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Title: Dispersive shock waves and modulation theory
Authors: El, G.A.
Hoefer, M.A.
Keywords: Whitham theory
Korteweg-de Vries equation
Nonlinear Schrödinger equation
Issue Date: 2016
Publisher: © Elsevier
Citation: EL, G.A. and HOEFER, M.A., 2016. Dispersive shock waves and modulation theory. Physica D: Nonlinear Phenomena, 333, pp. 11-65.
Abstract: 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.
Description: This paper was accepted for publication in the journal Physica D: Nonlinear Phenomena and the definitive published version is available at http://dx.doi.org/10.1016/j.physd.2016.04.006
Sponsor: This work was supported by the Royal Society International Exchanges Scheme IE131353 (both authors) and NSF CAREER DMS-1255422 (MAH).
Version: Accepted for publication
DOI: 10.1016/j.physd.2016.04.006
URI: https://dspace.lboro.ac.uk/2134/21161
Publisher Link: http://dx.doi.org/10.1016/j.physd.2016.04.006
ISSN: 1872-8022
Appears in Collections:Published Articles (Maths)

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