Traveling Wave Solutions of the Extended Calogero-Bogoyavlenskii-Schiff Equation

Traveling wave solutions of the extended Calogero–Bogoyavlenskii–Schiff equation (ECBS) is investigated. The (G′ G ⁄ )-expansion method is applied to find the explicit solutions. New traveling wave solutions of the ECBS equation are obtained. The solitary wave solutions show one-soliton, N-solitons, kink and periodic waves. Some of the resulting solutions are plotted. Keywords— Extended Calogero-Bogoyavlenskii-Schiff equation; (G′ G ⁄ )-expansion method; Traveling wave solutions.


INTRODUCTION
Nonlinear integrable equations are playing a substantial role in modeling most of the scientific and engineering applications [1][2][3][4][5], such as propagation of shallow water waves, optical fibers, condensed matter, electromagnetic, plasma and fluid mechanics. The study of higher dimensional nonlinear integrable equations is one of the most important recent studies [6][7][8]. Large assortments of mathematical methods of solutions are demoralized in studying these equations, some of these methods are the ( ′⁄ )-expansion method [9][10][11], exponential function method [12,13], Lax pairs [14,15], Extended homoclinic test [16,17], Hirota bilinear method [18,19], Darboux transformation [20,21], sine-cosine and tanh-coth methods [22]. Bogoyavlenskii and Schiff described the interaction of a Riemann waves along two spatial dimensions, by the nonlinear integrable equation Calogero-Bogoyavlenskii-Schiff (GBS) equation [23,24]. Riemann waves' dynamics is one of the most important applications of physics and engineering; such as tsunami and tidal in rivers, magentosound waves in plasmas, internal waves in oceans and optical tsunami in fibers.
Song et al. [25], introduced an integrable (2 + 1)dimensional equation. This equation describes the interaction between Riemann wave promulgated along the y-axis and long wave promulgated along the x-axis as; is a function of space variables x, y and temporal variable t. Equation (1) is also obtained by unifying two directional generalization of the potential KdV equation and Calogero-Bogoyavlenskii-Shiff equation [26][27][28]. The explicit N-soliton solutions of equation (1) is obtained in [25]. Wazwaz [18], extended equation (1)  = 0 (2) He in [18], showed that the extension term does not disturb the integrability of the Song's equation (1), and found its Nsoliton solutions using the simplified Hirota's method.
The purpose of this paper is to find the traveling wave solutions of the extended Calogero-Bogoyavlenskii-Schiff equation (2), using the ( ′⁄ ) expansion method. The paper is organized as follows: Section II, is devoted to summarize the ( ′⁄ ) expansion method. In section III, the ( ′⁄ ) expansion method is applied to the extended Calogero-Bogoyavlenskii-Schiff equation. Section IV, is devoted to discuss the results. The paper ends with conclusions in Section V.
II. MATHEMATICAL METHOD ( ′⁄ )-expansion method is excessively used in finding the traveling wave solutions of nonlinear evolution equations [29][30][31][32]. It can be summarized as; for any partial differential equation (PDE); P(u, ux, uy, ut, uxx, uxy,…) = 0 (3) Where p is a polynomial in u and its partial derivatives. Suppose the solution of the partial differential equation The constant c is the velocity of the travelling wave. The PDE (3) is reduced to a nonlinear ODE; which can be integrated many times with setting the constants of integration equal to zero for simplicity. Consider the traveling wave solution of the final ODE in the form; Where G = G() satisfies the second order linear ODE; Where, ′ = , ′′ = 2 2 , ai ,  and µ are real constants to be determined. The positive integer m is determined through balancing the highest order linear and nonlinear terms' derivatives appearing in the ODE. Substitute (5) and (6) into the final ODE, then collect all terms with the same order of ( ′⁄ ) and set each coefficient to zero yield a set of algebraic equations for ai, c, µ and .

V. CONCLUSIONS
In this paper, the ( ′⁄ )-expansion method is effectively employed to reveal many explicit solutions for the extended Calogero-Bogoyavlenskii-Schiff equation. The presented solutions include a variety of one-soliton, kink, periodic and N-soliton solutions. These aggregations of traveling wave solutions are applicable for different conditions. The results help in studying the wave behaviors in several applications as, deep oceans and nonlinear optics.