Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations

Pages: 1789 - 1811, Volume 36, Issue 4, April 2016      doi:10.3934/dcds.2016.36.1789

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Santosh Bhattarai - Trocaire College, Mathematics Department, 360 Choate Ave, Buffalo, NY 14220, United States (email)

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.

Keywords:  Nonlinear Schrödinger system, solitary waves, $L^2$ normalized solutions, ground states, positive solutions, stability.
Mathematics Subject Classification:  35Q55, 35B35, 35A15.

Received: July 2014;      Revised: July 2015;      Available Online: September 2015.