Some new solutions of the (1+1)-dimensional PDE via the improved (G′/G)-expansion method

Abstract

Abstract. In this article, an improved '/ G G -expansion method is implemented for the simplified Modified CamassaHolm (MCH) equation involving parameters, with an aim to construct many new traveling wave solutions. In this method, second order linear ordinary differential equation with constant coefficients has been implemented as an auxiliary equation. The generated solutions including solitons and periodic solutions are demonstrated by the hyperbolic function, the trigonometric function and the rational forms. If the parameters take particular values, the solutions become in special functional form. Moreover, it is worth mentioning that, some of our solutions are in good agreement with already published results in the open literature by setting appropriate values of constants, which proves our other solutions. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple 13.

Exact soliton solutions of the generalized Gross-Pitaevskii equation based on expansion method

“…We are particularly interested in the bright soliton solution given in (25a) for μ < 0 and c 0 = −1. In this case, with k(t) = k, g(t) = g as constants, from previous analysis (19a) and (23), it is not hard to see that p(t)σ (t) = κ is constant, q(t) = t( is constant), the solution (25) in this situation is of bright soliton type(|ψ(x, t)| ∼ sech 1/γ ( √ −μ(κ x + t − ξ 0 ))) moving with constant speed /κ. When g(t)(or k(t))depends on t, p(t)σ (t), which indicates the compressing ratio of the travelling wave along the x-direction, generally is not constant and the format of q(t)/(p(t)σ (t)) is complicated, which mean that there is shape variation in the traveling wave and the wave peak move with acceleration.…”

“…[24][25][26][27][28][29][30][31][32] Our problem solving strategy is specially tailored to the particular features of GGPE under investigation, while many previous efforts aiming at finding exact analytical solution for GPE are confined to somewhat subcategory of the work examined here in this paper. For example, one typical work focus on the nonlinear Schrödinger equation of GPE type without external harmonic potential, and although not stated explicitly in the work, 6 there are relationship formula connecting the variable coefficients of the equation terms, the interdependence is necessary in order to ensure the generation of consistent ODEs that are derived from G /G-expansion method of original version.…”

Exponential B-spline collocation solutions to the Gardner equation

“…Projective Riccati equations also enable to generate some hyperbolic type solitary wave and periodic solutions expressed in the finite series form [9]. These type solutions can also be constructed by generalized form of the (G /G)−expansion [10,11,12], exp − function [13] methods. A bunch of traveling wave type exact solutions to the Gardner equation are determined by using various hyperbolic ansatzes [14].…”

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

“…Till now, many researchers have investigated the solutions for KdV‐mKdV equation. The notable works of Naher and Abdullah , Bekir , Triki et al . , Gómez Sierra et al .…”

“…Till now, many researchers have investigated the solutions for KdV-mKdV equation. The notable works of Naher and Abdullah [30], Bekir [31], Triki et al [32], Gómez Sierra et al [33], Zhang and Tian [34], Krishnan and Peng [35], Yang and Tang [36], Lu and Shi [37] are involving the exact travelling wave solutions for KdV-mKdV equation.…”