Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

Abstract

In this paper, a direct method called negative power expansion (NPE) method is presented and extended to construct exact solutions of nonlinear mathematical physical equations. The presented NPE method is also effective for the coupled, variable-coefficient and some other special types of equations. To illustrate the effectiveness, the (2 + 1)-dimensional dispersive long wave (DLW) equations, Maccari's equations, Tzitzeica-Dodd-Bullough (TDB) equation, Sawada-Kotera (SK) equation with variable coefficients and two lattice equations are considered. As a result, some exact solutions are obtained including traveling wave solutions, non-traveling wave solutions and semi-discrete solutions. This paper shows that the NPE method is a simple and effective method for solving nonlinear equations in mathematical physics.

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

“…Also, the exact solutions of these equations are crucial to studying the propagation of Rossby waves [8][9][10][11]. So, many methods have been proposed on how to solve the exact solutions to nonlinear equations, for instance, the Hirota method [12][13][14], the Jacobi elliptic function expansion method [15][16][17][18], the G′/G-expansion method [19][20][21][22][23], the Exp (− Φ(ξ))-expansion method [24,25], the generalised exponential rational function method [26][27][28], the negative power expansion method [29], the hyperbolic function expansion method [30][31][32][33], the extended sub-equation method [34], (ω/g)-expansion method [35], the improved sub-ODE method [36], the Riccati-Bernoulli sub-ODE method [37][38][39][40], the Lie symmetry technique [41][42][43][44][45][46][47], the fractional sub-equation [48] etc. Tese are valid methods and tools for computing nonlinear equations.…”

“…Zkan et al used the improved tan (φ/2) expansion method to obtain the exact solution to the (2 + 1)-dimensional KdV equation and explained the infuence of diferent parameters on wave propagation through three-dimensional graphs and tables [69]. Compared to the negative power expansion method [29], the extended subequation method [34], and the improved sub-ODE method [36], we can obtain more formal solutions using the improved tan (φ/2) expansion method, which is one of the efcient mathematical methods and tools that are widely used and easy to implement.…”