Abstract
This project studies the linear stability of solitary wave solutions to Hamiltonian PDEs. It seeks
to do this by formulating the linear stability problem in terms of the Evans function, the zeroes
of which relate to eigenvalues of the stability problem. The objective is to derive a formula that
can be used to prove instability by using geometric properties of the solitary wave. This means
that instability can be deduced without having an exact expression for the solitary wave. The
approach to achieving this starts by introducing a new class of Hamiltonian PDEs based on a
Dirac-type operator that can model a variety of wave equations. The Dirac-type operator allows
for a connection to Clifford algebras to be made. Then, Evans functions are constructed for the
cases of one and two space dimensions. The main result is expressions for the second derivatives
of these Evans functions using 3 properties of the underlying solitary wave equation: transversality,
momentum and asymptotics of the solitary wave. The theory is then illustrated on a simple example
in each case.