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Analysis of solitary waves for some long-wave water wave models

Jin, Jie (2021) Analysis of solitary waves for some long-wave water wave models. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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The analysis of solitary waves is an important topic in studying the dynamic of water wave models. The thesis will be divided into two parts:
In the first part, we focus on the solitary waves solutions to the Boussinesq abcd system. The Boussinesq abcd system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters abcd satisfy one constraint. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to b not equal to d ). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves. Specifically, we first construct solutions bifurcating from the stationary waves, and obtain a global continuous curve of solutions that exhibits a loss of ellipticity in the limit. The second family of solutions bifurcate from the classical Boussinesq supercritical waves. We show that the curve associated to the second class either undergoes a loss of ellipticity in the limit or becomes arbitrarily close to having a stagnation point.
In the second part, we consider the Camassa-Holm-Kadomtsev-Petviashvili-I equation (CH-KP-I), which is a two dimensional generalization of the Camassa-Holm equation (CH). We prove transverse instability of the line solitary waves under periodic transverse perturbations. The proof is based on the framework of Rousset-Tzvetkov. Due to the high nonlinearity, our proof requires necessary modification.
In more detail, we first establish the linear instability of the line solitary waves. Then through an approximation procedure, we prove that the linear effect actually dominates the nonlinear behavior.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Jin, Jiejij50@pitt.edujij500000-0001-8975-2724
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairChen,
Committee MemberJiang,
Committee MemberWang,
Committee MemberPego,
Date: 8 October 2021
Date Type: Publication
Defense Date: 4 May 2021
Approval Date: 8 October 2021
Submission Date: 6 May 2021
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 74
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Boussinesq abcd system, Solitary waves, Analytic global bifurcation, CH-KP-Iequation, Transverse instability
Date Deposited: 08 Oct 2021 19:11
Last Modified: 08 Oct 2021 19:11

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