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Mathematics Colloquium: Numerical computation of the Evans function & stability of small deformations of an elastic rod
4:10 pm, Neill Hall 5W
Abstract: The Evans function is an analytic function that can be used to investigate the spectral stability of one-dimensional localized solutions to nonlinear PDEs. It is defined in such a way that its zeros on the right complex plane are eigenvalues of the corresponding linearized operator. The Evans function is a Wronskian-like function and can in principle be evaluated by a shooting method. In this situation,however, numerically evaluated solutions corresponding to linearly independent initial conditions tend to become linearly dependent, which may lead to spurious zeros of the Evans function. I will argue that for localized solutions that are sufficiently narrow, it is possible to accurately compute the Evans function using a shooting method. As an example, I will consider the spectral stability of a two-parameter family of traveling wave solutions to two coupled nonlinear Klein-Gordon equations. These are envelope equations that model the dynamics of small deformations of an elastic rod near a writhing bifurcation. Numerical results are in agreement with analytical results obtained by S. Lafortune and J. Lega.