Travelling Wave Solutions for the Modified Korteweg-De-Viress equation and the generalized Nonlinear Schrödinger equation

Abstract

The sine - Gordon equation appears in the propagation of fluxions in Josephson Junctions between two superconductors. It also appears in many scientific fields such as the notions, and dislocation in crystals where sin (u) is due to periodic structure of rows of atom. The term sin (u) is the Josephson current across on insulator between two superconductors. In this thesis we formally drive exact travelling wave solutions for the modified Kortewey - De – Vriss (MKdv – sine - Gordon) equation. The proposed analysis depends mainly on a variable separated ODE method. Two distinct sets of exact solitary wave solutions, that posse’s distinct physical structure, are formally derived for each equation. The derived solutions include other results and introduce entirely new solutions. The capability of extended , and Exp – function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by different methods. To obtain the single – solution for the equation, the extended and sine – cosine method is used. Furthermore, for this nonlinear evolution equation the Exp – function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp – function method provides powerful evolution equations in mathematical physics.