A Hamiltonian perturbation theory for coherent structures illustrated to wave problems


Groesen, E. van (1994) A Hamiltonian perturbation theory for coherent structures illustrated to wave problems. In: A. Mielke & K. Kirchgässner (Eds.), Proceedings of the IUTAM/ISIMM Symposium Structure and Dynamics of Nonlinear Waves in Fluids. World Scientific, Singapore, pp. 99-116. ISBN 9789810221249

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Abstract:In wave problems with a Hamiltonian structure and some symmetry property, the relative equilibria are the physical coherent structures. They appear as families of states, parameterized by physical observables connected to the symmetry. A new abstract (and general) result about the relation between the kernel of the linearized Hamiltonian flow and its adjoint is the basis of a Hamiltonian perturbation theory. In this theory, the effect of a perturbation of the system is decomposed in an effect within the class of coherent structures and a transversal deviation. It is shown that, under mild conditions, the transversal deviation remains small (of the order of the pertnrbation) while the main effect is a (quasi-static) succession of various coherent structures. This means that, in a good approximation, the coherent structures can be used as base functions, and that the evolution of the parameters, as a set of collective coordinates, specifies the dynamics.
Two examples illustrate the general result: dissipative perturbations of the Korteweg-de Vries equation, and a spatially inhomogeneous Sine-Gordon equation.
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Electrical Engineering, Mathematics and Computer Science (EEMCS)
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