Mel'nikov analysis of a symmetry-breaking perturbation of the NLS equation

Annalisa Calini

College of Charleston, Mathematics, Room 203, Maybank Hall,
165 Calhoun St Charleston, SC 29424-0001, USA

Constance M. Schober

Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, USA

Abstract: The effects of loss of symmetry due to noneven initial conditions on the chaotic dynamics of a Hamiltonian perturbation of the Nonlinear Schrödinger (NLS) equation were first numerically studied by Ablowitz, Herbst and Schober, where it was observed that temporally irregular evolution can occur even in the absence of homoclinic crossings. In this article we introduce a symmetry-breaking damped-driven perturbation of the NLS equation in order to develop a Mel'nikov analysis of the noneven chaotic regime. We obtain the following results. 1) Spatial symmetry breaking within the chaotic regime causes the wave form to exhibit a more complex dynamics than in the previous studies: center-wing jumping (which characterizes the even chaotic dynamics) about a shifted lattice site alternates (at random times) with the occurrence of modulated travelling wave solutions whose velocity changes sign in a temporally random fashion. 2) We give a heuristic description of the geometry of the full phase space (with no evennessimposed) and compute Mel'nikov-type measurements in terms of the complex gradient of the Floquet discriminant. The Mel'nikov analysis yields explicit conditions for the onset of chaotic dynamics which are consistent with the numerical observations; in particular, the imaginary part of the Mel'nikov integral appears to be correlated withpurely noneven features of the chaotic wave form.