical and
statistical features can be analytically described  in terms  of
random solutions of the dispersionless (nonlinear geometric optics) system \eqref{shallow}. As a result, we have derived a simple  asymptotic formula describing the
evolution of the normalized fourth moment of the random wave field. This formula, applied to the problem of the 1D-NLSE evolution of random field having initial Gaussian statistics, describes  the initial stage of the formation of heavy tails of the PDF of the field amplitude in the focusing case and the formation of low tails in the defocusing case.

    Recently, an exact and general identity that relates the changes in the statistical properties of the wave field to the changes of its Fourier spectrum has been derived by using the Hamiltonian structure of 1D-NLSE~\cite{Onorato:16}. In other words, the knowledge of the fourth order moment also provides the description of spectral properties. The general description of the stationary state of integrable turbulence and the theoretical prediction of the fourth order moment is still an open fundamental question.    In the weakly nonlinear regime, the wave turbulence approach provides  a statistical description of the nonlinear propagation of random wave fields in 1DNLSE systems~\cite{Janssen:03,Mori:06,Soh:10,Suret:11}.  Recently, using an approach based on the so-called large deviation theory, it has been  shown that rogue waves obey a large deviation principle, i.e. the heavy tails of the PDF of the random wave field are  dominated by single realizations \cite{Dematteis:18, Dematteis:18:NLS}. This  approach is very promising but  does not provide a simple formula for the evolution of the statistics. In this article we have demonstrated that the semi-classical approach is an extremely powerful tool enabling one to describe in a simple way the early stage of integrable turbulence {\it in the strongly nonlinear/ small dispersion} regime.  The proposed methodology  can be applied to the description of partially coherent random nonlinear waves described by other integrable equations, including shallow water waves described by the KdV equation and its extensions. In particular, the pre-breaking statistics of bi-directional random shallow water waves is equivalent to that described by the defocusing 1D-NLSE and studied in this paper.

The semi-classical approach to the statistics of random waves in integrable systems is general and can be used beyond the short-time asymptotic regime. It is known very well that, in the semi-classical limit the evolution of nonlinear dispersive waves after the gradient catastrophe point is described by the so-called Whitham modulation equations \cite{whitham74} governing the behaviour of the averaged integrals of motion, and replacing the dispersionless system \eqref{shallow} (see \cite{kamchatnov_nonlinear_2000, El:2016, el_dam_2016, tovbis_semiclassical_2016}  and references therein for the application of the Whitham theory to the defocusing and focusing 1D-NLSE).   Such an extension of the proposed method to longer times  is very promising but also highly nontrivial.

\section{Appendix: Computation of $\mathcal{O}(t^4)$ corrections for the case of Gaussian statistics at $t=0$} \label{appendixa}

Here we provide the simplified expressions for the three terms that are found in the
integral giving the coefficient of the $O(t^4)$ term in Eq. (\ref{eq:k4_t}).
To obtain these expressions, we assume Gaussian statistics at the initial time and
use Eq. (\ref{eq:2nd-order}) and Eq. (\ref{eq:4th-order}) to obtain

\begin{widetext}

\begin{equation}
  \begin{aligned}
    -\frac{1}{3\langle N\rangle^2L} \int_{0}^{L}\langle\rho_0^2\rho_{0x}\rho_{0xxx}\rangle dx &= -\frac{1}{3\langle N\rangle^2}\Big(\frac{2 i \pi}{L}\Big)^4 \times\sum_{k_1,...k_6} \langle\psi_{0k_1}\psi_{0k_2}\psi_{0k_3}\psi_{0k_4}\psi_{0k_5}^*\psi_{0k_6}^*\psi_{0k_7}^*\psi_{0k_8}^* \rangle       \\ \times &\delta_{k_5+k_6+k_7+k_8}^{k_1+k_2+k_3+k_4}(k_3-k_7)(k_4-k_8)^3 =48 \langle H_{\rm L}\rangle^2+4\langle N\rangle\langle\sum_k\Big(\frac{2 \pi k}{L}\Big)^4|\psi_{0k}|^2\rangle
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
  -\frac{17}{12\langle N\rangle^2L}\int_{0}^{L}\langle \rho_0\rho_{0x}^2\rho_{0xx}\rangle dx &= -\frac{17}{12\langle N\rangle^2}\Big(\frac{2 i \pi}{L}\Big)^4 \times\sum_{k_1,...k_6}\langle\psi_{0k_1}\psi_{0k_2}\psi_{0k_3}\psi_{0k_4}\psi_{0k_5}^*\psi_{0k_6}^*\psi_{0k_7}^*\psi_{0k_8}^* \rangle\delta_{k_5+k_6+k_7+k_8}^{k_1+k_2+k_3+k_4}  \\
  \times &(k_2-k_6)(k_3-k_7)(k_4-k_8)^2= \frac{136}{3} \langle H_{\rm L}\rangle^2
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
-\frac{1}{4\langle N\rangle ^2L}\int_{0}^{L}\langle \rho_{0x}^4\rangle dx &=-\frac{1}{4\langle N\rangle^2} \Big(\frac{2 i \pi}{L}\Big)^4  \times \sum_{k_1,...k_6}\langle\psi_{0k_1}\psi_{0k_2}\psi_{0k_3}\psi_{0k_4}\psi_{0k_5}^*\psi_{0k_6}^*\psi_{0k_7}^*\psi_{0k_8}^* \rangle \delta_{k_5+k_6+k_7+k_8}^{k_1+k_2+k_3+k_4}   \\ \times& (k_1-k_5)(k_2-k_6)(k_3-k_7)(k_4-k_8)=-24 \langle H_{\rm L}\rangle^2
\end{aligned}
\end{equation}

