Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

Santosh  Bhattarai

doi:10.3934/dcds.2016.36.1789

Article Contents

2016, Volume 36, Issue 4: 1789-1811. Doi: 10.3934/dcds.2016.36.1789

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Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

Santosh Bhattarai^1,

1.
Trocaire College, Mathematics Department, 360 Choate Ave, Buffalo, NY 14220

Received: June 30, 2014

Revised: June 30, 2015

Published: May 2017

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Abstract

In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schrödinger system \begin{equation*} 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, \end{equation*} 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 previously known results. In all variational methods used previously to study the stability of solitary waves, which we are aware of, the constraint functionals 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.

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Mathematics Subject Classification: 35Q55, 35B35, 35A15.

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