Spectral stability of solutions to Whitham-type equations with a nonlocal kernel
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Full Text E-thesis
Date
2024
Authors
McCormack, Kathleen Tamarisk
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Publisher
University College Cork
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Abstract
This thesis aims to provide a clear and reproducible outline of finding travelling wave solutions to certain nonlinear partial differential equations. This is done using a spectral cosine collocation method, and the stability of the travelling wave solutions found is then assessed using the Fourier-Floquet-Hill method. This is first done on a number of examples. Next, a variation of the Whitham equation which uses a nonlocal kernel with a parameter ν, that determines the strength of the dispersion, was solved and its stability spectra analyzed. This model is expected to have both stable continuous solutions and unstable breaking solutions, depending on the parameter ν. The solution method worked well across all examples and replicated prior results. It was less effective for the cases of the nonlocal modified dispersion model where breaking solutions would occur. This is due to a limitation in the assumptions of the Fourier method. The stability analysis aligned with previous findings for the cubic vortical Whitham equation and showed that the solutions that were found for the nonlocal modified dispersion model were stable. Further research is needed to assess the stability of the breaking solutions, which this solution method did not capture.
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Keywords
Whitham equation , Spectral stability , Nonlocal kernels , KdV-type equations , Cubic-vortical Whitham equation
Citation
McCormack, K. T. 2024. Spectral stability of solutions to Whitham-type equations with a nonlocal kernel. MSc Thesis, University College Cork.