Nonlinear and spectral stability of periodic traveling wave solutions for a nonlinear Schrödinger system. (English) Zbl 1228.76031

Pastor, Ademir

Differ. Integral Equ. 23, No. 1-2, 125-154 (2010).

Summary: This paper is concerned with nonlinear and spectral stability of periodic traveling wave solutions for a nonlinear Schrödinger type system arising in nonlinear optics. We prove the existence of two smooth curves of periodic solutions depending on conoidal-type functions. In the framework established by M. Grillakis, J. Shatah and W. Strauss, we prove a stability result under perturbations having the same minimal wavelength and zero mean over their fundamental period. By using the so-called Bloch wave decomposition theory, we show spectral stability for a general class of periodic solutions.

MSC:

76B25	Solitary waves for incompressible inviscid fluids
35C07	Traveling wave solutions
35Q55	NLS equations (nonlinear Schrödinger equations)

Keywords:

traveling wave solution; nonlinear Schrödinger type system; periodic solution