STRONGLY NONLINEAR MODAL EQUATIONS FOR NEARLY INTEGRABLE PDES

The purpose of this paper is the derivation of reduced, finite-dimensi onal dynamical systems that govern the near-integrable modulations of N-phase, spatially periodic, integrable wavetrains. The small paramete r in this perturbation theory is the size of the nonintegrable perturb ation in the equation. rather than the amplitude of the solution, whic h is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivat ion we present relies heavily on the integrability of the underlying P DE and applies, in general, to any N-phase periodic wavetrain. For spe cific applications, however, a numerical pretest is applied to fix the truncation order N. We present one example of the reduction philosoph y with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numer ical study.