Dispersive hydrodynamics: Preface

Biondini, Gino; El, Gennady; Hoefer, Mark; Miller, Peter D.

doi:10.1016/j.physd.2016.07.002

Cited by 45 publications

(62 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From an analytical point of view, one such question is whether there are any rigorous conditions for the global existence of solutions of the KP-Whitham system (1.4) which generalize those available for the KdV-Whitham system (namely, the result that if the ICs for the Riemann invariants r 1 , r 2 , r 3 are non-decreasing, the KdV-Whitham system admits a global solution, as a consequence of the the sorting property of the velocities V 1 , V 2 , V 3 .) 7. From a practical point of view, an opportunity of future study will be to perform careful numerical simulations of the KP-Whitham system (1.4) with a variety of ICs (especially ones that cannot be reduced to one-dimensional cases) and carry out a detailed comparison with the original PDE (i.e., the KP equation).…”

mentioning

confidence: 99%

Whitham modulation theory for the Kadomtsev– Petviashvili equation

2017

View full text Add to dashboard Cite

Abstract. The genus-1 KP-Whitham system is derived for both variants of the KadomtsevPetviashvili (KP) equation; namely the KPI and KPII equations. The basic properties of the KPWhitham system, including symmetries, exact reductions, and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

show abstract

mentioning

confidence: 99%

Whitham modulation theory for the Kadomtsev– Petviashvili equation

2017

View full text Add to dashboard Cite

show abstract

“…This separation of scales suggests the usefulness of an asymptotic WKBtype approach to the theoretical understanding of the arising dynamics. In this section, we show that the mathematical framework of dispersive hydrodynamics [73], a semi-classical theory of nonlinear dispersive waves, provides some insightful interpretation of the experimentally observed multi-scale coherent structures.…”

Section: Dispersive Focusing Dam Break Flows: Semi-classical Theorymentioning

confidence: 99%

From modulational instability to focusing dam breaks in water waves

et al. 2020

View full text Add to dashboard Cite

We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability but exhibits a specific, strongly nonlinear modulation, which propagates from the edges of the wavepacket towards the center with finite speed. Using numerical tools of nonlinear spectral analysis of experimental data we identify the observed envelope wave structures with focusing dispersive dam break flows, a peculiar type of dispersive shock waves recently described in the framework of the semi-classical limit of the 1D focusing nonlinear Schrödinger equation (1D-NLSE). Our experimental results are shown to be in a good quantitative agreement with the predictions of the semi-classical 1D-NLSE theory. This is the first observation of the persisting dispersive shock wave dynamics in a modulationally unstable water wave system.

show abstract

“…for Equation (1). Our consideration will be based upon the fundamental assumption that the initial step (3) is regularized in the long-time limit by the emergence of three distinct regions in the -upper half space-time plane so that the solution is given by two constant states = − and = + that are separated by an expanding DSW region ( Figure 2).…”

Section: Modulation Equations and The Matching Regularization Of The mentioning

confidence: 99%

“…Let be a small parameter characterizing the wave amplitude. We seek the solution of the dispersive hydrodynamic equation (1) in the form of an asymptotic expansion about the constant 0 for a nearly monochromatic wave with the dominant carrier wavenumber…”

Section: Small Amplitude Dsw Regime and The Nls Equationmentioning

confidence: 99%

Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

Congy

Hoefer

et al. 2018

Stud Appl Math

Self Cite

View full text Add to dashboard Cite

The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude-frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, that is, this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge. K E Y W O R D S asymptotic analysis, nonlinear waves, partial differential equationsThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

show abstract

Dispersive hydrodynamics: Preface

Cited by 45 publications

References 53 publications

Whitham modulation theory for the Kadomtsev– Petviashvili equation

Whitham modulation theory for the Kadomtsev– Petviashvili equation

From modulational instability to focusing dam breaks in water waves

Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

Contact Info

Product

Resources

About