Soliton propagation in nonuniform media

Abstract

The inverse-spectral-transform method of solution is shown to be applicable to the physically interesting problem of the nonlinear Schrodinger equation with a general "potential" term, iq, +q +2[~q~-Fix)]q=0. The method determines the class of solutions that are symmetric or antisymmetric in x. This is done with the help of a modification of the Ablowitz-Kaup-Newell-Segur and Zakharov-Shabat (AKNS-ZS) formalism incorporating an xand t-dependent eigenvalue parameter g, together with a transformation of variables. In certain physical applications, F(x) describes the inhomogeneity of the medium in which nonlinear wave propagation occurs. The functions F(x) for which the equation is amenable to solution by our method are shown to fall into two classes, depending on whether or not g is explicitly t dependent. If it is, we show that F{x)must be a general quadratic function of x. An explicit solution q(x, t) is written down and interpreted for a parabolic potential barrier. If g is independent of t, we find that localized solutions with static envelopes can exist for certain other functional forms of F(x). Finally, we comment on the extension of the analysis to explicitly time-dependent potentials or inhomogeneities F(x,t).

Soliton stability criterion for generalized nonlinear Schrödinger equations

“…Here two solitons with asymmetric shapes form a bound state in which their shapes and their separation distance oscillate. We observe this in numerical simulations, using an IC obtained by modifying a result for the case of a parabolic potential barrier [19], where two asymmetric lumps move away from each other, become smaller, and vanish eventually.…”

“…In Ref. [19], only the case of a parabolic potential barrier (μλ < 0, λ 0 = 0, μ 0 = 0) was explicitly considered, and it resulted in the solution…”

“…A solution with two asymmetric lumps was obtained by an IST by Balakrishnan [19] who investigated the GNLSE,…”

Negative and positive dust grain effect on the modulation instability of an intense laser propagating in a hot magnetoplasma

“…It is a classical field equation whose important applications are in the propagation of EMWs in nonlinear optical fibers and planar waveguides [59] and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the meanfield regime [60]. Additionally, it can be revealed in the studies of small-amplitude gravity waves on the surface of deep zero-viscosity water [59], Langmuir waves in hot plasmas [59], propagation of plane-diffracted wave beams in the focusing areas of the ionosphere [61] and propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains [62]. The MI for the right-and left-hand circularly polarized EMW can be obtained using the usual method introduced by Shukla et al [46].…”

An Approximation Method for the Scattering Data of One-Dimensional Soliton Equations under Arbitrary Rapidly Decreasing Initial Pulses

“…It is known that the AKNS-ZS formulation for the NLSE can be extended to a situation where such a potential term exists. 36) As long as we choose suitable initial conditions, by transforming a spatial variable x to a new one and reconstructing a new scattering potential from a given initial wave packet, we can use the same procedure as that demonstrated in previous sections. Nevertheless, it is unclear whether all initial conditions with a rapidly decreasing property work well in this extended scheme.…”