Stability of normalized solitary waves for three coupled nonlinear Schrodinger equations

TL;DR

This paper proves the existence and stability of a three-parameter family of solitary waves in a symmetric three-coupled nonlinear Schrödinger system, using a novel variational approach with independent mass constraints.

Contribution

It introduces a new variational method with independent constraints to establish existence and stability of solitary waves in a coupled NLS system, improving previous results.

Findings

Existence of stable solitary waves with three independent parameters.

Development of a variational approach with separate mass constraints.

Enhanced understanding of multi-component nonlinear Schrödinger systems.

Abstract

In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schr\"odinger 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.