Abstract

Dispersive shock waves (DSW’s) are described by a slowly varying one-phase periodic wave train that connects solitons at one edge to linear dispersive waves at the other. DSW ’s are associated with equations which are ‘regularized’ by a small dispersive term. This is different from viscous shock waves (VSW’s) which are characterized by rapid localized changes in the underlying physical quantities. Equations with VSW ’s are regularized by a small viscous term. Recent laboratory experiments in Bose-Einstein condensates and nonlinear optics have demonstrated both single phase DSW and complex-interacting multiphase behavior. Numerical and analytical results explain many of the observed phenomena. If time permits, a reformulation of the water wave equations in terms of a novel nonlocal system will be discussed. Special cases include shallow and deep water asymptotic reductions. Localized waves including two dimensional water wave lumps to the nonlinear water wave equations with sufficient surface tension are numerically obtained.