### 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 | |

State | Published - 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Theoretical and Mathematical Physics*,

*180*(1), 759-764. https://doi.org/10.1007/s11232-014-0177-7

**Five-wave classical scattering matrix and integrable equations.** / Zakharov, Vladimir E; Odesskii, A. V.; Cisternino, M.; Onorato, M.

Research output: Contribution to journal › Article

*Theoretical and Mathematical Physics*, vol. 180, no. 1, pp. 759-764. https://doi.org/10.1007/s11232-014-0177-7

}

TY - JOUR

T1 - Five-wave classical scattering matrix and integrable equations

AU - Zakharov, Vladimir E

AU - Odesskii, A. V.

AU - Cisternino, M.

AU - Onorato, M.

PY - 2014

Y1 - 2014

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 - integrability

KW - intermediate long-wave equation

KW - Korteweg-de Vries equation

KW - scattering matrix

UR - http://www.scopus.com/inward/record.url?scp=84905641883&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84905641883&partnerID=8YFLogxK

U2 - 10.1007/s11232-014-0177-7

DO - 10.1007/s11232-014-0177-7

M3 - Article

VL - 180

SP - 759

EP - 764

JO - Theoretical and Mathematical Physics(Russian Federation)

JF - Theoretical and Mathematical Physics(Russian Federation)

SN - 0040-5779

IS - 1

ER -