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Stability of solitary waves and weak rotation limit for the Ostrovsky equation. (English) Zbl 1064.35148
This paper is concerned with the Ostrovsky equation, which can be written in the form \((u_t - \beta u_{xxx} + (u^2)_x)_x = \gamma u\) for some constants \(\beta, \gamma \in \mathbb R\) with \(\gamma >0\). This equation describes the propagation of long internal surface waves in shallow water in the presence of a rotation. The authors consider a model where dispersion is considered but dissipation is neglected. They classify the existence and nonexistence of the dispersion parameter and prove that the set of solitary waves is stable with respect to perturbation in the case of positive dispersion. They also investigate the issue of passing to the limit as the rotation parameter tends to zero for solutions of the Cauchy problem on a bounded time interval.

MSC:
35Q35 PDEs in connection with fluid mechanics
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35K35 Initial-boundary value problems for higher-order parabolic equations
35Q51 Soliton equations
76B55 Internal waves for incompressible inviscid fluids
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