General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger equation

General soliton solutions to a reverse-time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one-soliton solutions and two-soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.     

Variable sinh-Gaussian solitons in nonlocal nonlinear Schrödinger equation

Based on the nonlocal nonlinear Schrödinger equation that governs phenomenologically the propagation of laser beams in nonlocal nonlinear media, we theoretically investigate the propagation of sinh-Gaussian beams (ShGBs). Mathematical expressions are derived to describe the beam propagation, the intensity distribution, the beam width, and the beam curvature radius of ShGBs. It is found that the propagation behavior of ShGBs is variable and closely related to the parameter of sinh function (PShF). If the PShF is small, the transverse pattern of ShGBs keeps invariant during propagation for a proper input power, which can be regarded as solitons. If the PShF is large, it varies periodically, which is similar to the evolution of temporal higher-order solitons in nonlinear optical fiber. Numerical simulations are carried out to illustrate the typical propagation characteristics.     

Autoresonance excitation of a soliton of the nonlinear Schrödinger equation

The action of an external parametric perturbation with slowly changing frequency on a soliton of the nonlinear Schrödinger equation is studied. Equations for the time evolution of the parameters of the perturbed soliton are derived. Conditions for the soliton phase locking are found, which relate the rate of change of the perturbation frequency, its amplitude, the wave number, and the phase to the initial data of the soliton. The cases when the initial amplitude of the soliton is small and when the amplitude of the soliton is of the order of unity are considered.     

Solitary wave solutions for nonlinear fractional Schrödinger equation in Gaussian nonlocal media

This article is devoted to the study of nonlinear fractional Schrödinger equation with a Gaussian nonlocal response. We firstly prove the existence of solitary wave solutions by using the variational method and Mountain Pass Theorem. Numerical simulations are presented to verify the findings of the existence theorem. And we also investigate the impacts of Gaussian nonlocal response and fractional-order derivatives on the solitary waves, which enable us to perform control experiments for the development of rogue waves in quantum mechanics and optics.     

Breathers in a damped-driven nonlinear Schrödinger equation

We construct time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation as solutions of a boundary value problem for the space-dependent Fourier coefficients.     

Characteristics of rogue waves on a soliton background in the general three-component nonlinear Schrödinger equation

Under investigation in this work is the general three-component nonlinear Schrödinger equation, which is an important integrable system. The new localized wave solutions of the equation are derived using a Darboux-dressing transformation with an asymptotic expansion. These localized waves display rogue waves on a multisoliton background. Furthermore, the main characteristics of the new localized wave solutions are analyzed with some graphics. Our results indicate that more abundant and novel localized waves may exist in the multi-component coupled equations than in the uncoupled ones.     

Hirota bilinearization of coupled higher-order nonlinear Schrödinger equation

In this paper, we present the Hirota bilinearization of the coupled Sasa–Satsuma equation. The procedure employed here generates a more general solution than the one reported earlier. We also discuss the soliton solutions of the equation and show that the solutions found earlier are only special cases of the solution discussed here.     

Expanding integrable models of the nonlinear Schrödinger equation

By using a few Lie algebras and the corresponding loop algebras, we establish some isospectral problems whose compatibility conditions give rise to a few various expanding integrable models (including integrable couplings) of the well-known nonlinear Schrödinger equation. The Hamiltonian forms of two of them are generated by making use of the variational identity. Finally, we propose an efficient method for generating a nonlinear integrable coupling of the nonlinear Schrödinger equation.     

On a class of nonlinear inhomogeneous Schrödinger equation

In this paper, we study the inhomogeneous Schrödinger equation $$i\varphi_{t}=-\triangle\varphi -|x|^{b}|\varphi|^{p-1}\varphi,\quad x\in \mathbb{R}^{N}.$$ By using variational methods and a refined interpolation inequality, we establish some simple but sharp conditions on the solutions which exist globally or blow up in a finite time. An interesting result is that we obtain the existence of global solution for arbitrarily large data.     

Next hierarchy of mixed derivative nonlinear Schrödinger equation

In this paper, we derive the next hierarchy of the mixed derivative nonlinear Schrödinger (MDNLS) equation. Considering the Wadati–Konno–Ichikawa eigen value problem, the Lax Pair for the above equation is explicitly constructed. Obtained results are in agreement with the results derived through other methods in the recent past. We also briefly discuss the construction of Bäcklund transformation.     

The elliptic-in-t solutions of the nonlinear Schrödinger equation

Four various anzatzes of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 188–200, May, 1996.Translated by V. I. Serdobol'skii.     

Multiplicity of positive solutions of a nonlinear Schrödinger equation

We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: where if N3 and p(1, ) if N=1,2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well := int a^–1(0) consisting of k components and the first eigenvalues of –+b(x) on _j under Dirichlet boundary condition are positive for all . Under these conditions we show that (PM) has at least 2^k–1 positive solutions for large . More precisely we show that for any given non-empty subset , (P) has a positive solutions u(x) for large . In addition for any sequence _n we can extract a subsequence _ni along which u_ni converges strongly in H¹(R^N). Moreover the limit function u(x)=lim_iu_ni satisfies (i) For jJ the restriction u|_j of u(x) to _j is a least energy solution of –v+b(x)v=v^p in _j and v=0 on _j. (ii) u(x)=0 for .Mathematics Subject Classifications (2000):35Q55, 35J20     

Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schrödinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

Multicomponent nonlinear Schrödinger equation in the case of nonzero boundary conditions

The inverse scattering method is applied to the analysis of the multicomponent nonlinear Schrödinger operator in the class ofs×s matrix-valued functionsq(x) such that . A number of peculiarities due to the zero boundary conditions are exhibited, including: a) the existence of several chemical potentials; b) the modifications in the integrals of motion and of the Poisson brackets between the scattering data.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 34–46, 1983.     

Multiple soliton solutions for a quasilinear Schrödinger equation

Using Morse theory, truncation arguments and an abstract critical point theorem, we obtain the existence of at least three or infinitely many nontrivial solutions for the following quasilinear Schrödinger equation in a bounded smooth domain

$$\left\{ {\begin{array}{*{20}{c}} { - {\Delta _p}u - \frac{p}{{{2^{p - 1}}}}u{\Delta _p}\left( {{u^2}} \right) = f\left( {x,u} \right)\;in\;\Omega } \\ {u = 0\;on\;\partial \Omega .} \end{array}} \right.$$

(0.1)

Our main results can be viewed as a partial extension of the results of Zhang et al. in [28] and Zhou and Wu in [29] concerning the the existence of solutions to (0.1) in the case of p = 2 and a recent result of Liu and Zhao in [21] two solutions are obtained for problem 0.1.

     

Superconvergence analysis of BDF-Galerkin FEM for nonlinear Schrödinger equation

Numerical Algorithms - A nonlinear iteration scheme for nonlinear Schrödinger equation with 2-step backward differential formula (BDF) finite element method (FEM) is proposed. Energy stability...     

The nonlinear Schrödinger equation with a potential

We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.