On the quasi-periodic solutions of generalized Kaup systems

Claudia Valls

doi:10.3934/dcds.2015.35.467

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2015, Volume 35, Issue 1: 467-482. Doi: 10.3934/dcds.2015.35.467

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On the quasi-periodic solutions of generalized Kaup systems

Claudia Valls^1,

1.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

Received: November 2013

Revised: May 2014

Published online: August 2014

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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.

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Mathematics Subject Classification: Primary: 34C05, 35Qxx; Secondary: 37Kxx.

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