The Improvedexp⁡-Φξ-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

Abstract

Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.

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

“…Therefore, the development of 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 improved exp(− (ζ ))-expansion method [1], the improved Kudryashov technique [2], the first integral technique [3], the generalized direct algebraic method [4], the improved Q-expansion method [5], the Jacobian elliptic functions technique [6], and more others. More information on various techniques and analytical solutions, including dark and light solitons, can be found in the references [7][8][9][10][11][12][13][14][15][16][17][18].…”

Simulation of Wave Solutions of a Mathematical Model Representing Communication Signals

“…The solutions obtained as a result of applying these techniques allow commenting on the behavior of mathematical models. Some of them are the (𝐺 ′ 𝐺 ⁄ )-expansion technique and its modifications (Wang et al, 2008;Naher, 2012;Naher and Abdullah, 2013;Akbar et al, 2016;Duran, 2020;, the (1/G')expansion method , sine-Gordon expansion method and (𝑚 + 𝐺 ′ 𝐺) ⁄ -expansion method (Ismael et al, 2020), the improved Bernoulli sub-equation function method Duran et al, 2021), the Riccati-Bernoulli sub-ODE method (Yang et al, 2015), the exp(−ϕ(ξ))-expansion method and its improved forms (Misirli and Gurefe, 2011;Arshed et al, 2019;Chen et al, 2019;Yel et al, 2019;Baskonus, 2021;, the generalized Kudryashov method (Demiray et al, 2015;Mahmud et al, 2017;Rahman et al, 2019), the new function method (Aktürk et al, 2017), the Hirota's bilinear transformation (Hietarinta, 2005), the Backlund transformation method (Hirota and Satsuma, 1977;Lu et al, 2006), rational sine-cosine method (Marwan et al 2011;Qawasmeh and Alquran, 2014) the tanh method and its various extension (Fan, 2000;Elwakil et al, 2005;Yang and Hon, 2006), the tanh-coth expansion method (Wazwaz, 2007a(Wazwaz, , 2007bParkes, 2010), the homotopy perturbation method (He, 2006a(He, , 2006b(He, , 2008Biazar et al, 2009), the simplified Hirota's method (Wazwaz, 2016), the extended sinh-Gordon equation expansion method (Kumar et al, 2018;…”

Solitary Wave Solutions for the Shallow Water Wave Equations and the Generalized Klein-Gordon Equation Using Exp(-ϕ(η) )-Expansion Method

“…Some of these well-known models are, as examples, the exp-functions method [9,10], the modified simple equation method [11], the Jacobi elliptic function expansion method [12,13], the Adomian decomposition method [14], the F-expansion method [15], the homogenous balance method [16], the ( / ) G G  -expansion method [17,18], the novel ( / ) G G  -expansion method [19-21], the new generalized ( / ) G G  -expansion method [22,23] and so on has been used to solve different types of nonlinear systems of partial differential equations (PDEs). Recently, the exp − ( ) -expansion method has become widely applied to construct for traveling wave solutions of nonlinear equations in science and engineering [24][25][26][27]. For example, this method has been utilized to construct traveling wave solutions of the Pochhammer-Chree equation by Nematollah et al [28] and Rashid et al [29] also have used this method for constructing traveling wave solutions of nonlinear evolution equations.…”