Constance SCHOBER

Dept. of Mathematics and Statistics
Old Dominion University 
Norfolk Virginia 23529-0077.
USA
schober@math.odu.edu

Chaotic Dynamics for Symmetry-Breaking Perturbations of Integrable Equations

Abstract : It is well known that for certain parameter regimes the periodic focusing NonLinear Schr\"{o}dinger equation (NLS) exhibits a chaotic response when the system is perturbed. When even symmetry is imposed, the dynamics is characterized by irregular center-wing flipping of the wave form and one can prove persistence of transversal intersections of the invariant manifolds as the main source of homoclinic chaos. In this talk we will examine evenness breaking perturbations of the NLS. Novel types of chaotic behavior are observed for the non-even case, including solutions which exhibit random flipping between a left and right modulated traveling wave. This behavior is related to an extra circle symmetry in the problem and to random variations of the sign of the linear momentum. A Mel'nikov analysis interpretation of this new phenomena is provided.