Weak and Strong Interactions between Internal Solitary Waves

Gear, John Anthony; Grimshaw, Roger

doi:10.1002/sapm1984703235

Cited by 215 publications

(200 citation statements)

References 20 publications

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“…The system (6) and (7) for λ ¼ −1 describes a two-layer liquid model studied in references [17][18][19].…”

Section: A Parametric Coupled Kdv Systemmentioning

confidence: 99%

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Singular Lagrangians and Its Corresponding Hamiltonian Structures

Restuccia¹,

Sotomayor²

2017

Lagrangian Mechanics

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We present a general procedure to obtain the Lagrangian and associated Hamiltonian structure for integrable systems of the Helmholtz type. We present the analysis for coupled Korteweg-de Vries systems that are extensions of the Korteweg-de Vries equation. Starting with the system of partial differential equations it is possible to follow the Helmholtz approach to construct one or more Lagrangians whose stationary points coincide with the original system. All the Lagrangians are singular. Following the Dirac approach, we obtain all the constraints of the formulation and construct the Poisson bracket on the physical phase space via the Dirac bracket. We show compatibility of some of these Poisson structures. We obtain the Gardner ε-deformation of these systems and construct a master Lagrangian which describe the coupled systems in the weak ε-limit and its modified version in the strong ε-limit.

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“…The system (6) and (7) for λ ¼ −1 describes a two-layer liquid model studied in references [17][18][19].…”

Section: A Parametric Coupled Kdv Systemmentioning

confidence: 99%

“…Besides the physical applications of coupled KdV systems at low energies [17][18][19], one of the Poisson structures of the KdV equation is related to the Virasoro algebra with central terms. The latest is a fundamental symmetry of string theory, a proposal for a consistent quantum gravity theory.…”

Section: Introductionmentioning

confidence: 99%

Singular Lagrangians and Its Corresponding Hamiltonian Structures

Restuccia¹,

Sotomayor²

2017

Lagrangian Mechanics

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“…We shall solve (12) in X. Due to Proposition 3.1, in order to apply a fixed point argument, it is sufficient to estimate the Y norm of the bilinear form B(u, v) in terms of the X norms of u and v. We have the following Proposition which will be proved in the next section.…”

Section: F(p(d)w)(ζ) = P(ζ)f(w)(ζ)mentioning

confidence: 99%

“…The model consists of two coupled Kadomtsev-Petviashvili (KP) equations (shallow fluid) or of two coupled "intermediate long wave" equations (ILW) (deep fluid). The interaction of internal gravity waves propagating in one horizontal direction has been considered by Gear and Grimshaw [12]. Rigorous mathematical results in this case have been obtained by Bona et al [4], Ash, Cohen and Wang [3], Albert, Bona and Saut [1] (cf.…”

Section: Introductionmentioning

confidence: 99%

On a model system for the oblique interaction of internal gravity waves

Saut¹,

Tzvetkov²

2000

ESAIM: M2AN

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Abstract.We give local and global well-posedness results for a system of two Kadomtsev-Petviashvili (KP) equations derived by R. Grimshaw and Y. Zhu to model the oblique interaction of weakly nonlinear, two dimensional, long internal waves in shallow fluids. We also prove a smoothing effect for the amplitudes of the interacting waves. We use the Fourier transform restriction norms introduced by J. Bourgain and the Strichartz estimates for the linear KP group. Finally we extend the result of [3] for lower order perturbation of the system in the absence of transverse effects.Mathematics Subject Classification. 35Q07, 35Q53, 76B15.

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“…When L ≡ 0, system (1.1) was derived by Gear and Grimshaw in [7] as a model to describe strong interactions of two long internal gravity waves in a stratified fluid, where the two waves are assumed to correspond to different modes of the linearized equations of motion. It has the structure of a pair of KdV equations with both linear and nonlinear coupling terms.…”

mentioning

confidence: 99%

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

2011

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Abstract. We study the stabilization of solutions of a coupled system of Kortewegde Vries (KdV) equations in a bounded interval under the effect of a localized damping term. We use multiplier techniques combined with the so-called "compactness-uniqueness argument". The problem is then reduced to proving a unique continuation property (UCP) for weak solutions. The exponential decay of solutions was previously obtained in Bisognin, Bisognin, and Menzala (2003) when the damping was effective simultaneously in neighborhoods of both extremes of the bounded interval. In this work we address the general case using a different approach to obtain the UCP and stabilize the system.

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Weak and Strong Interactions between Internal Solitary Waves

Cited by 215 publications

References 20 publications

Singular Lagrangians and Its Corresponding Hamiltonian Structures

Singular Lagrangians and Its Corresponding Hamiltonian Structures

On a model system for the oblique interaction of internal gravity waves

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

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