Dispersive shock waves and modulation theory

• Published in: Physica. D 2016-10, Vol.333, p.11-65
• Main Authors: El, G.A., Hoefer, M.A.
• Format: Article
• Language: English
• Subjects: Korteweg–de Vries equation; Nonlinear Schrödinger equation; Whitham theory
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2016.04.006

There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G.B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs. •Dispersive shock waves (DSWs) are fundamental, coherent nonlinear wave structures.•Both integrable and non-integrable systems exhibit DSWs.•Nonlinear modulation theory provides a mathematical framework for DSW description.•DSWs are ubiquitous in physical applications.•Summarizes the developments over the past 50 years on DSW modulation theory.