Nonlinear stability of periodic traveling wave solutions to the Schrodinger and the modified Korteweg-de Vries equations

This work is concerned with stability properties of periodic traveling waves solutions of the focusing Schrodinger equation iu(t) + u(xx) + vertical bar u vertical bar(2)u = 0 posed in R, and the modified Korteweg-de Vries equation u(t) + 2u(2)u(x) + u(xxx) = 0 posed in R. Our principal goal in this paper is the study of positive periodic wave solutions of the equation phi(omega)'' + phi(3)(omega) - omega phi omega = 0, called dnoidal waves. A proof of the existence of a smooth curve of solutions with a fixed fundamental period L, omega is an element of (2 pi(2)/L-2, + infinity) -> phi omega is an element of H-per(infinity) ([0, L]), is given. It is also shown that these solutions are nonlinearly stable in the energy space H-per(1) ([0, L]) and unstable by perturbations with p period 2L in the case of the Schrodinger equation. (c) 2007 Elsevier Inc. All rights reserved.