Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Wen-Xiu Ma, Ph.D.
Thomas J. Bieske, Ph.D.
Arthur A. Danielyan, Ph.D.
Sherwin Kouchekian, Ph.D.
Seung-Yeop Lee, Ph.D.
Hamiltonian structure, Lump solution, Riemann-Hilbert approach, Soliton hierarchy, Soliton solution
In the first part of this dissertation we introduce two matrix iso-spectral problems, a Kaup-Newell type and a generalization of the Dirac spectral problem, associated with the three-dimensional real Lie algebras sl(2;R) and so(3;R), respectively. Through zero curvature equations, we furnish two soliton hierarchies. Hamiltonian structures for the resulting hierarchies are formulated by adopting
the trace identity. In addition, we prove that each of the soliton hierarchies has a bi-Hamiltonian structure which leads to the integrability in the Liouville sense. The motivation of the first part is to construct soliton hierarchies with infinitely many commuting symmetries and conservation laws.
The second part of the dissertation is dedicated to the investigation of exact solutions to some nonlinear evolution equations. We find lump solutions and lump-type solutions to a (2+1)-dimensional 5th-order KdV-like equation and a (3+1)-dimensional Jimbo–Miwa-like equation, respectively. Moreover, we explore interaction solutions of lump-type solutions with kink solutions and resonance
stripe solitons solutions for the Jimbo–Miwa-like equation. Finally, we consider a Riemann-Hilbert problem for a coupled complex modified-KdV system and present its N-soliton solutions.
Scholar Commons Citation
Batwa, Sumayah A., "Lump Solutions and Riemann-Hilbert Approach to Soliton Equations" (2018). Graduate Theses and Dissertations.