# Coherent Structures in the Dynamics of Perturbed Hamiltonian Systems

## Gianne Derks

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.

Comments and suggestions to Gianne Derks

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