American Institute of Mathematical Sciences

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2004, 10(3): 731-753. doi: 10.3934/dcds.2004.10.731

Cauchy problem for the Ostrovsky equation

V. Varlamov 1, and  Yue Liu 2,

1. 

Department of Mathematics, University of Texas - Pan American, Edinburg, TX 78539-2999, United States
2. 

Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, United States

Received  September 2002 Revised  August 2003 Published  January 2004

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 $t\rightarrow \infty, x/t=$const. For the nonlinear problem solutions are constructed in the form of a series and the analogous long-time asymptotics is obtained.
Keywords: Cauchy problem, long–time asymptotics., Ostrovsky equation, construction of solutions.
Mathematics Subject Classification: 35B40, 35B65, 35C10, 35C20, 35E1.
Citation: V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731
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