Solitons and Other Exact Solutions for Two Nonlinear PDEs in Mathematical Physics Using the Generalized Projective Riccati Equations Method

Abstract

We apply the generalized projective Riccati equations method with the aid of Maple software to construct many new soliton and periodic solutions with parameters for two higher-order nonlinear partial differential equations (PDEs), namely, the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity and the nonlinear quantum Zakharov-Kuznetsov (QZK) equation. The obtained exact solutions include kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and periodic solutions. The given nonlinear PDEs have been derived and can be reduced to nonlinear ordinary differential equations (ODEs) using a simple transformation. A comparison of our new results with the well-known results is made. Also, we drew some graphs of the exact solutions using Maple. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.

Optical soliton solutions to the $(2+1)$-dimensional Chaffee–Infante equation and the dimensionless form of the Zakharov equation

“…Closed form solitary wave solutions provide better internal information about those phenomena. Therefore, considerable efforts have been made by many mathematicians and physical scientists to obtain closed form wave solutions of such NLEEs and a number of powerful and efficient methods, such as the Bäcklund transformation method [1], the first integral method [2], the modified simple equation method [3,4], the Exp-function method [5], the (G /G)-expansion method [6], the sine-cosine method [7], the modified Kudryashov method [8], the homogeneous balance method [9], the F-expansion method [10], the variational iteration method [11], the tanh-function method [12], the Adomian decomposition method [13], the projective Riccati equation method [14], the homotopy analysis method [15], and the (G /G, 1/G)expansion method [16] have been developed.…”

The (G′/G)-expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines

“…In the recent years, investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena in such as fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology and so on. Several methods for finding the exact solutions to nonlinear equations in mathematical physics have been presented, such as the inverse scattering method [1], the Hirota bilinear transform method [2], the truncated Painlevé expansion method [3,4], the Bäcklund transform method [5,6], the exp-function method [7][8][9], the tanh-function method [10][11][12], the Jacobi elliptic function expansion method [13][14][15], the (G /G)-expansion method [16][17][18][19][20], the (G /G,1/G)-expansion method [21][22][23], the generalized projective Riccati equations method [24][25][26] and so on.…”

Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity

“…Introduction. In 2018, [7] studied the exact solutions for the following nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity:…”

“…For finding many new exact solutions, equation (1) has been studied in [10] using five different techniques, namely, the G ′ G -expansion method, the improved Sub-ODE method, the extended auxiliary equation method, the new mapping method, and the Jacobi elliptic function method. In [7], the authors applied the generalized projective Riccati equations method to find some new soliton and periodic solutions. Let q(x, t) = φ(ξ)e iQ(x,t) ,…”

“…Because the three papers [9], [10] and [7] did not study the dynamical behavior of system (8). In this paper, it is different from these references that we use the method of dynamical systems to investigate the bifurcations of the phase portraits of system (8) and find all possible exact solutions of system (8) depending on the changes of the parameter group (β, γ, k 1 ) and m. The results of this paper provide more complete understanding for the traveling wave solutions of equation (1).…”