DSW_review.pdf (11.31 MB)
Dispersive shock waves and modulation theory
journal contribution
posted on 2016-05-10, 12:41 authored by Gennady El, M.A. HoeferThere 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.
Funding
This work was supported by the Royal Society International Exchanges Scheme IE131353 (both authors) and NSF CAREER DMS-1255422 (MAH).
History
School
- Science
Department
- Mathematical Sciences
Published in
Physica D: Nonlinear PhenomenaCitation
EL, G.A. and HOEFER, M.A., 2016. Dispersive shock waves and modulation theory. Physica D: Nonlinear Phenomena, 333, pp. 11-65.Publisher
© ElsevierVersion
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Acceptance date
2016-04-05Publication date
2016-04-20Notes
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.2016.04.006ISSN
1872-8022Publisher version
Language
- en