On soliton dynamics in nonlinear schrödinger equations

In this paper we describe our results on dynamics of solitons in generalized nonlinear Schr?dinger equations with external potentials (potNLS’s) in all dimensions except for 2. We also outline some of the ideas of the proofs involved. The detailed discussion of the results as well as the proofs are presented in [GS2]. For a certain class of nonlinearities a potNLS has solutions which are periodic in time and exponentially decaying in space and which are centered near different critical points of the potential. We call those solutions which are centered near the minima and which minimize energy restricted to the 2-unit sphere, trapped solitons or just solitons. Our results show that, under certain conditions on the potentials and initial conditions, trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position plus a small fluctuation. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton’s center and momentum) is close to Newton’s equation but with a dissipative term due to radiation of the energy to infinity. This paper is part of the first author’s PhD thesis. Both authors are supported by NSERC under Grant NA7901. Received: January 2006; Revision: April 2006; Accepted: April 2006     

Stability and convergence of difference scheme for nonlinear evolutionary type equations

A finite difference scheme is derived for the initial-boundary problem for the nonlinear equation system $$\frac{\partial u}{\partial t}=A\frac{\partial^{2}u}{\partial x^{2}}+f(u),$$ where A is a complex diagonal matrix, f is a complex vector function. The stability and convergence in discrete L ^∞-norm of proposed Crank-Nicolson type finite difference schemes is proved. No restrictions on the ratio of time and space grid steps are assumed. Some numerical experiments have been conducted in order to validate the theoretical results.     

Stability and convergence of Dufort-Frankel-type difference schemes for a nonlinear Schrödinger-type equation

We consider a first boundary problem for the nonlinear Schrödinger equation $$\frac{{\partial u}}{{\partial t}} = ia\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u,u*)u.$$ The convergence of a three-layer explicit difference scheme in theC andW ₁ ² norms is proved. The stability of the scheme with respect to the initial data in the same norms is proved. To justify the convergence and stability we use grid analogues of the energy-preservation laws and grid multiplicative inequalities. The relation 2|a|τ/h ²≤ν<1 is assumed for the grid stepsizes.     

Measures not charging semipolars and equations of schrödinger type

SupposeX is a Borel right process andm is a -finite excessive measure forX. Given a positive measure not chargingm-semipolars we associate an exact multiplicative functionalM(). No finiteness assumptions are made on . Given two such measures and ,M()=M() if and only if and agree on all finely open measurable sets. The equation (q–L)u+u=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(). Both uniqueness and existence are considered.Supported in part by NSF Grant DMS 92-24990.     

Explicit second-order accurate schemes for the nonlinear Schrödinger equations

We consider three-level explicit schemes for solving the nonlinear variable coefficient Schrödinger-type equation. Using spectral and energy methods we establish the stability and convergence of these schemes. The existence of discrete conservation laws is investigated. General results are applied for the DuFort-Frankel and leap-frog diffenrence schemes.     

On the compact difference scheme for the two-dimensional coupled nonlinear Schrödinger equations

A high-order finite difference method for the two-dimensional coupled nonlinear Schrödinger equations is considered. The proposed scheme is proved to preserve the total mass and energy in a discrete sense and the solvability of the scheme is shown by using a fixed point theorem. By converting the scheme in the point-wise form into a matrix–vector form, we use the standard energy method to establish the optimal error estimate of the proposed scheme in the discrete L²-norm. The convergence order is proved to be of a fourth-order in space and a second-order in time, respectively. Finally, some numerical examples are given in order to confirm our theoretical results for the numerical method. The numerical results are compared with exact solutions and other existing method. The comparison between our numerical results and those of Sun and Wangreveals that our method improves the accuracy of space and time directions.     

On a class of nonlinear Schrödinger equations

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:

On the Hyers–Ulam stability of certain nonautonomous and nonlinear difference equations

Aequationes mathematicae - This article is devoted to the study of certain nonautonomous and nonlinear difference equations of higher order. Our main objective is to formulate sufficient conditions...     

Global existence of small solutions to quadratic nonlinear schrödinger equations

Abstract. We study the global existence of small solutions to some quadratic nonlinear Schrödinger equations in three or four space dimensions.     

On energy stability for the coupled nonlinear Schrödinger system

In this short note, we prove that smooth solutions to the coupled nonlinear Schr?dinger system with coercive polynomial nonlinearities are unique among distributional solutions enjoying the energy inequality. The argument also yields the stability of classical solutions in the energy norm. The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002