A study on traveling wave solutions in the shallow-water-type system
Abstract
The study of water waves reveals the physical principles of many phenomena of scientific and engineering interest. In this dissertation I consider three models: two-component Camassa-Holm system(2CH), generalized two-component Camassa-Holm equation(g2CH) and rotation-Camassa-Holm equation(R-CH). In the first part, we consider the stability of the Camassa-Holm peakons and antipeakons in the dynamics of the two-component Camassa-Holm system. The second part shows that the train of $N$-smooth traveling waves of this system is dynamically stable to perturbations in energy space with a range of parameters. In the third part, we formally derive the simplified phenomenological models with the Coriolis effect due to the Earth's rotation and justify rigorously that the solutions of these models are well approximated by the solutions of the rotation-Camassa-Holm equation. Furthermore, we demonstrate nonexistence of the Camassa-Holm-type peaked solution and classify various localized traveling-wave solutions to the rotation-Camassa-Holm equation.