Cauchy problem for the Ostrovsky equation

Abstract

Considered herein is an initial-value problem for the Ostrovsky equationthat arises in modelling the unidirectional propagation of long waves in arotating homogeneous incompressible fluid. Nonlinearity and dispersion aretaken into account, but dissipation is ignored. Local- and global-in-timesolvability is investigated. For the case of positive dispersion afundamental solution of the Cauchy problem for the linear equation isconstructed, and its asymptotics is calculated as $t\rightarrow \infty, x/t=$const. For the nonlinear problem solutions are constructed in theform of a series and the analogous long-time asymptotics is obtained.