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

The effects of loss of symmetry due to noneven initial conditions on the chaotic dynamics of a Hamiltonian perturbation of the nonlinear Schrödinger equation (NLS) were first numerically studied in [Phys. A 228 (1996)], where it was observed that temporally irregular evolutions 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 [Phys. A 228 (1996)]: center-wing jumping (which characterizes the even chaotic dynamics) about a shifted lattice site alternates (at random times) with the occurrence of modulated traveling 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 evenness imposed) 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 with purely noneven features of the chaotic wave form.