Thursday, May 22

3:00 PM-5:00 PM Ballroom II - Level B

Melnikov Methods for Partial Differential Equations

In the past two years, several important breakthroughs have been made in applying the Method of Melnikov to demonstrating the existence of transverse homoclinic orbits and chaotic dynamics in near-integrable partial differential equations. This minisymposium will present some of them. The presented work deals with conservative and dissipative perturbations of the nonlinear Schrödinger equation, and the Maxwell-Bloch equations that describe the dynamics of light in laser cavities. The techniques used in investigating these problems are the inverse spectral theory for integrable partial differential equations with periodic boundary conditions, Backlund transformations and the dressing method, invariant manifold theory in infinite-dimensional settings, k normal forms, geometric singular perturbation theory and the theory of orbits homoclinic to resonance bands, the energy-phase method for detecting multi-pulse homoclinic orbits, and numerical nonlinear spectral analysis. Careful comparisons with numerical simulations will also be presented.

Organizers: Constance M. Schober, Old Dominion University; and Gregor Kovacic, Rensselaer Polytechnic Institute

3:00 The Nonlinear Schrödinger Equation: Asymmetric Perturbations, Traveling Waves and Chaotic Structures
Constance M. Schober, Organizer
3:30 Homoclinic Orbits in the Maxwell-Bloch Equations of Nonlinear Laser Optics
Gregor Kovacic, Organizer; Jiyue J. Li and Victor Roytburd, Rensselaer Polytechnic Institute; and Thomas A. Wettergren, Naval Undersea Warfare Center
4:00 The Fate of Traveling Waves of the Nonlinear Schrödinger Equation under Ginzburg-Landau Type Perturbations
C. David Levermore, Gustavo Cruz-Pacheco, and Benjamin P. Luce, Los Alamos National Laboratory
4:30 Smale Horseshoes in Perturbed Nonlinear Schrödinger Equations
Charles Y. Li, Massachusetts Institute of Technology

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TMP, 4/4/97