Graduation Year


Document Type




Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Wen-Xiu Ma, Ph.D.

Co-Major Professor

Wei Chen, Ph.D.

Committee Member

Leslaw Skrzypek, Ph.D.

Committee Member

Yuncheng You, Ph.D.

Committee Member

Sherwin Kouchekian, Ph.D.


Soliton hierarchy, Hamiltonian structure, bilinear form, lumps, complexitons, algebro-geometric solutions


In chapter 2, we study two Kaup-Newell-type matrix spectral problems, derive their soliton hierarchies within the zero curvature formulation, and furnish their bi-Hamiltonian structures by the trace identity to show that they are integrable in the Liouville sense. In chapter 5, we obtain the Riemann theta function representation of solutions for the first hierarchy of generalized Kaup-Newell systems.

In chapter 3, using Hirota bilinear forms, we discuss positive quadratic polynomial solutions to generalized bilinear equations, which generate lump or lump-type solutions to nonlinear evolution equations, and propose an algorithm for computing higher-order lump or lump-type solutions. In chapter 4, we study mixed exponential and trigonometric wave solutions (called complexitons) to general bilinear equations, and propose two methods to find complexitons to generalized bilinear equations. We also succeed in proving that by choosing suitable complex coefficients in soliton solutions, multi-complexitons are actually real wave solutions from complex soliton solutions and establish the linear superposition principle for complexion solutions.

In each chapter, we present computational examples.

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Mathematics Commons