Stability of normalized solitary waves for three coupled nonlinear Schrodinger equations

Abstract

In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schrödinger system \[ i\partial_t u_{j}+\partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj} |u_k|^{p}\right)|u_j|^{p-2}u_j = 0, \ j=1,2,3, \] where $u_j$ are complex-valued functions of $(x,t)\in {\mathbb{R}}^2$ and $a_{kj}$ are positive constants satisfying $a_{kj}=a_{jk}$ (symmetric attractive case). Our approach improves many of the previous known results. In all methods used previously to study solitary waves, which we are aware of, the variational problem has consisted of finding the extremum of an energy functional subject to the constraints that were not independently chosen. Here we study a problem of minimizing the energy functional subject to three independent $L^2$ mass constraints and establish existence and stability results for a true three-parameter family of solitary waves.

28 pages. Analogous results on $L^2$ normalized solutions for 2-coupled nonlinear Schrödinger system are proved in our earlier work arXiv:1406.2418

References in corpus (1)

• Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities

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