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K.R. Khusnutdinova, K.R. Moore, and D.E. Pelinovsky

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Validity of the weakly nonlinear solution
of the Cauchy problem for the Boussinesq-type equation

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Studies in Applied Mathematics 133, 52-83 (2014)

**Abstract:**

WWe 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 counter-propagating 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 time scales 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.

**Keywords**:

Regularized Boussinesq-type equation, justification of amplitude equations,
energy method, Ostrovsky equation, numerical approximations.