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Initial-value problem for coupled Boussinesq equations and

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The Pennsylvania State University CiteSeerX Archives
Keywords
Coupled Boussinesq equations
Ostrovsky equation
Asymptotic multiple-scales expansions
Averaging
Initial-value problem

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URI
http://hdl.handle.net/20.500.12424/1031929
Online Access
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.768.9152
http://arxiv.org/pdf/1104.3432.pdf
Abstract
We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scales expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.
Date
2016-08-18
Type
text
Identifier
oai:CiteSeerX.psu:10.1.1.768.9152
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.768.9152
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Metadata may be used without restrictions as long as the oai identifier remains attached to it.
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