Five-wave classical scattering matrix and integrable equations

V. E. Zakharov; A. V. Odesskii; M. Cisternino; M. Onorato

doi:10.1007/s11232-014-0177-7

Five-wave classical scattering matrix and integrable equations

V. E. Zakharov, A. V. Odesskii, M. Cisternino, M. Onorato

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.

Original language	English (US)
Pages (from-to)	759-764
Number of pages	6
Journal	Theoretical and Mathematical Physics
Volume	180
Issue number	1
DOIs	https://doi.org/10.1007/s11232-014-0177-7
State	Published - Aug 2014

Keywords

Benjamin-Ono equation
Korteweg-de Vries equation
integrability
intermediate long-wave equation
scattering matrix

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s11232-014-0177-7

Cite this

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title = "Five-wave classical scattering matrix and integrable equations",

abstract = "We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.",

keywords = "Benjamin-Ono equation, Korteweg-de Vries equation, integrability, intermediate long-wave equation, scattering matrix",

author = "Zakharov, {V. E.} and Odesskii, {A. V.} and M. Cisternino and M. Onorato",

note = "Funding Information: This work was supported in part by the National Science Foundation (Grant No. NSF40CE-1130450), the Russian Federation (Contract No. 11.934.31.0035 signed 25 November 2010), the ONR (Grant No. 214 N000141010991, M. O.), and MIUR (Grant No. PRIN 2012BFNWZ2, M. O.).",

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language = "English (US)",

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AU - Zakharov, V. E.

AU - Odesskii, A. V.

AU - Cisternino, M.

AU - Onorato, M.

N1 - Funding Information: This work was supported in part by the National Science Foundation (Grant No. NSF40CE-1130450), the Russian Federation (Contract No. 11.934.31.0035 signed 25 November 2010), the ONR (Grant No. 214 N000141010991, M. O.), and MIUR (Grant No. PRIN 2012BFNWZ2, M. O.).

PY - 2014/8

Y1 - 2014/8

N2 - We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.

AB - We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.

KW - Benjamin-Ono equation

KW - Korteweg-de Vries equation

KW - integrability

KW - intermediate long-wave equation

KW - scattering matrix

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JO - Theoretical and Mathematical Physics(Russian Federation)

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Five-wave classical scattering matrix and integrable equations

Abstract

Keywords

ASJC Scopus subject areas

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