#### Graduation Year

2019

#### Document Type

Dissertation

#### Degree

Ph.D.

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Wen-Xiu Ma, Ph.D.

#### Committee Member

Yuncheng You, Ph.D.

#### Committee Member

Sherwin Kouchekian, Ph.D.

#### Committee Member

Razvan Teodorescu, Ph.D.

#### Committee Member

Leslaw Skrzypek, Ph.D.

#### Keywords

Hamiltonian Operator, Hamiltonian Structure, Integrable Systems, Matrix Spectral Problem, Riemann-Hilbert Problem

#### Abstract

We begin this dissertation by presenting a brief introduction to the theory of solitons and integrability (plus some classical methods applied in this field) in Chapter 1, mainly using the Korteweg-de Vries equation as a typical model. At the end of this Chapter a mathematical framework of notations and terminologies is established for the whole dissertation.

In Chapter 2, we first introduce two specific matrix spectral problems (with 3 potentials) associated with matrix Lie algebras $\mbox{sl}(2;\mathbb{R})$ and $\mbox{so}(3;\mathbb{R})$, respectively; and then we engender two soliton hierarchies. The computation and analysis of their Hamiltonian structures based on the trace identity affirms that the obtained hierarchies are Liouville integrable. This chapter shows the entire process of how a soliton hierarchy is engendered by starting from a proper matrix spectral problem.

In Chapter 3, at first we elucidate the Gauge equivalence among three types $u$-linear Hamiltonian operators, and construct then the corresponding B\"acklund transformations among them explicitly. Next we derive the if-and-only-if conditions under which the linear coupling of the discussed u-linear operators and matrix differential operators with constant coefficients is still Hamiltonian. Very amazingly, the derived conditions show that the resulting Hamiltonian operators is truncated only up to the 3rd differential order. Finally, a few relevant examples of integrable hierarchies are illustrated.

In Chapter, 4 we first present a generalized modified Korteweg-de Vries hierarchy. Then for one of the equations in this hierarchy, we build the associated Riemann-Hilbert problems with some equivalent spectral problems. Next, computation of soliton solutions is performed by reducing the Riemann-Hilbert problems to those with identity jump matrix, i.e., those correspond to reflectionless inverse scattering problems. Finally a special reduction of the original matrix spectral problem will be briefly discussed.

#### Scholar Commons Citation

Gu, Xiang, "Hamiltonian structures and Riemann-Hilbert problems of integrable systems" (2018). *Graduate Theses and Dissertations.*

https://scholarcommons.usf.edu/etd/7677