| Abstract: |
| In this talk, we study a higher-order modified Camassa-Holm equation, which can be viewed as a natural generalization of the modified Camassa-Holm equation. It has been known that the modified Camassa-Holm equation enjoys several nontrivial properties, of different character than those of the Camassa-Holm equation. Our primary goal is to understand how higher-order nonlinearities affect the dispersive dynamics. First, we investigate invariant properties and conservation laws of the equation, and establish the local well-posedness of the Cauchy problem in Besov and Sobolve spaces. Second, we study the orbital stability of peaked traveling wave solutions of the equation. Finally, We study the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution. |
|