Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg–de Vries equations

This work is concerned with stability properties of periodic traveling waves solutions of the focusing Schrödinger equationiut+uxx+|u|2u=0iut+uxx+|u|2u=0 posed in RR, and the modified Korteweg–de Vries equationut+3u2ux+uxxx=0ut+3u2ux+uxxx=0 posed in RR. Our principal goal in this paper is the study of positive periodic wave solutions of the equation ϕω″+ϕω3−ωϕω=0, called dnoidal waves. A proof of the existence of a smooth curve of solutions with a fixed fundamental period L  , ω∈(2π2/L2,+∞)→ϕω∈Hper∞([0,L]), is given. It is also shown that these solutions are nonlinearly stable in the energy space Hper1([0,L]) and unstable by perturbations with period 2L in the case of the Schrödinger equation.