Complex short pulse equation and its integrable discretizations

Authors

Raul M. Guajardo, University of Texas-Pan American

Date of Award

5-2015

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Dr. Baofeng Feng

Second Advisor

Dr. Andras Balogh

Third Advisor

Dr. Virgil U. Pierce

Abstract

In this thesis, we are mainly concerned with the complex short pulse (CSP) equation which was proposed in Physica D [1]. We present the Lax pair of the CSP equation and show the compatibility condition gives the CSP equation. Therefore, the integrability is confirmed. Then, a set of bilinear equations is proposed which yields the CSP equation through the hodograph transformation. Based on the bilinear form, general N-soliton solution is given in determinant form. Regarding two-soliton solution, a bound state can be formed if the velocities of two solitons are equal. Furthermore, a semi- and fully discrete analogues are constructed by using Hirota’s bilinear method and defining discrete hodograph transformations. General N-soliton solutions are also presented for both the semi-discrete and fully discrete versions of the CSP equation. The Lax pair for the semi-discrete CSP equation is found and therefore the Lax integrability is confirmed.

Comments

Copyright 2015 Raul M. Guajardo. All Rights Reserved.

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Granting Institution

University of Texas-Pan American