Abstract
The focus of this thesis is on the study of modulation of symmetric Hamiltonian ODEs and
PDEs. We consider equations which can be written as a symmetric Hamiltonian gradient system,
derived from a Lagrangian. The system has conservation laws associated and some of
these conserved quantities will play a role in the theory developed. The symmetry of the system
defines a particular type of solution called relative equilibria. Modulation of these solutions and
back substitution into the governing equations leads to universal form equations, such as the
KdV equation, for the modulated wave number.
The modulation of the relative equilibria is reliant on a property of the relative equilibria holding.
Namely that they are degenerate. This degeneracy is related to conserved quantities of
the overall Hamiltonian system. When the relative equilibria are degenerate we can derive the
normal form through modulation in the usual way.
In the case of non-degenerate relative equilibria we use Whitham modulation to derive characteristics.
We can then re-modulate with our space variable changed to be in a frame travelling
at the speed of one of these characteristics. Thus working around the fact that our relative
equilibria are non-degenerate.
We can apply this theory of re-modulation to pattern formation. If our base state to be modulated
is a basic roll pattern, then we will derive KdV through the re-modulation in a frame
of reference at a new angle. Thus producing modulated patterns at a second angle. As the
modulation equation is the KdV equation which is well known to have soliton solutions, a third
angle at which solitons form can also be considered.
The relative equilibria solutions we are considering are an overall class of solutions. A particular
type of relative equilibria which are of interest, especially in the ODE case within this thesis,
are relative periodic orbits. These are simply periodic orbits when considered from the frame
of reference relative to the group. So we have a periodic orbit in a base space before the group
action is considered. This implies a form of lifting action, called a connection, from the periodic
orbit to the full system in the group frame. Associated with this connection is a change in angle
called holonomy which we also investigate to understand the geometry of relative periodic orbits
further.