Modulational stability of ground states of nonlinear Schrödinger equations. (English) Zbl 0583.35028

Weinstein, Michael I.

SIAM J. Math. Anal. 16, 472-491 (1985).

The author studies the stability of ground state solitary wave solutions of the initial value problem for the nonlinear Schrödinger equation (NLS) \[ 2i\phi_ t+\Delta \phi +| \phi |^{2\sigma}\phi =0,\quad t>0,\quad \phi (x,0)=R(x),\quad x\in {\mathbb{R}}^ N, \] under small perturbations in both the nonlinear interaction and initial data. R(x) is a real, positive radial solution of the time-independent equations, where \(0<\sigma <2/(N-2),\) and called the ground state. Then the NLS equation has, by its scaling properties, a \((2N+2)\)-parameter family of solutions obtained from the ground state R(x). It is shown that if \(\sigma <2/N\), this ground state family is stable for large times under the small perturbations concerned, modulo time-dependent adjustments of the \(2N+2\) free parameters. These parameters satisfy a system of \(2N+2\) nonlinear ordinary differential equations, called the modulation equations, which govern the amplitude, phase, position and speed of the dominant solitary wave part of the solution.

Reviewer: T.Ichinose

MSC:

35J10	Schrödinger operator, Schrödinger equation
35J60	Nonlinear elliptic equations
35A08	Fundamental solutions to PDEs
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

positive solution; nonlinear Schrödinger equation; perturbations; modulation equations; solitary wave