The nonlinear Schrödinger equation: Asymmetric perturbations, traveling waves and chaotic structures

It is well known that for certain parameter regimes the periodic focusing Non-linear Schrödinger (NLS) equation exhibits a chaotic response when the system is perturbed. When even symmetry is imposed the mechanism for chaotic behavior is due to the symmetric subspace being separated by homoclinic manifolds into disjoint invariant regions. For the even case the transition to chaotic behavior has been correlated with the crossings of critical level sets of the constants of motion (homoclinic crossings). Using inverse spectral theory, it is shown here that in the symmetric case the homoclinic manifolds do not separate the full NLS phase space. Consequently the mechanism of homoclinic chaos due to homoclinic crossings is lost. Near integrable dynamics, when no symmetry constraints are imposed, are examined and an example of a temporal irregular solution that exhibits random flipping between left and right traveling waves is provided.