A study on traveling wave solutions in the shallow-water-type system

Author

Ting Luo

Graduation Semester and Year

2017

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Yue Liu

Second Advisor

Jianzhong Su

Third Advisor

Gaik Ambartsoumian

Fourth Advisor

Guojun Liao

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.

Keywords

Traveling wave, Shallow water

Disciplines

Mathematics | Physical Sciences and Mathematics

License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

Comments

Degree granted by The University of Texas at Arlington

Recommended Citation

Luo, Ting, "A study on traveling wave solutions in the shallow-water-type system" (2017). Mathematics Dissertations. 204.
https://mavmatrix.uta.edu/math_dissertations/204