New extended generalized Kudryashov method for solving three nonlinear partial differential equations

Abstract. New extended generalized Kudryashov method is proposed in this paper for the first time. Many solitons and other solutions of three nonlinear partial differential equations (PDEs), namely, the (1+1)-dimensional improved perturbed nonlinear Schrödinger equation with anti-cubic nonlinearity, the (2+1)-dimensional Davey–Sterwatson (DS) equation and the (3+1)-dimensional modified Zakharov–Kuznetsov (mZK) equation of ion-acoustic waves in a magnetized plasma have been presented. Comparing our new results with the well-known results are given. Our results in this article emphasize that the used method gives a vast applicability for handling other nonlinear partial differential equations in mathematical physics.

The objective of this article is to use a new extended generalized Kudryashov method, for the first time, to construct new exact solutions of the following three nonlinear partial differential equations (PDEs).
(I) The (1 + 1)-dimensional improved perturbed nonlinear Schrödinger equation with anti-cubic nonlinearity [17]: where i = √ −1, a, b, b 1 , b 2 , b 3 , α, λ and υ 1 are real constants. The independent variables x and t represent spatial and temporal variables, respectively. The dependent variable E(x, t) is the complex valued wave profile for the (1 + 1)-dimensional improved perturbed nonlinear Schrödinger equation with anti-cubic nonlinearity. Here the coefficients a and b represent the improved term that introduces stability to the NLS equation and the usual group velocity dispersion (GVD), respectively. The nonlinearities stem out from the coefficients of b j for j = 1, 2, 3, where b 1 gives the effect of anti-cubic nonlinearity, b 2 is the coefficient of Kerr law nonlinearity, and b 3 is the coefficient of pseudo-quintic nonlinearity, respectively. The parameters α and λ represent the intermodal dispersion and the self-steepening perturbation term, respectively. Finally, υ 1 is the nonlinear dispersion coefficient. If b 1 = 0, there is no anti-cubic nonlinearity. which has been discussed in [17] using the soliton ansatz method.
(II) The (2 + 1)-dimensional Davey-Sterwatson (DS) equation [19,26,35,45]: where λ is a real constant. The case σ = 1 is called the DS-I equation, while σ = i is the DS-II equation. The parameter λ characterizes the focusing or defocusing case. The Davey-Stewartson I and II are two well-known examples of integrable equations in two space dimensions, which arise as higher dimensional generalizations of the nonlinear Schrodinger (NLS) equation [19]. They appear in many applications, for example, in the description of gravity-capillarity surface wave packets in the limit of the shallow water. Davey and Stewartson first derived their model in the context of water waves from purely physical considerations. In the context, u(x, y, t) is the amplitude of a surface wave packet, while v(x, y) represents the velocity potential of the mean flow interacting with the surface wave [19]. Equation (2) has been discussed in [35] using the numerical schemes method, in [45] -using the homotopy perturbation method, in [19] -using the multiple scales method and in [26] -using the first-integral method.
(III) The (3 + 1)-dimensional modified ZK equation of ion-acoustic waves in a magnetized plasma [36]: where c is a positive real constant. Equation (3) has been discussed by Munro and Parkes [36], where they showed that if the electrons are nonisothermal, then the govering equation of the ZK equation is a modified form refereed to as the mZK equation (3), also they showed that the reductive perturbation method leads to a modified Zakharov-Kuznetsov (mZK) equation. This article is organised as follows: In Section 2, we give the description of a new extended generalized Kudryashov method for the first time. In Sections 3, 4 and 5, we solve Eqs. (1), (2) and (3) using the proposed method described in Section 2. In Section 6, the graphical representations for some solutions of Eqs. (1), (2) and (3) are plotted. In Sections 7, conclusions are illustrated. To our knowledge, Eqs. (1), (2) and (3) are not discussed before using the proposed method obtained in the next section.

Description of a new extended generalized Kudryashov method
Consider the following nonlinear PDE: where P is a polynomial in u and its partial derivatives in which the highest-order derivatives and the nonlinear terms are involved. According to the well-known generalized Kudryashov method [25,30,41] and with reference to [38], we can propose the main steps of a new extended generalized Kudryashov method for the first time as follows: Step 1. We use the traveling wave transformation u(x, y, z, t) = u(ξ), ξ = l 1 x + l 2 y + l 3 z − l 4 t, where l 1 , l 2 , l 3 and l 4 are a nonzero constants, to reduce Eq. (4) to the following nonlinear ordinary differential equation (ODE): where H is a polynomial in u(ξ) and its total derivatives u (ξ), u (ξ) and so on, where = d/dξ.
Step 2. We assume that the formal solution of the ODE (5) can be written in the following rational form: where such that α s and β m = 0 and http://www.journals.vu.lt/nonlinear-analysis where exp a (pξ) = a pξ and p is a positive integer. The function Q is the solution of the first-order differential equation From (6) and (8) we have and so on.
The obtained solutions will be depended on the symmetrical hyperbolic Fibonacci functions given in [1] and [37]. The symmetrical Fibonacci sine, cosine, tangent and cotangent functions are defined as 3 On solving Eq. (1) (1) (1) using the new extended generalized Kudryashov method In this section, we use the above method describing in Section 2 for solving Eq. (1). To this aim, we assume that Eq. (1) has the formal solution where ψ(ξ) and χ(ξ) are real functions of ξ, while ω, k and are real constants. Substituting (12) into Eq. (1) and separating the real and the imaginary parts, we have the two nonlinear ODEs: and To solve the above coupled pair of Eqs. (13) and (14), we introduce the ansatz: where β and γ are constants. Inserting (15) into Eq. (14), we obtain Substituting (15) along with (16) into Eq. (13), we have the nonlinear ODE where the coefficients A 1 , A 2 , A 3 and A 4 are given by , where g(ξ) is a positive function of ξ. Substituting (18) into (17), we have the new equation By balancing gg with g 4 in (19), the following formula is obtained: Let us now discuss the following cases.
provided A 4 < 0. In this case, from (7), (12), (18), (26) and (32) we deduce that Eq. (1) has the solution Equation (1) has the symmetrical Fibonacci cotangent function solutions and From (33) we deduce that Eq. (1) has the dark soliton solution and from (34) we deduce that Eq. (1) has the singular soliton solution provided A 4 < 0 and A 3 > 0. Simliarly, we can find many other solutions by choosing another values for s, m and p.

On solving Eq. (2) (2) (2) using the new extended generalized Kudryashov method
In this section, we use the above method describing in Section 2 for solving Eq. (2). To this aim, we assume that Eq. (2) has the formal solution and where u(ξ), η(x, y, t) and v(ξ) are all real functions, while α, k and l are real constants. Substituting (35) and (36) into Eq. (2) yield the following system of ODEs: Integrating (38) with respect to ξ, we obtain where ε is the constant of integration, and α 2 = ±1/4. Substituting (39) into (37), we have By balancing u with u 3 in (40), the following formula is obtained: Let us now discuss the following cases.
Case 1. If we choose p = 1 and m = 1, then s = 2. Thus, we deduce from (6) that where α 0 , α 1 , α 2 , β 0 and β 1 are real constants to be determined such that α 2 and β 1 = 0. Substituting (41) along with (8) into Eq. (40), collecting the coefficients of each power of Q i (i = 0, 1, . . . , 6) and setting each of these coefficients to zero, we obtain a system of algebraic equations, which can be solved by Maple, we obtain the following results.