**
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.