Graduation Year

2022

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

Myung Kim, Ph.D.

Committee Member

Mohamed Elhamdadi, Ph.D.

Committee Member

Sherwin Kouchekian, Ph.D.

Committee Member

Arthur Danielyan, Ph.D.

Committee Member

Ivan Rothstein, Ph.D.

Keywords

Nonlocal hierarchy, Nonlocal Sasa-Satsuma, Reverse-spacetime, Riemann-Hilbert problem, Soliton dynamics, Soliton solution

Abstract

For many years, the study of integrable systems has been one of the most fascinating branches of mathematics and has been thought to be an interesting area for both mathematicians and physicists alike.Many natural phenomena can be predicted by using integrable systems, particularly by studying their different solutions, as well as analyzing and exploring their properties and structures. They are commonly found in nonlinear optics, plasmas, ocean and water waves, gravitational fields, and fluid dynamics. Typical examples of integrable systems include the Korteweg-de Vries (KdV) equation, the nonlinear Schrödinger (NLS) equation, and the Kadomtsev-Petviashvili (KP) equation. Solitons are intrinsic solutions for these equations, and various types of solitons can be obtained, such as bright and dark solitons, lump and rogue waves, and breathers.

In the dissertation, we present and investigate a novel nonlocal nonlinear reverse-spacetime Sasa-Satsuma equation, which is a KdV-type equation. Furthermore, we analyze it and determine its Hamiltonian structure. This equation is derived from an AKNS spectral problem involving a nonlocal 5 by 5 matrix. Also, as part of our investigation, we also develop a higher-order nonlocal reverse-time NLS-type equation that originates from the analysis of a local 4 by 4 matrix spectral problem. By using vectors lying in the kernel of the Jost solutions, we can generate soliton solutions for the nonlocal Sasa-Satsuma and the nonlocal NLS-type equations using the Riemann-Hilbert problem with the real line being the contour.

When the reflection coefficients vanish, the jump matrix is taken to be the identity matrix, which provides explicit soliton solutions through the corresponding Riemann-Hilbert problem. This allows us to explore the dynamical behaviors for both equations; the nonlocal reverse-spacetime Sasa-Satsuma equation and the higher-order nonlocal reverse-time NLS-type equation. These dynamical behaviors depend on the configuration of the eigenvalues in the spectral plane and how they are chosen, sometimes under specific conditions.

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