| Abstract: |
| The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, perhaps the most striking is the fact that it admits weak multi-soliton solutions - `peakons` - with a peaked shape corresponding to a discontinuous first derivative. There exists a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions.
In this talk, we establish information about the spectral stability/instability of those solutions. By spectral stability analysis, we mean the analysis of the spectrum of the operator arising from the linearised equation. Furthermore, we extend our analysis to the peakon solutions of the Novikov equation.
The computation of the spectrum differs from similar problems due to the following two facts: the solutions to the equations are not analytical, and the linear operator is nonlocal as it contains integral terms. |
|