Mathematical Modelling

On a nonlinear problem of the breaking water waves

M. Kirane^a, B. T. Torebek<sup>bc

a</sup> Université de La Rochelle, La Rochelle, France
^b Al-Farabi Kazakh National University, Almaty, Kazakhstan
^c Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

Abstract: The paper is devoted to the initial boundary value problem for the Korteweg-de Vries–Benjamin–Bona–Mahony equation in a finite domain. This particular problem arises from the phenomenon of long wave with small amplitude in fluid. For certain initial-boundary problems for the Korteweg-de Vries–Benjamin–Bona–Mahony equation, we obtain the conditions of blowing-up of global and travelling wave solutions in finite time. The proof of the results is based on the nonlinear capacity method. In closing, we provide the exact and numerical examples.

Keywords: breaking waves, Korteweg-de Vries–Benjamin–Bona–Mahony equation, blow-up of solution, initial-boundary problems.

UDC: 517.95

MSC: 35K55, 35R11

Received: 09.10.2018

Language: English

DOI: 10.14529/mmp190203

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