Whiskered Tori for Integrable Pde’s: Chaotic Behavior in Near Integrable Pde’s

Abstract

This article is a summary of our numerical and theoretical studies (which were done in various collaborations with Alan Bishop, Nick Ercolani, Greg Forest, and Steve Wiggins) of near integrable nonlinear wave equations under periodic boundary conditions. Two examples, a damped driven sine-Gordon equation and a perturbed nonlinear Schrodinger equation, are discussed in detail. The article begins with a thorough description of numerical experiments on the two systems in a parameter regime for which the response is spatially coherent, yet temporally chaotic. In addition to the description of this qualitative behavior in the pde’s, numerical and statistical issues are emphasized. Next, the spectral transform for the integrable nonlinear Schrodinger equation is developed in sufficient detail for use in both theoretical and numerical analysis of the perturbed system. This integrable theory includes the introduction of a Morse function which unveils a hyperbolic or saddle structure in the constants of the motion, the association of this saddle structure with complex double periodic eigenvalues for the spectral transform, and the use of Backlund transformations to produce from these complex double points analytical representations of homoclinic orbits and whiskered tori. Next, the spectral transform is used as a numerical diagnostic to monitor the chaotic attractors in the perturbed system. Finally, a Melnikov analysis of a perturbed model system is described. This geometric perturbation theory is based upon the analytical representations of whiskered tori in the nearby integrable system. Open problems are discussed throughout the text and summarized in the conclusion.