THE NONLINEAR SCHRODINGER-EQUATION - ASYMMETRIC PERTURBATIONS, TRAVELING WAVES AND CHAOTIC STRUCTURES

It is well known that for certain parameter regimes the periodic focus ing Non-linear Schrodinger (NLS) equation exhibits a chaotic response when the system is perturbed. When even symmetry is imposed the mechan ism for chaotic behavior is due to the symmetric subspace being separa ted by homoclinic manifolds into disjoint invariant regions. For the e ven case the transition to chaotic behavior has been correlated with t he crossings of critical level sets of the constants of motion (homocl inic crossings). Using inverse spectral theory, it is shown here that in the symmetric case the homoclinic manifolds do not separate the ful l NLS phase space. Consequently the mechanism of homoclinic chaos due to homoclinic crossings is lost. Near integrable dynamics, when no sym metry constraints are imposed, are examined and an example of a tempor al irregular solution that exhibits random flipping between left and r ight traveling waves is provided.