Explore Optical Solitary Wave Solutions of the kp Equation by Recent Approaches

Hammad Alotaibi

doi:10.3390/cryst12020159

Abstract

The study of nonlinear evolution equations is a subject of active interest in different fields including physics, chemistry, and engineering. The exact solutions to nonlinear evolution equations provide insightful details and physical descriptions into many problems of interest that govern the real world. The Kadomtsev–Petviashvili (kp) equation, which has been widely used as a model to describe the nonlinear wave and the dynamics of soliton in the field of plasma physics and fluid dynamics, is discussed in this article in order to obtain solitary solutions and explore their physical properties. We obtain several new optical traveling wave solutions in the form of trigonometric, hyperbolic, and rational functions using two separate direct methods: the (w/g)-expansion approach and the Addendum to Kudryashov method (akm). The nonlinear partial differential equation (nlpde) is reduced into an ordinary differential equation (ode) via a wave transformation. The derived optical solutions are graphically illustrated using Maple 15 software for specific parameter values. The traveling wave solutions discovered in this work can be viewed as an example of solutions that can empower us with great flexibility in the systematic analysis and explanation of complex phenomena that arise in a variety of problems, including protein chemistry, fluid mechanics, plasma physics, optical fibers, and shallow water wave propagation.

Highlights

The study of traveling wave solutions is critical for understanding and quantitatively describing a variety of nonlinear phenomena
Complex phenomena in nature generally involve nonlinear properties that can be characterized by nonlinear partial differential equation (NLPDE) s where solitary and soliton solutions may appear [1,2]
Solitary and soliton waves solutions which are solutions of specific NLPDEs are considered to be a special classes of traveling wave solutions that have distinct features

Summary

Introduction

The study of traveling wave solutions is critical for understanding and quantitatively describing a variety of nonlinear phenomena. We present three expansion methods; namely, the modified ( g0 /g2 )-expansion approach and ( g0 )-expansion approach, and the generalized simple (w/g)-expansion approach to explore new optical solitary wave solutions for nonlinear equations via KP. This subsection aims to explore the traveling wave solutions of Equation (9) using the generalized simple (w/g)-expansion approach. Setting the function w = g0 , and the constants a = −μ, b = −λ and c = −1 in the auxiliary Equation (7), the solution (5) has the following form: g0 g.

Methodology of AKM

Graphical Representations of Traveling Wave Solutions for KP Equation

Conclusions tions for s shas forNPDE