A Mathematical Study of Nonlinear Dispersive Partial Differential Equations for Solitons

Abstract

An effort has been made to find out the exact solitary wave solutions of several PDEs. These partial differential equations are globally accepted as the governing equations of the fields they are related to. Finding their new exact solutions is not only exciting mathematically. It is also a service towards a better understanding of the phenomena themselves. In recent decades, nonlinear wave phenomena and their governing equations have attracted great attention. They arise in several fields of science and engineering including hydrodynamics, condensed matter physics, Bose-Einstein-condensation, nonlinear optics, Josephson junctions, biophysics, field theory etc. To solve these partial differential equations (PDEs), some methods and tools have been developed in the past. This work is mainly concerned with a specific class of exact solutions, called the solitary wave solutions. These types of solutions are of great interest for scientists and engineers working in the fields of fiber optics and wave propagation etc. We have found new soliton solutions to some well-known nonlinear PDEs such as the , Camassa-Holm-ΚP equations, generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation, generalized Zakharov-Kuznetsov(ZK) equation, perturbed nonlinear Schrödinger’s equation in the form of Kerr law nonlinearity(NLSE) equation (2+1)dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) wave equation, (3+1)-dimensional generalized Kadomtsev-Petviashvili (KP) equation, modified Kortewgde Vries-Zakharov-Kuznetsov (KdV-ZK) equation, fourth order Boussinesq equation, Landau-Ginburg-Higgs equation, Cahn-Allen equation, two-mode modified Korteweg-de Vries (TmKdV) equation and nonlinear Schrödinger equation. These new solutions are more generalized and are supposed to give better simulations of the real-world problems to which the PDEs correspond. It is also expected from our study that the computer simulators used to mimic the wave propagation can be enhanced on the basis of our exact solutions. We have used some transformations to convert the PDEs, at hand, to the corresponding ODEs. To extract solitary wave solutions to these PDEs, certain balancing principles are implemented. The shapes and behaviors of these solutions are simulated graphically with the help of the mathematical software Maple. Descriptions of the parameters and the values that have been used to simulate the waves are also provided. We have not only determined some totally novel solutions to the equations under consideration, but also, we have managed to generalize many already existing solutions.