This thesis investigates both theoretically and numerically the stability of travelling
wave solutions using Fredholm determinants, on the real line. We identify a class of
travelling wave problems for which the corresponding integral operators are of trace
class. Based on the geometrical interpretation of the Evans function, we give an alternative
proof connecting it to (modified) Fredholm determinants. We then extend
that connection to the case of front waves by constructing an appropriate integral
operator. In the context of numerical evaluation of Fredholm determinants, we prove
the uniform convergence associated with the modified/regularised Fredholm determinants
which generalises Bornemann's result on this topic. Unlike in Bornemann's
result, we do not assume continuity but only integrability with respect to the second
argument of the kernel functions. In support to our theory, we present some numerical
results. We show how to compute higher order determinants numerically, in particular
for integral operators belonging to classes I3 and I4 of the Schatten-von Neumann
set. Finally, we numerically compute Fredholm determinants for some travelling wave
problems e.g. the `good' Boussinesq equation and the fth-order KdV equation.UK EPSRC (Engineering and Physical Sciences Research Council) grant EP/G03613
