Coherent Structures in the Dynamics of Perturbed Hamiltonian Systems

Gianne Derks


Summary

In this thesis we consider two aspects of perturbed Hamiltonian systems by using special solutions of the unperturbed Hamiltonian system. These special solutions exist in Hamiltonian systems with an additional constant of motion and are known as relative equilibria. In the applications they often refer to socalled coherent structures.

The first aspect is the use of relative equilibria to approximate solutions starting near a relative equilibrium, in a Hamiltonian system with dissipation. The approximation consists of a quasi-static succession of relative equilibria. The main contribution in this thesis is to investigate the conditions for which the quasi-static approximation can be justified, even in case the deviation from the initial state, caused by the small perturbation, is large.

The general theory to be developed will be illustrated for two specific examples: the damped spherical pendulum and the uniformly damped Korteweg-de Vries equation with periodic boundary conditions. The relative equilibria of the Korteweg-de Vries equation are space-periodic coherent structures, known as cnoidal waves. These waves have a soliton-like behaviour. We show that the distance between a solution and a quasi-static approximation is at each instant of the order of the damping rate times the norm of the cnoidal wave.

A second aspect is investigated when a combination of dissipation and ``self-excitation'' is added to a one dimensional Hamiltonian wave equation. Explicitly, we consider the periodic Korteweg-de Vries equation with a small viscous damping and a small inverse uniform damping added. The cnoidal waves are travelling wave solutions, and the persistence of such waves under the perturbation is investigated. the self-excitation term is small compared to the dissipative term, and in fact the travelling waves are attractive. The existence of the travelling wave solutions is also proven analytically. If the self-excitation increases, the travelling wave solutions become unstable and modulated travelling waves can be observed.


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Last modified: Tue Jun 22 17:36:05 BST 1999