Exact Traveling Wave Solutions of the Gardner Equation by the Improved tanΘϑ-Expansion Method and the Wave Ansatz Method

Abstract

Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved tanΘϑ-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.

Abundance of Exact Solutions of a Nonlinear Forced (2 + 1)-Dimensional Zakharov–Kuznetsov Equation for Rossby Waves

“…Manafan et al proposed a new method to solve nonlinear partial diferential equations, namely, the improved tan (φ/2) expansion method [49]. With the help of this method, many classical nonlinear partial diferential equations have been investigated and abundant exact solutions have been obtained [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69]. Mohyud-Din and Irshad used this method to construct an exact solution for the generalised KP equation and explained that it can provide better help for the study of generalised KP equations [60].…”

On some soliton structures for the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in mathematical physics

“…Many physical processes in the nonlinear sciences, including signal processing, condensed matter physics, acoustics, theoretical physics, and optical continuum creation, depend on solitonic behavior [14, 15]. The soliton solutions of several types of NLSEs have been effectively described by numerous computer analyses over the past couple of years, including the extended sinh‐Gordon equation expansion method [16], the extended Jacobi elliptic approach [17], the modified auxiliary equation mapping method [18], the modify extended direct algebraic technique [19], the inverse scattering transformation method [20], extended rational sine‐cosine method [21], the semi‐inverse variational principle [22], the generalized exponential rational function method [23, 24], the improved $$$ \tan \left(\phi \right) $$$‐expansion method [25], the extended sinh‐Gordon expansion method [26], and many more [27–30].…”