Differential and Integral Equations

A rigorous link between KP and a Benney-Luke equation

Lionel Paumond

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Abstract

Kadomtsev-Petviashvili (KP) equations model weakly nonlinear dispersive waves, which are essentially unidimensional, when weak transverse effects are taken into account. These equations can be formally obtained by asymptotic methods from the Euler equations, but there is no rigorous justification yet. In this paper we consider an intermediate equation (BL) derived first by Benney and Luke. (BL) reduces formally to (KP) if we seek waves propagating essentially in one direction, with a weak variations in time and in the transverse direction measured by a small parameter ${\varepsilon}$. We show rigorously that the $L^2(\mathbb R^2)$-norm of the difference between the amplitude of the wave given by (KP) and the one given by (BL) is of order $\mathcal{O}({\varepsilon}^{3/4})$ during a growing with ${\varepsilon}$ time.

Article information

Source
Differential Integral Equations, Volume 16, Number 9 (2003), 1039-1064.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060556

Mathematical Reviews number (MathSciNet)
MR1989540

Zentralblatt MATH identifier
1056.76014

Subjects
Primary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]
Secondary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Citation

Paumond, Lionel. A rigorous link between KP and a Benney-Luke equation. Differential Integral Equations 16 (2003), no. 9, 1039--1064. https://projecteuclid.org/euclid.die/1356060556


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