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D'Alembert-type solution of the Cauchy problem for the Boussinesq-Klein-Gordon equation

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posted on 21.03.2019 by Karima Khusnutdinova, Matt Tranter
In this paper we construct a weakly-nonlinear d'Alembert-type solution of the Cauchy problem for the Boussinesq-Klein-Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localised solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple-scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert-type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right- and left-propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have non-zero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localised solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have non-zero mean values. Importantly, in all cases the initial conditions for the leading-order Ostrovsky equations by construction have zero mean, while initial conditions for the Boussinesq-Klein-Gordon equation may have non-zero mean values.

Funding

KRK is grateful to the Erwin Schrodinger Institute (ESI) in Vienna, Austria for the financial support of her participation in the programme “Mathematical Aspects of Physical Oceanography” where some parts of this paper have been discussed and developed. MRT acknowledges the support of the UK Engineering and Physical Sciences Research Council (EPSRC) during his PhD studentship.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Studies in Applied Mathematics

Citation

KHUSNUTDINOVA, K.R. and TRANTER, M.R., 2019. D'Alembert-type solution of the Cauchy problem for the Boussinesq-Klein-Gordon equation. Studies in Applied Mathematics, 142 (4), pp. 551-585.

Publisher

© Wiley

Version

AM (Accepted Manuscript)

Publisher statement

This is the peer reviewed version of the following article: KHUSNUTDINOVA, K.R. and TRANTER, M.R., 2019. D'Alembert-type solution of the Cauchy problem for the Boussinesq-Klein-Gordon equation. Studies in Applied Mathematics, 142 (4), pp. 551-585, which has been published in final form at https://doi.org/10.1111/sapm.12263. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.

Acceptance date

08/03/2019

Publication date

2019-03-28

ISSN

0022-2526

eISSN

1467-9590

Language

en

Exports