The main objective of this proposal is to enable a two-week collaborative work of Dr Gennady El (Loughborough University, UK) and Dr Mark Hoefer (University of Colorado, Boulder, USA) on the research monograph « Dispersive Shock Waves » for the Cambridge University Press (CUP) series « Cambridge Monographs in Mechanics ».

The aim of this research monograph is to provide a comprehensive description of the mathematical theory of dispersive shock waves (DSWs) and the physical applications of this theory. A DSW is an expanding, modulated nonlinear wavetrain connecting two disparate states, the dispersive analogue of a dissipative, classical shock. Generally, the generation of DSWs represents a universal mechanism to resolve unphysical hydrodynamic singularities in dispersive media. Their fundamental role in such media is similar to that of viscous shock waves in classical gas and fluid dynamics. Physical manifestations of DSWs range from undular bores on shallow water to nonlinear diffraction patterns in laser and atom (matter wave) optics. The mathematical description of DSWs involves a synthesis of methods from hyperbolic quasilinear systems, asymptotics, and soliton theory. The principal tool is nonlinear wave modulation theory, often referred to as Whitham averaging.

The time is ripe for this monograph. While the mathematical principles underlying the DSW theory were developed in the sixties and seventies, no comprehensive collection of the developments since exists. The authors’ fundamental contributions to DSW theory provide them with a unique perspective that will permeate the text. From clear descriptions of the present state of knowledge to identified directions of future research, the authors aim to make this unified work both a reference and a stimulant for further theoretical and experimental investigations.
As the first comprehensive text on the mathematics and physics of DSWs the monograph will have a major impact on a broad international community of researchers involved in nonlinear wave studies.