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

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
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