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Stability of traveling-wave solutions for a Schrödinger system with power-type nonlinearities. (English) Zbl 1303.35102
Summary: In this article, we consider the Schrodinger system with power-type nonlinearities, \[ i\frac{\partial}{\partial t}u_j+ \Delta u_j + a|u_j|^{2p-2} u_j + \sum_{k=1, k\neq j}^m b |u_k|^{p}|u_j|^{p-2} u_j=0; \quad x\in \mathbb{R}^N, \] where \(j=1,\dots,m, u_j\) are complex-valued functions of \((x,t)\in \mathbb{R}^{N+1}\), a,b are real numbers. It is shown that when \(b>0\), and \(a+(m-1)b>0\), for a certain range of p, traveling-wave solutions of this system exist, and are orbitally stable.

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
35B35 Stability in context of PDEs
35Q35 PDEs in connection with fluid mechanics
35C08 Soliton solutions
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