Stability and bifurcation of periodic travelling waves in a derivative non-linear Schrödinger equation

Abstract

We study periodic travelling wave solutions of a derivative non-linear Schrödinger equation and show the existence of infinitely many families of semi-trivial solutions (Theorem 2). Each of the families constitutes a branch of travelling waves corresponding to a non-zero integer called the winding number. A sufficient condition for the orbital stability of travelling waves on the branches with positive winding number is given in terms of the wave speed and winding number of the solution (Theorem 3). Bifurcation points are found on each semi-trivial branch of travelling wave solutions (Theorem 4), and the qualitative, and approximately quantitative, orbitshapes of the bifurcated solutions are given. The stability of the semi-trivial solutions under subharmonic perturbations is studied in Theorem 6, and subharmonic bifurcations are established in Theorem 7.