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the distortion of pulses propagating through waveguides arises due to dispersive spreading
. however , if the waveguides is made using materials the refractive index of which depends nonlinearly on the intensity of the propagating pulses , it will be possible to compensate the effect of dispersion by a pulse - narrowing effect @xcite . a typical example in respect of this
is provided by optical fibers . here
the weak nonlinearity of the index of refraction arising due to kerr effect produces a self modulation which in turn causes steepening of the wave .
the dynamical interplay between the dispersive and narrowing effects produces the so called optical solitons . for an isotropic medium with cubic nonlinearity
the complex optical wave field @xmath0 is governed by the nonlinear schrdinger equation ( nls )    @xmath1    here @xmath2 stands for time and @xmath3 , the coordinate along the fiber .
the suffixes @xmath3 and @xmath2 of @xmath4 denote partial derivatives with respect to these variables . in particular , @xmath5 .
the main reason for the robustness of the solitons described by @xmath6 is that the wave numbers of the solitons lie in a range that is forbidden for the linear dispersive waves @xcite . as a result
, linear wave can not be in resonance with the soliton to receive energy from the latter .
we then have a stable soliton .
the propagation of picosecond pulses is well described by the nls equation in @xmath6 which accounts for only the second - order dispersion and self modulation . but for femtosecond pulses another important physical effect namely , the third - order dispersion comes into play . in this case
the appropriate evolution equation for the pulse propagation is given by @xcite @xmath7 understandably , the parameter @xmath8 is a measure of the perturbation caused by the third - order dispersion . due to the effect of perturbation
a soliton could have wave numbers residing inside the linear spectrum of the system .
such a soliton is called an embedded soliton @xcite .
clearly , the embedded solitons can be in resonance with the perturbation to generate continuous wave radiation for subsequent decay . despite this , in certain parameter regimes these solitons can be stable @xcite . in view of this
the stability analysis of embedded solitons arising in the solutions of higher - order nls equation has been subject of considerable current interest .
the present work is an effort in this direction .
studies in embedded solitons @xcite within the framework of a third - order nls equation indicate that these soliton exits as a continuous family .
the members of the family being characterized by their propagation velocities are connected through a gauge transformation @xcite .
we demonstrate the existance of such a family by using a straightforward analytical model and argue that embedded solitons can be fully stable despite their tendency to shed radiation .
we believe that such an approach has distinct advantages to deal with physical problems because many unknown effects are then readily expressed and evaluated . to achieve our goal
we shall derive a variational approach involving trial functions in order to describe the main characteristics of the embedded solitons as determined by the third - order nls equation in @xmath9 .
although the chosen trial functions have a specific form , the shape parameters are allowed to evolve as the solitons propagate . with this assumtion
the evolution equation under consideration simplifies to a reduced lagrangian problem @xcite .
we shall try to study the stability of embedded solitons by taking variations of this lagrangian with respect to the shape parameters .
for the nls equation in @xmath6 there exists a well - defined spectral problem @xcite such that one can write a closed form analytical solution of it in terms of sech functions .
as opposed to this , the third - order nls equation can not be solved analytically . keeping this in view we look for a variational treatment of @xmath9 to find an accurate approximation solution . for our treatment
we will essentially rely on a ritz optimization procedure @xcite applied to the lagrangian function for the nls equation in @xmath9 .
we have found that the action functional @xmath10 with the lagrangian density @xmath11 reproduces the third - order nls equation via the hamilton s variational principle .    in the ritz optimization procedure , the first variation of the action functional is made to vanish for a suitable chosen trial function . in analogy with the well - known solution of @xmath6
we introduce a sech type trial function @xmath12\times \,\,\,\,\,\,\nonumber \\\exp i\left [ v(x)(t - y(x))+\frac{b(x)}{2a(x)}\left ( t - y(x)\right)^2+\sigma(x ) \right ]   .\end{aligned}\ ] ] here @xmath13 , @xmath14 , @xmath15 , @xmath16 , @xmath17 , and @xmath18 are real functions of @xmath3 .
