Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

Камчатнов, А. М.; Kuo, Yan-Min; Lin, Tai-Chia; Horng, Tzyy-Leng; Gou, Shih-Chuan; Clift, R; El, Gennady; Grimshaw, Roger

doi:10.1017/jfm.2013.556

Cited by 18 publications

(19 citation statements)

References 28 publications

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“…The detailed description of kink or CDSW emergence in Riemann problem solutions of the Gardner equation was carried out in [214] for both signs of µ. These Gardner equation results were applied to the resonant generation of internal undular bores in stratified fluid flows over topography [222] (see Sec. 8).…”

Section: Non-classical Dswsmentioning

confidence: 99%

“…This behavior is observed with repulsive barriers as well as with attractive potentials. generation of internal waves in the framework of the forced Gardner equation was studied in [222], where, due to nonconvexity of the hyperbolic flux (see Sec. 5), a rich variety of transcritical regimes were shown to be possible, including non-classical DSWs (kinks, contact DSWs) along with standard, KdV type, DSWs.…”

Section: Resonant Generation Of Dsws In Forced Systemsmentioning

confidence: 99%

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Dispersive shock waves and modulation theory

Hoefer

2016

Physica D: Nonlinear Phenomena

292

590

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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 Schr\"{o}dinger 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.Comment: review article, 68 pages, 52 figure

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Section: Non-classical Dswsmentioning

confidence: 99%

Section: Resonant Generation Of Dsws In Forced Systemsmentioning

confidence: 99%

Dispersive shock waves and modulation theory

Hoefer

2016

Physica D: Nonlinear Phenomena

292

590

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“…A complete classification of the Riemann problem solutions for the Gardner equation, a combined KdV-mKdV equation reducible to mKdV, was constructed in the recent paper [54] for both signs of µ. Application of the results for the Gardner equation to the resonant generation of internal undular bores in stratified fluid flows over topography was considered in [55].…”

Section: Overview Of This Workmentioning

confidence: 99%

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

El¹,

Hoefer²,

Shearer³

2017

SIAM Rev.

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107

150

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Abstract. We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg-de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

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“…The algebraic method developed here is different from construction of multi-soliton solutions on the background of quasi-periodic solutions developed in [20] by using commutation methods. It is also different from other analytical techniques for explicit characterization of eigenvalues related to the periodic solutions of the mKdV equation in the Whitham modulation theory [26,27] (see also [25,33] for earlier works).…”

Section: Introductionmentioning

confidence: 97%

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

Chen¹,

Pelinovsky²

2019

J Nonlinear Sci

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We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there exist three symmetric pairs of simple eigenvalues away from the imaginary axis, and we give a new representation of the second non-periodic solution to the Lax equations for the same eigenvalues. We show that Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background, which are either brought from infinity by propagating algebraic solitons or formed in a finite region of the time-space plane.

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Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

Cited by 18 publications

References 28 publications

Dispersive shock waves and modulation theory

Dispersive shock waves and modulation theory

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

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