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Radiating solitary waves in coupled Boussinesq equations

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posted on 28.04.2017 by Roger Grimshaw, Karima Khusnutdinova, Kieron R. Moore
In this paper we are concerned with the analytical description of radiating solitary wave solutions of coupled regularised Boussinesq equations. This type of solution consists of a leading solitary wave with a small-amplitude co-propagating oscillatory tail, and emerges from a pure solitary wave solution of a symmetric reduction of the full system. We construct an asymptotic solution, where the leading order approximation in both components is obtained as a particular solution of the regularised Boussinesq equations in the symmetric case. At the next order, the system uncouples into two linear non-homogeneous ordinary differential equations with variable coefficients, one correcting the localised part of the solution, which we find analytically, and the other describing the co-propagating oscillatory tail. This latter equation is a fourth order ordinary differential equation and is solved approximately by two different methods, each exploiting the assumption that the leading solitary wave has a small amplitude, and thus enabling an explicit estimate for the amplitude of the oscillating tail. These estimates are compared with corresponding numerical simulations.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

The IMA Journal of Applied Mathematics

Citation

GRIMSHAW, R.H.J., KHUSNUTDINOVA, K.R. and MOORE, K.R., 2017. Radiating solitary waves in coupled Boussinesq equations. IMA Journal of Applied Mathematics, 82 (4), pp. 802–820.

Publisher

© the Authors. Published by Oxford University Press

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Acceptance date

21/04/2017

Publication date

2017

Notes

This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record GRIMSHAW, R.H.J., KHUSNUTDINOVA, K.R. and MOORE, K.R., 2017. Radiating solitary waves in coupled Boussinesq equations. IMA Journal of Applied Mathematics, 82 (4), pp. 802–820 is available online at: https://doi.org/10.1093/imamat/hxx014.

ISSN

0272-4960

eISSN

1464-3634

Language

en

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