Dispersive shock waves and modulation theory

2016/02/29 by G. A. Él, G. A. El, Mark A. Hoefer +1 · 10 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Classical mechanics #Context (archaeology) #Excitation #Field (mathematics) #Geology #Integrable system #Mathematical physics #Mathematics #Mechanics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum mechanics #Shock (circulatory) #Shock wave #Statistical physics #Theoretical physics #nlin.PS #nlin.SI #physics.flu-dyn

paper · pdf · doi:10.1016/j.physd.2016.04.006

published as Physica D 333, 11-65 (2016) · review article, 68 pages, 52 figures

arxiv created 2016/04/12 · openalex publication_date 2016/04/30 · arxiv updated 2016/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

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Cited by

• Sharp Upper Bound for Amplitudes of Finite-Gap Solutions of the Modified Korteweg-de Vries Equation
• Traveling and Dispersive Shock Waves in a Two-Dimensional Fermi-Pasta-Ulam-Tsingou Lattice
• On the quasi-continuum approximation of some localized patterns in the FPUT lattice
• Modulation theory for lumps and interactions between lumps and a mean field in the Kadomtsev-Petviashvili equation
• Exact Charge, Current, and Velocity Fields of Interacting Korteweg-de Vries Solitons
• On the Riemann problem for the Adlam-Allen model
• A study of variational single solitary waves governed by the conservative-extended KdV equation with applications to shallow water dispersive shocks
• Nonlinear dispersive waves in the discrete modified KdV equation
• From Convex to Non-convex: Evolution of Rarefaction-dispersive shock interactions and the Influence of Non-convexity
• The co-directional overtaking collision problem between a dispersive shock wave and a rarefaction wave for the Hirota equation

Related

• Shock waves in dispersive hydrodynamics with non-convex dispersion
• On shallow water non-convex dispersive hydrodynamics: the extended KdV model
• Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure
• Solitonic dispersive hydrodynamics: theory and observation
• Dispersive, superfluid-like shock waves in nonlinear optics