\end{widetext}

\begin{acknowledgments}
 This work  was  supported by EPSRC grant
EP/R00515X/1 (GE) and Dstl grant DSTLX-1000116851(GR, GE, SR). It has also been partially supported by the Agence Nationale de la Recherche through the LABEX CEMPI project (ANR-11-LABX-0007)   and by the Ministry of Higher Education and Research, Hauts-De-France Regional Council and European Regional Development Fund (ERDF) through the Contrat de
Projets Etat-R\'egion (CPER Photonics for Society P4S). GE and GR  thank the PhLAM laboratory at the University of Lille for hospitality and partial financial support (GE).
\end{acknowledgments}


Thank you for dressing in colours of the Gambian flag on Friday. £210 was raised to buy an Ipad for our Gambian school.
We continue to have a strong link with a school called Bakau Newtown in The Gambia. Every half term both schools send examples of their learning to one another to develop a deeper understanding of the wider world. Over the past few years we have raised money to buy a new roof for their school, new sinks, for reading books and stationary. There is a display in the school hall which shows examples of what recent work has been completed.
Please click on the links below to find out more about projects we have worked together on.
Just before Christmas the Y6 Committee organised a toy sale to raise funds for our Gambian school. Thank you to all of you who brought old toys and money to buy new ones as a total of £33.40 was raised.


Career Point School of Computer Application provides a stimulating academic environment that amalgamates the best of conventional and modern pedagogy, rich and industry-relevant curriculum, through competent faculty staff and intensive teaching-learning processes and continuous evaluation system.
The curriculum offers a wholesome coverage of technical subjects and specializations in programming, web applications, and networking. It is further strengthened with practical, projects and industry training. The Courses in Computer Applications, not only aim at conceptual and technical aspects of computer technology and application, but also focus on the overall development of personality of individual students, to perform successfully across cultures and teams. All this is ensured through an effective curriculum design.


On paper, Nadine Shah seems something of an outsider. She was born to Pakistani and Norwegian parents, but grew up in Whitburn, south Tyneside, where she was the "foreign kid". Among the north-east music collective – including Field Music, Frankie and the Heartstrings, Maxïmo Park and the Futureheads – she's a lone female, who makes brooding, theatrical post-punk as opposed to the rest of the scene's buoyant art-pop. Then, there's the rest of the alternative musical landscape in 2015, which remains dominated by the same skinny-white-man cliche that allowed Johnny Borrell to ride a motorbike through Kirsten Dunst's house at the height of indie's mid-00s tight-jeaned boom. Shah is the anti-Borrell. For starters, she wouldn't be seen dead in white denim.
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It was Nina Simone – a singer with a similar sense of melancholy and low vocal range – who first inspired Shah to be an artist. Simone's androgynous voice encouraged Shah to switch from replicating the pretty, overembellished warbles of Mariah Carey and the R&B divas and to embrace her own accent's beautiful, warm vowel sounds. That desire to capture the truth is something she still relishes, from her critique of a former hipster boyfriend's disingenuous fondness for Kerouac on Fool, to her scathing disregard for fancy knickers ("Now you're dolled up at the party/ Hoping there's someone to meet/ You are snatching and you're grabbing/ You've got lace on underneath", she sings on Living, while during The Gin One she sniffs at the woman who "threw the cottons out, and now my drawers are lined in lace").
However, people are buying her second album, Fast Food, which came out in April. In the first week of the Official Vinyl Chart, it debuted at No 9, and Shah sold more on that format than Brit Critics Choice winner James Bay. For an independent, uncompromising albums artist such as Shah, the revolt against instant-consumption culture is hugely valuable. But she hopes that her presence in the mainstream will benefit more than her own career.
The same can't be said for her place in the homogenous indie scene, though. Does she agree that indie is a Caucasian preserve?
"Yeah, because my ears prick up when I hear an Asian surname or one that's a bit different," she says. She can count only a few artists who divert from the stereotype, too: newcomer Ben Khan, Joe Mirza of the Slow Revolt and Bat For Lashes' Natasha Khan. Is this the industry's fault?
As we begin to meander into Shah's fondness for ill-fated romance, hipsters and her mother's disdain for her last album, a thought hits suddenly. By doing this interview, she's avoided the rigmarole of helping her band unload the gear into the venue. "Ha, yes!" she says triumphantly, before going out for a celebratory fag. Looks like there's a bit of Borrell in everyone, after all.
Fast Food is out now on Apollo.


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