Stability of the line soliton of the KP-II equation under periodic transverse perturbations

Mizumachi, Tetsu; Tzvetkov, Nikolay

doi:10.1007/s00208-011-0654-3

Cited by 53 publications

(58 citation statements)

References 31 publications

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“…As far as stability issues are concerned, Mizumachi and Tzvetkov [17] proved that the KdV line soliton is stable under the flow generated by the KP-II equation on L 2 (R × T) for any speed c > 0. Regarding the KP-I equation, Rousset and Tzvetkov [21] proved that Q c is orbitally unstable in E 1 (R×T) under the KP-I flow constructed on Z 2 (R×T) in [8], whenever c > c * = 4/ √ 3, and that it is orbitally stable if c < c * .…”

Section: Stability Resultsmentioning

confidence: 99%

Global well-posedness of partially periodic KP-I equation in the energy space and application

Robert

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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In this article, we address the Cauchy problem for the KP-I equationfor functions periodic in y. We prove global well-posedness of this problem for any data in the energy spaceWe then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.From the work of Benjamin [2], we know that these solutions are orbitally stable in H 1 (R) under the flow generated by the KdV equation (1.2), meaning that every solution of (1.2) with initial data close to Q c in H 1 (R) remains close in H 1 (R) to the Q c -orbit (under the action of translations) at any time t > 0.Looking at (1.1), we see that every solution of the KdV equation (1.2) is a solution of the KP equations (1.1), seen as a function independent of y. It is then a natural question to ask whether Q c is orbitally stable or unstable under the flow generated by (1.1). In order to do so, we first need a global well-posedness theory for (1.1) in a space containing Q c . In particular, this rules out any well-posedness result in an anisotropic Sobolev space H s1,s2 (R 2 ). A more suited space to look for is the energy space for functions periodic in y :where T = R/2πZ. Indeed, due to the Hamiltonian structure of (1.1), the mass M(u)(t) := R×T u 2 (t, x, y)dxdy (1.4) and the energy E(u)(t) := R×T (∂ x u) 2 (t, x, y) + (∂ −1 x ∂ y u) 2 (t, x, y) −

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Section: Stability Resultsmentioning

confidence: 99%

Global well-posedness of partially periodic KP-I equation in the energy space and application

Robert

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…VIII-3 [46]. Finally, we recall the L 2 -stability result for solitary waves of the cubic NLS proved by Mizumachi and Pelinovsky in [45].…”

Section: The Kdv Case: L 2 Stabilitymentioning

confidence: 87%

H 1 -stability of mKdV multi-kinks

Muñoz¹

2012

Journées Équations Aux Dérivées Partielles

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“…Recall that the KdV soliton is transversally L 2 stable with respect to weak transverse perturbations described by the KP II equation ( [195,194]). This result is unconditional since it was established in [207] that the Cauchy problem for KP II is globally well-posed in H s (R × T), s ≥ 0, or for all initial data of the form u 0 + ψ c where u 0 ∈ H s (R 2 ), s ≥ 0 and ψ c (x − ct, y) is a solution of the KP-II equation such that for every σ ≥ 0, (1 − ∂ 2…”

Section: Variamentioning

confidence: 99%

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

Saut

2019

Fields Institute Communications

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This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their (rigorous) derivations we will review old and recent results on the Cauchy problem, comparing those obtained by IST and PDE techniques and also results more connected to the physical origin of the equations. We will consider mainly the Cauchy problem on the whole real line with only a few comments on the periodic case. We will also briefly discuss some close relevant problems in particular the higher order extensions and the two-dimensional (KP like) versions of the BO and ILW equations.Remark 3. The above derivation was performed for purely gravity waves. Surface tension effects result in adding a third order dispersive term in the asymptotic models. One gets for instance the so-called Benjamin equation (see [30,31]):where δ > 0 measures the capillary effects. This equation, which is in some sense close to the KdV equation, is not known to be integrable. Its solitary waves, the existence of which was proven in [30,31] by the degree-theoretic approach, present oscillatory tails. We refer to [145] for the Cauchy problem in L 2 and to [12,21] for further results on the existence and stability of solitary wave solutions and to [46] for numerical simulations.In presence of surface tension, the ILW equation has to be modified in the same way. We are not aware of mathematical results on the resulting equation.Remark 4. The BO equation was fully justified in [219] as a model of long internal waves in a two-fluid system by taking into account the influence of the surface

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Stability of the line soliton of the KP-II equation under periodic transverse perturbations

Abstract: We prove the nonlinear stability of the KdV solitary waves considered as solutions of the KP-II equation, with respect to periodic transverse perturbations. Our proof uses a Miura transform which sends the solutions of an mKP-II equation to solutions of the KP-II equation.

Cited by 53 publications

References 31 publications

Global well-posedness of partially periodic KP-I equation in the energy space and application

Global well-posedness of partially periodic KP-I equation in the energy space and application

H 1 -stability of mKdV multi-kinks

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

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