Stability theory for periodic pulse train solutions of the nonlinear Schrodinger equation

The problem of the stability of periodic and quasiperiodic trainsof soliton pulses in the nonlinear Schrödinger equationis examined using linearized perturbation theory. When the quasiperiodicsoliton pulse train is subjected to perturbations of positionor phase, there are both stable and unstable regions of theparameter space. The stability exponents of these perturbationsare determined in the asymptotic case of large separation betweenthe solitons.     

Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity

In this paper, we consider the nonlinear Schr\"{o}dinger equation with an anti-cubic nonlinearity. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system under different parameter conditions. Corresponding to different level curves defined by the Hamiltonian, we derive all exact explicit parametric representations of the bounded solutions (including periodic peakon solutions, periodic solutions, homoclinic solutions, heteroclinic solutions and compacton solutions).     

Solitons and other solutions to a new coupled nonlinear Schrodinger type equation

In this paper, the first integral method combined with Liu's theorem is applied to integrate a new coupled nonlinear Schrodinger type equation. Using this combination, more new exact traveling wave solutions are obtained for the considered equation using ideas from the theory of commutative algebra. In addition, more solutions are also obtained via the application of semi-inverse variational principle due to Ji-Huan He. The used approaches with the help of symbolic computations via Mathematica 9, may provide a straightforward effective and powerful mathematical tools for solving nonlinear partial differential equations in mathematical physics.     

Existence and orbital stability of periodic wave solutions for the nonlinear Schrodinger equation

In this paper, we study the existence and orbital stability of periodic wave solutions or the Schrödinger equation. The existence of periodic wave solution is obtained by using the phase portrait analytical technique. The stability approach is based on the theory developed by Angulo for periodic eigenvalue problems. A crucial condition of orbital stability of periodic wave solutions is proved by using qualitative theory of ordinal differential equations. The results presented in this paper improve the previous approach, because the proving approach does not dependent on complete elliptic integral of first kind and second kind.     

The existence of global solutions for the fourth-order nonlinear Schrodinger equations

In this paper, the problem of a class of multidimensional fourth-order nonlinear Schr\"{o}dinger equation including the derivatives of the unknown function in the nonlinear term is studied, and the existence of global weak solutions of nonlinear Schr\"{o}dinger equation is proved by the Galerkin method according to the different values of $\lambda$.     

Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms

We report exact bright and dark solitary wave solution of the nonlinear Schrodinger equation (NLSE) in cubic–quintic non-Kerr medium adopting phase–amplitude ansatz method. We have found the solitary wave parameters along with the constraints under which bright or dark solitons may exist in such a media. Furthermore, we have also studied the modulation instability analysis both in anomalous and normal dispersion regime. The role of fourth order dispersion, cubic–quintic nonlinear parameter and self-steeping parameter on modulation instability gain has been investigated.     

一个二维非线性Schrodinger方程的整体适定性

利用高低频技术证明了当初值属于Hs(R2),s>(16)/(17)时二维五次非线性Schrodinger方程的整体适定性.     

Some exact solutions of the variable coefficient Schrodinger equation

By constructing appropriate transformations, we find some exact solutions of variable coefficient Schrodinger equation, which include elliptic function and trigonometric solutions and so on.     

Solitary wave interaction for a higher-order nonlinear Schrodinger equation

Solitary wave interaction for a higher-order version of thenonlinear Schrödinger (NLS) equation is examined. An asymptotictransformation is used to transform a higher-order NLS equationto a higher-order member of the NLS integrable hierarchy, ifan algebraic relationship between the higher-order coefficientsis satisfied. The transformation is used to derive the higher-orderone- and two-soliton solutions; in general, the N-soliton solutioncan be derived. It is shown that the higher-order collisionis asymptotically elastic and analytical expressions are foundfor the higher-order phase and coordinate shifts. Numericalsimulations of the interaction of two higher-order solitarywaves are also performed. Two examples are considered, one satisfiesthe algebraic relationship derived from asymptotic theory, andthe other does not. For the example which satisfies the algebraicrelationship, the numerical results confirm that the collisionis elastic. The numerical and theoretical predictions for thehigher-order phase and coordinate shifts are also in strongagreement. For the example which does not satisfy the algebraicrelationship, the numerical results show that the collisionis inelastic and radiation is shed by the solitary wave collision.As the bed of radiation shed by the waves decays very slowly(like t^–), it is computationally infeasible to calculatethe final phase and coordinate shifts for the inelastic example.An asymptotic conservation law is derived and used to test thefinite-difference scheme for the numerical solutions.     

Maximal estimates for solutions to the nonelliptic Schrodinger equation

Maximal estimates are studied for solutions to an initial valueproblem for the nonelliptic Schrödinger equation. A resultof Rogers, Vargas and Vega is extended.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

Infinitely many solutions for a class of sublinear Schrodinger equation

In this paper, we investigate the Schr\"{o}dinger equation, which satisfies that the potential is asymptotical 0 at infinity in some measure-theoretic and the nonlinearity is sublinear growth. By using variant symmetric mountain lemma, we obtain infinitely many solutions for the problem. Moreover, if the nonlinearity is locally sublinear defined for $|u|$ small, we can also get the same result. In which, we show that these solutions tend to zero in $L^{\infty}(\mathbb{R}^{N})$ by the Br\"{e}zis-Kato estimate.     

一类非线性延迟微分方程数值解的振动性分析

本文考虑一类非线性延迟微分方程-带有单调造血率的造血模型数值解的振动性.通过研究特征方程根的情况得到数值解振动的条件并且讨论了非振动的数值解的一些性质.为了更有力的说明我们的结果,最后给出了相应的算例.     

A geometric characterization of the nonlinear Schrodinger equation and its applications

We prove that the nonlinear Schrodinger equation of attractive type (NLS+ describes just spher-ical surfaces (SS) and the nonlinear Schrodinger equation of repulsive type (NLS-) determines only pseudo-spherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equationsthere exists a local gauge transformation (despite of changing the independent variables or not) which trans-forms a solution of one into any solution of the other, it is impossible to have such a gauge transformationbetween the NLS+ and the NLS-.     

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.     

The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation

Abstract In [3] Dias and Figueira have reported that the square of the solution for the nonlinear Dirac equation satisfies the linear wave equation in one space dimension. So the aim of this paper is to proceed with their work and to clarify a structure of the nonlinear Dirac equation. The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation are obtained. Keywords: Nonlinear Dirac equation, Dirac-Klein-Gordon equation, Pauli matrix Mathematics Subject Classification (2000): 35C05, 35L45     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

On kink and anti-kink wave solutions of Schrodinger equation with distributed delay

This paper deals with existence problem of traveling wave solutions of a class of nonlinear Schr\"{o}dinger equation having distributed delay with a strong generic kernal. By using the geometric singular perturbation theory and the Melnikov function method, we establish results of the existence of kink and anti-kink wave solutions of the nonlinear Schr\"{o}dinger equation with time delay when the average delay is sufficiently small.