Singular Limits of Dispersive Waves

The subject, of "Singular Limits of Dispersive vVaves" had its modern origins in the 1960's when Whitham introduced the first systematic approach to the asymptotic analysis of nonlinear wavepackds. Initially developed through a variational principle applied to the modulation of families of traveling wave solutions, he soon realized that an efficient derivation of modulation eq'uations could b(' accomplished by av eraging local conservation laws. He carried out this analysis for a wide variety of dispersive nonlinear wave equations including the nonlinear Klein Gordon, KdV, and NLS equations. The seminal work of Gardner, Greene, Kruskal and Miura led to the discovery of partial differential equations which are completely integrable through inverse spectral transforms. This provided a larger framework in which to develop modulation theory. In particular, one could consider the local modulation of families of quasiperiodic so lutions with an arbitrary number ofphases. extending the sillglf' phase traveling waves treated Ly Vhitham. The first to extend vVhitham's ideas to the mllltiphase setting were Flaschka, Forest and lvIcLaughlin, who derived N-phase modulation equations for the KdV equation. By using geometric techniques from the theory of Riemann surfaces they presented these equations in Riemann invariant form and demonstrated their hyperbolicity.

Zusammenfassung
Proceedings of a NATO ARW and of a Chaos, Order, and Patterns Panel sponsored workshop held in Lyons, France, July 8-12, 1991

Inhalt
Whitham Modulation Equations and Their Exact Solutions.- The Whitham Equation and Shocks in the Toda Lattice.- Semiclassical Behavior in the NLS Equation: Optical Shocks-Focusing Instabilities.- A Numerical Study of Nearly Integrable Modulation Equations.- The Quasiclassical Limit of the Inverse Scattering Method.- Solving Dispersionless Lax Equations.- Nonisospectral Symmetries of the KdV Equations and the Corresponding Symmetries of the Whitham Equations.- Breaking Problem in Dispersive Hydrodynamics.- Whitham Deformations of Two-Dimensional Lioville Tori.- Higher Rank Darboux Transformations.- On the Initial Value Problem of the Whitham Averaged System.- On the Integrability of the Averaged KdV and Benney Equations.- Explicit Construction of the Lax-Levermore Minimizer for the KdV Zero Dispersion Limit.- Dispersionless Limit of Integrable Systems in 2+1 Dimensions.- Asymptotics and Limits.- KdV Equation with Nontrivial Boundary Conditions at x ? ±?.- Long-time Asymptotics for the Autocorrelation Function of the Transverse Ising Chain at the Critical Magnetic Field.- Resonances in Multifrequency Averaging Theory.- Billiards Systems and the Transport Equation.- The Behavior of Solutions of the NLS Equation in the Semiclassical Limit.- Critical and Subcritical Cases of the Toda Shock Problem.- Existence and Regularity of Dispersive Waves.- Geometric Phases and Monodromy at Singularities.- Nonlinear Waves and the 1:1:2 Resonance.- Defects of One-Dimensional Vortex Lattices.- Oscillations Arising in Numerical Experiments.- On the Hierarchy of the Generalized KdV Equations.- A New Theory of Shock Dynamics.