Towards a rigorous derivation of the fifth order KP equation

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg-de Vries solitary waves, the present study is concerned with a distinct class of gravity-capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney-Roskes-Davey-Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev-Petviashvili equation, a model for weakly nonlinear gravity-capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.

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“…A systematic derivation was presented in Paumond (2005). Here, we shall work with the fifth-order KP equation (for B!1/3) in the normalized form…”

Section: Preliminariesmentioning

confidence: 99%

On the stability of lumps and wave collapse in water waves

Akylas

Cho

2008

Phil. Trans. R. Soc. A.

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“…A rigorous proof of this derivation is known in the one dimensional case (see [8,10,17]) and consistency results for the two dimensional problem can be found in [7,11]. It is quite natural to link rigorously KP to intermediate models, that is done in [4,9] and specially in [14,15] (1) and (2) are considered.…”

Section: Introductionmentioning

confidence: 88%

“…Precisely, in this special case, (BL) is not linearly well-posed. Then, we can derive (see [14]) an equation still valid when we suppose that σ is equal or close to 1/3,…”

Section: Introductionmentioning

confidence: 99%

“…This model becomes linearly well-posed even if σ is close or equal to 1/3. For the study of the Cauchy problem for these nonlinear equations we infer to [14,15].…”

Section: Introductionmentioning

confidence: 99%

“…We recall that if σ > 1/3, this equation is KP-I, if σ < 1/3, it is KP-II. By setting a − b = σ − 1/3 = θµ with θ ∈ R independent of µ and doing exactly the same in (2) we obtain the fifth order KP-I equation (see [14])…”

Section: Introductionmentioning

confidence: 99%

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Numerical comparisons of two long-wave limit models

Labbé¹,

Paumond²

2004

ESAIM: M2AN

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Abstract. The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039-1064; Pego and Quintero, Physica D 132 (1999) 476-496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically the link between (KP) and (BL) and we point out the coupling effects emerging by considering two solitary waves propagating in two opposite directions.

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On the nonlinear scattering and well-posedness for a 2D-Boussinesq–Benney–Luke type system

Quintero

Montes

2016

Journal of Differential Equations

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Towards a rigorous derivation of the fifth order KP equation

Cited by 13 publications

References 15 publications

On the stability of lumps and wave collapse in water waves

On the stability of lumps and wave collapse in water waves

Numerical comparisons of two long-wave limit models

On the nonlinear scattering and well-posedness for a 2D-Boussinesq–Benney–Luke type system

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