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We consider the existence and stability of traveling waves of a generalized Ostrovsky equation ( u t − β u x x x − f ( u ) x ) x = γ u , where the nonlinearity f ( u ) satisfies a power-like scaling condition. We prove that there exist ground state solutions which minimize the action among all nontrivial solutions and use this variational characterization to study their stability. We also introduce a numerical method for computing ground states based on their variational properties. The class of nonlinearities considered includes sums and differences of distinct powers.

Studied here is the Ostrovsky, Stepanyams and Tsimring (OST) equation u t + u x x x − e H ( u + u x x ) x + u u x = 0 . It is showed that the associated initial value problem is locally analytically well-posed in the critical Sobolev spaces H − 3 ∕ 2 ( R ) . This improves the previous results on this equation in the Sobolev spaces.

In this paper, we consider the modified Ostrovsky, Stepanyams and Tsimring equation u t + u x x x − η ( H u x + H u x x x ) + u 2 u x = 0 . We prove that the associated initial value problem is locally well-posed in Sobolev spaces H s ( R ) for s > − 1 ∕ 2 . We also prove that our result is sharp in the sense that the flow map of this equation fails to be C 3 in H s ( R ) for s − 1 ∕ 2 . Moreover, we prove that for any s > 1 ∕ 2 and T > 0 , its solution converges in C ( [ 0 , T ] ; H s ( R ) ) to that of the mKdV equation if η tends to 0.

Considered here is the Schrodinger-improved Boussinesq system. First we prove local and global well-posedness in the energy space for the periodic initial-value problem. The proof combines a Strichartz-type estimate with the contraction mapping principle. Second we establish the existence and orbital stability of periodic and solitary traveling-wave solutions. The stability results are set out in the context of abstract Hamiltonian systems.