the parameters @xmath13 , @xmath14 and @xmath16 are related to the three lowest - order moments of the @xmath4 envelope and represent respectively its amplitude , central position and width respectively .
the other parameters @xmath18 , @xmath15 and @xmath17 stand for the phase , velocity ( centre of the soliton ) and frequency chirp .
understandably , these parameters will all vary with the distance of propagation . using @xmath19 in @xmath20
we get @xmath21 where    @xmath22    @xmath23    @xmath24sech^2\left(\frac{t - y}{a}\right)\ ] ]    and @xmath25    here the subscript @xmath26 on @xmath27 indicates that we have inserted the @xmath28 ansatz for @xmath0 into the lagrangian density . in terms of @xmath29
the variational principle implied by @xmath30 leads to @xmath31 with @xmath32 we have found that result for @xmath33 is given by    @xmath34
from the vanishing conditions of the variationals @xmath35 , @xmath36 , @xmath37 , @xmath38 , @xmath39 and @xmath40 we obtain the following equations .
@xmath41    @xmath42    @xmath43    @xmath44    @xmath45    and @xmath46    from @xmath47    @xmath48    use of @xmath49 in @xmath50 gives @xmath51 again @xmath49 and @xmath52 give @xmath53 similarly from @xmath49 and @xmath54 we have @xmath55    subtracting @xmath56 from @xmath57 we write @xmath58 from @xmath49 , @xmath59 and @xmath60 we get a second - order differtial equation for the pulse width @xmath61 the first integral of @xmath62 gives @xmath63 where the potential field @xmath64 the constant of integration @xmath65 is determined by the initial conditions of @xmath62 .
first from the initial conditions of @xmath62 written as @xmath66 and @xmath67 , we have @xmath68 using @xmath69 in @xmath70 we write @xmath71 we now introduce a normalized pulse width @xmath72 to rewrite @xmath73 as    @xmath74    with    @xmath75    and    @xmath76    the potential function @xmath77 vanishes for two values of @xmath78 , namely , @xmath79 and @xmath80 implying that there is a minimum in between these points .
the point at which @xmath77 is minimum can be obtained from @xmath81 as @xmath82 in the limit when @xmath83 we have @xmath84 and @xmath85 . in this case
the potential well degenerates into a single point such that a particle released at that point will stay there . in the context of this work , this signifies that a wave pulse for which @xmath83 propagates with unchanged shape as a cosequence of exact balance between nonlinear and dispersive effects .
interestingly , we now have a continuous family of soliton solutions because @xmath86 depends on the continuous variable @xmath87 .
this implies that in the neighbourhood of each embedded soliton there is another one having a slightly higher or lower energy .
a particular embedded soliton when perturbed may always relax to the adjacent one .
this makes an embedded soliton fully stable .    in fig .
1 we plot the potential function @xmath77 as a function of @xmath78 for @xmath88 .
each of these potentials has @xmath89 and is , therefore , associated with an embedded soliton .
the solid curve @xmath90 is the potential function corresponding to the usual second - order nls equation while the dotted curves represent the potential functions of the embedded solitons as found in the solution of third - order nls equation . for @xmath91 ,
the potential curves for the embedded solitons lie above the solid curve for @xmath92 . for @xmath93
the dotted curves fall below the solid curve .
we have drawn only a few curves for discrete values of @xmath94 . however , a dense set of curves can be drwan by varying @xmath94 continuously .
this work forms the part of a research project f.10 - 10/2003(sr ) supported by the university grants commission , govt . of india .
one of the authors ( sga ) is thankful to the ugc , govt .
of india for a research fellowship .
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we derive a straightforward variational method to construct embedded soliton solutions of the third - order nonlinear schdinger equation and analytically demonstrate that these solitons exist as a continuous family . 
 we argue that a particular embedded soliton when perturbed may always relax to the adjacent one so as to make it fully stable .