The Cauchy Problem and Stability of Solitary-Wave Solutions for RLW–KP-Type Equations

Bona, Jerry L.; Liu, Yue; Tom, Michael M.

doi:10.1006/jdeq.2002.4171

Cited by 26 publications

(41 citation statements)

References 33 publications

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“…Since the argument employed is similar to the fKPII case we will give a sketch of the main differences between both cases. Note that due to Remark 3.2, we may assume that 4 3 ≤ α ≤ 2. As in the previous case we have that…”

Section: Global Weak Solutions For Fkp-i Equationsmentioning

confidence: 99%

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The Cauchy Problem for the Fractional Kadomtsev--Petviashvili Equations

Linares¹,

Pilod²,

Saut³

2018

SIAM J. Math. Anal.

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Section: Global Weak Solutions For Fkp-i Equationsmentioning

confidence: 99%

“…Remark 2.2. Solutions in the class H k,0 (R 2 ) can be obtained when α = 2, κ = 1, that is for the classical KP-II equation and also when α > 4 3 for the fKP-II equation (see [14]). We do not know if they exist in the range 0 < α < 4 3 .…”

mentioning

confidence: 99%

The Cauchy Problem for the Fractional Kadomtsev--Petviashvili Equations

Linares¹,

Pilod²,

Saut³

2018

SIAM J. Math. Anal.

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“…The Cauchy problems for those regularized KP equations has been studied in [108], for initial data in the natural energy space. We refer also to [16] for more regular initial data and for a stability analysis of the solitary waves in the focusing case.…”

Section: 7mentioning

confidence: 99%

Numerical Study of Blow up and Stability of Solutions of Generalized Kadomtsev–Petviashvili Equations

Klein

Saut

2012

J Nonlinear Sci

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“…Pour l'équation de KP-II, Bourgain [8] montre que pour tout donnée initiale prise dans H m (R 2 ), avec m ≥ 0, il existe une unique solution de (2.1)-(2.3) globale en temps. Le problème de Cauchy pour l'équation de KP-BBM-II est traité dans les articles de Bona et al [7], ou Saut et Tzvetkov [18], ils montrent que pour toute donnée initiale prise dans le sous-espace de L 2 (R 2 ) muni de la norme (||u|| 2 + ||u x || 2 ) 1/2 , il existe une unique solution de (2.2)-(2.3) globale en temps.…”

Section: Problèmes De Cauchyunclassified

Comparaison entre modèles d'ondes de surface en dimension 2

Mammeri¹

2007

ESAIM: M2AN

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Résumé. Partant du principe de conservation de la masse et du principe fondamental de la dynamique, on retrouve l'équation d'Euler nous permettant de décrire les modèles asymptotiques de propagation d'ondes dans des eaux peu profondes en dimension 1. Pour décrire la propagation des ondes en dimension 2, Kadomtsev et Petviashvili [Sov. Phys. Dokady 15 (1970) Abstract. On the basis of the principle of conservation of mass and fundamental principle of dynamics, we find the Euler equation enabling us to describe the asymptotic models of waves propagation in shallow water in dimension 1. To describe the waves propagation in dimension 2, a linear perturbation of the KdV equation is used by Kadomtsev and Petviashvili [Sov. Phys. Dokady 15 (1970) [J. Fluid Mech. 92 (1979) 691-715]. We will insist, in same manner, on the fact that the equations of KP-BBM can be also obtained starting from the Euler equation, and up to what point they describe the physical model. In a second time, we take again the method introduced in the article of Bona et al. [Lect. Appl. Math. 20 (1983) 235-267] in which the solutions of long water waves in dimension 1, namely the solutions of KdV and BBM, are compared, to show here that the solutions of KP-II and KP-BBM-II are close for a time scale inversely proportional to the waves amplitude. From the point of view of modelling, it will be clear according to the first part, that only the model described by KP-BBM-II is well posed, and since from the physical point of view, KP-II and KP-BBM-II describe the small amplitude long waves when the surface tension is neglected, it is interesting to compare them. Moreover, we will see that the method used here remains valid for the periodic problems.

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The Cauchy Problem and Stability of Solitary-Wave Solutions for RLW–KP-Type Equations

Cited by 26 publications

References 33 publications

The Cauchy Problem for the Fractional Kadomtsev--Petviashvili Equations

The Cauchy Problem for the Fractional Kadomtsev--Petviashvili Equations

Numerical Study of Blow up and Stability of Solutions of Generalized Kadomtsev–Petviashvili Equations

Comparaison entre modèles d'ondes de surface en dimension 2

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