Extended generalized $$(Zakh\frac{G^{\prime }}{G})$$ ( Z a k h G ′ G ) -expansion method for solving the nonlinear quantum Zakharov–Kuznetsov equation

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

“…is a unknown function, P is a polynomial in V and its partial derivatives in which the highestorder derivatives and nonlinear terms are involved. Let us now give the main steps of the (G'/G)-expansion method [16][17][18][19][20]:…”

“…Step 5. It is well-known that Equations (2.5)-(2.7) have many families of solutions obtained in [16][17][18][19][20].…”

“…With reference to solving Equation (2.6) [20], we deduce that the Jacobi elliptic functions solutions and other exact solutions of Equation (1.1) as follows:…”

“…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.…”

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

“…Nonlinear waves appear in various scientific fields, especially in physics such as fluid mechanics, plasma physics, optical fibers, and solid state physics. In recent years, many powerful tools have been established to determine soliton and periodic wave solutions of nonlinear PDEs, such as the ( / )-expansion method [1][2][3][4][5][6], the extended auxiliary equation method [7,8], the new mapping method [9][10][11], the generalized projective Riccati equations method [12][13][14][15][16][17], and the ( / , 1/ )-expansion method [18]. Conte and Musette [12] presented an indirect method to find solitary wave solutions of some nonlinear PDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation [19].…”