Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On the quasi-periodic solutions of generalized Kaup systems

Pages: 467 - 482, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.467

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Claudia Valls - Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal (email)

Abstract: In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Kap system given by \[ \partial_t u = \partial_x v - 2 \partial_x(v^3), \quad \partial_t v = \partial_x u - \frac 1 3 \partial_{xxx} u. \] It is proved that the above equation admits small-amplitude solutions that are quasiperiodic in time and that correspond to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof relies on the Hamiltonian formulation of the problem, the study of its Birkhoff normal form and an infinite dimensional KAM theorem. This is the abstract of your paper and it should not exceed.

Keywords:  Water waves, Kaup system, Hamiltonian formalism, KAM, Birkhoff normal form.
Mathematics Subject Classification:  Primary: 34C05, 35Qxx; Secondary: 37Kxx.

Received: November 2013;      Revised: May 2014;      Available Online: August 2014.