Fredholm determinants for the stability of travelling waves
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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.