Theses and Dissertations

Date of Award

5-2024

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Bao-Feng Feng

Second Advisor

Sergey Grigorian

Third Advisor

Bingyuan Liu

Abstract

In this thesis, we systematically study a vector complex modified Kordeweg-de Vries equation by combining Hirota's bilinear method and the the Kadomtsev–Petviashvili (KP) reduction method. This vector nonlinear equation is a multi-component generalization of the well-known modified Kordeweg-de Vries (mKdV) equation and can be reduced to the known Hirota equation, Sasa-Satsuma (SS) equation, Sasa-Satsuma-mKdV equation as well as coupled Sasa-Satsuma equation. First, we bilinearize the vector complex mKdV equation under both the zero and nonzero boundary conditions by introducing auxiliary tau functions. Then, starting from two sets of bilinear equations of multi-component KP hierarchy and single-component KP-Toda hierarchy with shifted singular points and the corresponding tau functions satisfying these bilinear equations, we derive the bright and dark soliton solutions via a series of reduction procedures including dimension reduction and complex conjugate reduction. The explicit forms of one- and two-bright and dark soliton solutions are presented and their asymptotic behavior are analyzed. It is quite interesting that the bright soliton can be classified into one-hump, two-hump and oscillated types while the dark soliton composes of regular dark, anti-dark, double hole, Mexican hat and anti-Mexican hat types.

By further imposing conditions on the solutions to the vector complex mKdV equation, the soliton solutions for the Hirota, Sasa-Satsuma, Sasa-Satsuma-mKdV and coupled Sasa-Satsuma equation are derived and are expected to find their physical applications.

Comments

Copyright 2024 Changyan Shi.

https://go.openathens.net/redirector/utrgv.edu?url=https://www.proquest.com/pqdtglobal1/dissertations-theses/study-on-vector-complex-modified-korteweg-de/docview/3085286376/sem-2?accountid=7119

Download

Included in

Mathematics Commons

Share

COinS