Radiating solitary waves in coupled Boussinesq equations
journal contributionposted on 2017-04-28, 15:27 authored by Roger Grimshaw, Karima KhusnutdinovaKarima 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.
- Mathematical Sciences
Published inThe IMA Journal of Applied Mathematics
CitationGRIMSHAW, 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
- AM (Accepted Manuscript)
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NotesThis 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.