Date of Award

2014

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Dennis Clougherty

Abstract

The Darboux transformation is one of the main techniques for finding solutions of integrable equations. The Darboux transformation is not only powerful in the construction of muilti-soliton solutions, recently, it is found that the Darboux transformation, after some modification, is also effective in generating the rogue wave solutions. In this thesis, we derive the rogue wave solutions for the Davey-Stewartson-II (DS-II) equation in terms of Darboux transformation. By taking the spectral function as the product of plane wave and rational function, we get the fundamental rogue wave solution and multi-rogue wave solutions via the normal Darboux transformation. Last but not least, we construct a generalized Darboux transformation for DS-II equation by using the limit process. As applications, we use the generalized Darboux transformation to derive the second-order rogue waves. In addition, an alternative way is applied to derive the N-fold Darboux transformation for the nonlinear Schrödinger (NLS) equation. One advantage of this method is that the proof for N-fold Darboux transformation is very straightforward.

Language

en

Number of Pages

57 p.

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