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Cauchy problem for the Ostrovsky equation. (English) Zbl 1059.35035
Summary: Considered herein is an initial-value problem for the Ostrovsky equation that arises in modelling the unidirectional propagation of long waves in a rotating homogeneous incompressible fluid. Nonlinearity and dispersion are taken into account, but dissipation is ignored. Local- and global-in-time solvability is investigated. For the case of positive dispersion a fundamental solution of the Cauchy problem for the linear equation is constructed, and its asymptotics is calculated as \(t\to\infty\), \(x/t=\text{const}\). For the nonlinear problem solutions are constructed in the form of a series and the analogous long-time asymptotics is obtained.

MSC:
35G25 Initial value problems for nonlinear higher-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35C10 Series solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
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