On the Whitham equations for the semiclassical limit of the defocusing nonlinear Schr�dinger equation

Tian, Fei-Ran; Ye, Jian

doi:10.1002/(sici)1097-0312(199906)52:6<655::aid-cpa1>3.0.co;2-a

Cited by 22 publications

(5 citation statements)

References 18 publications

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“…Thus, the auxiliary functions needed to solve the hodograph transform are the same for the defocusing NLS equation (δ = 0) and for the Hirota equation and the modifications of both derivative-type NLS equations (δ = 0). A similar fortuitous cancellation occurs for modulation equations within the KdV hierarchy [33,36]. Since the Euler-Poisson-Darboux equation for the defocusing NLS equation, the Hirota equation and the modified derivative-type NLS equations are the same, the Riemann-Green functions needed to construct the auxiliary functions are identical in all four cases.…”

Section: Discussionmentioning

confidence: 74%

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On the exact solution for smooth pulses of the defocusing nonlinear Schrödinger modulation equations prior to breaking

2009

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Section: Discussionmentioning

confidence: 74%

“…An alternative exact representation for the evolution of monotone data is presented in [33], subject to some technical restrictions on the monotone data, both pre-and post-breaking. Once again, our analysis applies to the more general case of smooth pulses prior to first breaking.…”

Section: Introductionmentioning

confidence: 99%

On the exact solution for smooth pulses of the defocusing nonlinear Schrödinger modulation equations prior to breaking

2009

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“…[32]). To the best of our knowledge, the reconstruction problem with fairly generic initial data was first solved for the Airy system in [31]. An analogous study for the deformed system with results of comparable generality would certainly be valuable but will be left to future work.…”

Section: Some Hodograph Solutionsmentioning

confidence: 99%

“…As well known, such a system displays a lot of 'good' properties. For instance, it is one of the few quasi-linear systems in N > 1 fields in which the Riemann procedure can be effectively carried out (see [31]) and Whitham equations can be quite explicitly solved. By means of the bi-Hamiltonian procedure, an explicit generating function for a set of polynomial constants of the motion can be provided.…”

Section: Introductionmentioning

confidence: 99%

Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach

2016

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The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite 2D channel. The Hamiltonian structure of the averaged equations is obtained directly from that of the Euler equations through the process of Hamiltonian reduction. Long-wave asymptotics together with the Boussinesq approximation of neglecting the fluids' inertia is then applied to reduce the leading order vertically averaged equations to the shallow-water Airy system, and thence, in a non-trivial way, to the dispersionless non-linear Schrödinger equation. The full non-Boussinesq system for the dispersionless limit can then be viewed as a deformation of this well known equation. In a perturbative study of this deformation, it is shown that at first order the deformed system possesses an infinite sequence of constants of the motion, thus casting this system within the framework of completely integrable equations. The Riemann invariants of the deformed model are then constructed, and some local solutions found by hodograph-like formulae for completely integrable systems are obtained. IntroductionAspects of the theory of two-layer stratified flows in an infinite 2D channel have been the subject of intense recent studies. Layer models are widely used in a variety of geophysical applications (going back to early references such as that by Long [20] in the framework of meteorology), and are of conceptual value for illustrating many fundamental properties of stratified fluid dynamics. A typical configuration is depicted in Figure 1, with an interface between the two fluid representing the sharp pycnocline between superficial ("fresh") water, labeled by the index 1, and deep ("salty") water, labeled by the 2-index. (Other relevant notation used throughout the paper is defined by the figure). Long internal waves in such systems were studied in, e.g., [10,11], by deriving the two-layer models (including dispersive terms) by the layer-averaging method (see e.g., [30]). Their dispersionless counterparts were more recently reconsidered in papers by Milewski, Tabak and collaborators [25,12]. These papers were mainly interested in studying the Kelvin-Helmholtz (KH) instability (see also [4]) viewed as hyperbolic vs. elliptic transition for the resulting quasi-linear equations of motion, and its relation to the well-posedness of the initial value problem for these equations. In particular, for the so-called Boussinesq approximation, which in this context consists of disregarding density differences in the inertial terms while retaining them in the buoyancy terms, the following conditions for shear-flow stability are all equivalent:i) The "standard" stability criterion expressed by the Richardson number for the linearized two-layer equations ([23, 3]) around a velocity jump.ii) The hyperbolicity (nonlinear) criterion for the reduced quasi-linear system of PDEs in the variables ξ, w (the difference between ...

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“…Nonlinear phenomena play a crucial role in various scientific fields such as fluid mechanics, optical fibers, solid state physics, chemical dynamics and geochemistry, and are modeled using nonlinear partial differential equations. Typical nonlinear dispersion equations include the Korteweg-de Vries (KdV) equation [1,2], the nonlinear Schrödinger (NLS) equation [3,4], the sine-Gordon equation [5,6], the Camassa-Holm equation [7,8], etc. Their precise solutions are of great significance for understanding the mechanisms and dynamic behaviors of complex physical phenomena.…”

Section: Introductionmentioning

confidence: 99%

The Whitham Modulation Solution of the Complex Modified KdV Equation

Zeng

Liu

2023

Mathematics

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This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.

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On the Whitham equations for the semiclassical limit of the defocusing nonlinear Schr�dinger equation

Cited by 22 publications

References 18 publications

On the exact solution for smooth pulses of the defocusing nonlinear Schrödinger modulation equations prior to breaking

On the exact solution for smooth pulses of the defocusing nonlinear Schrödinger modulation equations prior to breaking

Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach

The Whitham Modulation Solution of the Complex Modified KdV Equation

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