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Validity of the weakly nonlinear solution of the cauchy problem for the boussinesq-type equation

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posted on 28.01.2016 by Karima Khusnutdinova, Kieron R. Moore, Dmitry Pelinovsky
We consider the initial-value problem for the regularized Boussinesq-type equation in the class of periodic functions. Validity of the weakly nonlinear solution, given in terms of two counterpropagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the timescales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis. © 2014 by the Massachusetts Institute of Technology.

Funding

The research was supported by the LMS Scheme 2 grant and by the Loughborough University School of Science small grant.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Studies in Applied Mathematics

Volume

133

Issue

1

Pages

52 - 83

Citation

KHUSNUTDINOVA, K.R., MOORE, K.R. and PELINOVSKY, D., 2014. Validity of the weakly nonlinear solution of the cauchy problem for the boussinesq-type equation. Studies in Applied Mathematics, 133(1), pp. 52-83.

Publisher

© Wiley

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/

Publication date

2014

Notes

This is the peer reviewed version of the following article: KHUSNUTDINOVA, K.R., MOORE, K.R. and PELINOVSKY, D., 2014. Validity of the weakly nonlinear solution of the cauchy problem for the boussinesq-type equation. Studies in Applied Mathematics, 133(1), pp. 52-83., which has been published in final form at http://dx.doi.org/10.1111/sapm.12034. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."

ISSN

0022-2526

eISSN

1467-9590

Language

en

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