J Austral Math Soc Ser B 31 pp397--424, 1990.

Nonlinear interaction of positive and negative energy modes in Hamiltonian systems

J. P. Denier and R. H. J. Grimshaw

Abstract

We consider the nonlinear evolution of a Hamiltonian system as the system passes through a linear resonance (as the system parameters vary). Two cases are considered. In the first case the linearized problem (at resonance) possess a full complement of normal mode solutions. This case is presented in the context of the interaction between modes which may have oppositely signed energy. The second case considered has an additional degeneracy in that the linearized problem (at resonance) has a single normal mode solution.
  Both cases are analysed using normal form theory and in both cases the systems governing the transition through resonance are shown to be completely integrable in the classical sense. Possible bifurcations as the resonance is traversed are discussed. Conditions for the existence of algebraic singularities at some finite positive time are also presented.

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Authors

J. P. Denier

Present address:

Department of Mathematics, Exeter University, North Park Road, Exeter, EX4 4QE, England.

R. H. J. Grimshaw

School of Mathematics, University of New South Wales, P. O. Box 1, Kensington, NSW 2033, Australia.