New Soliton Wave Solutions to a Nonlinear Equation Arising in Plasma Physics

Abstract

The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics, physics, and engineering disciplines. This article intends to analyze several traveling wave solutions for the modified regularized long-wave (MRLW) equation using several approaches, namely, the generalized algebraic method, the Jacobian elliptic functions technique, and the improved Q-expansion strategy. We successfully obtain analytical solutions consisting of rational, trigonometric, and hyperbolic structures. The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation. The adaptive moving mesh method evenly distributes the points on the high error areas. This method perfectly and strongly reduces the error. We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used. To better understand the considered equation's physical meaning, we present some 2D and 3D figures. The exact and numerical approaches are efficient, powerful, and versatile for establishing novel bright, dark, bell-kink-type, and periodic traveling wave solutions for nonlinear PDEs.

Constructions of the soliton solutions to coupled nonlinear Schrödinger equation with advanced mathematical techniques

“…Our analysis covers the range of t from 0 to 10 and x from −5 to 3. We transform the coupled Nonlinear Schrödinger (NLS) equation into a system of real and imaginary equations described in (5). We then explore the analytical solutions of this system using the generalized tanh method.…”

“…Nonetheless, nonlinear PDEs, because of their complexity, frequently require simpler analytical solutions. Traveling wave methods involve employing specific strategies to identify exact solutions for particular PDEs by focusing on solutions that display distinct traveling wave characteristics, such as the Kudryashov approach, 3,4 the improved Q-expansion strategy, 5 the ( G ′ G )-expansion method, 6 the ( G ′ G ′ + G+ A )-expansion technique, 7 and more. These techniques help convert a given partial differential equation (PDE) into a more straightforward ordinary differential equation (ODE) that can be easily solved.…”

Optical Solitons with Differential Group Delay and Inter-Modal Dispersion Singlet

“…In this section, the extended form of the simple equation method (ESEM) is introduced to obtain the traveling wave solutions [5][6].…”

“…The traveling wave approach involves employing specific strategies to identify exact solutions for particular NLEEs by focusing on solutions that display distinct traveling wave characteristics. Examples include Kudryashov's approach [3][4], the improved Q-expansion strategy [5], the sine-Gordon expansion method [6], the modified simple equation method [7], the generalized exponential rational function method [8], the Riccati equation method [9], the auxiliary equation method [10], the unified method [11], the improved F-expansion technique [12], the exp(-ζ(ξ)) expansion technique [13], the Khater method [14], and the homotopy analysis transform method (Hatm) [15].…”

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

“…Therefore, developing fundamental and systematic methods for deriving analytical solutions to PDEs has become a popular and fascinating subject for most scholars. Among these techniques, we propose the Kudryashov approach [1,2], the improved Q-expansion strategy [3,4], the G G -expansion method [5], and the Jacobi elliptic expansion [6,7]. These methods are handy for transforming a given PDE into a more straightforward ordinary differential equation (ODE) that can be more easily solved.…”