The Multiple Exp-Function Method and the Linear Superposition Principle for Solving the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation

Abstract

In this article, the multiple exp-function method and the linear superposition principle are employed for constructing the exact solutions and the solitary wave solutions for the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. With help of Maple and by using the multiple exp-method, we can get exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions. Furthermore, we apply the linear superposition principle to find n-wave solutions of the CBS equation. Two cases with specific values of the involved parameters are plotted for each two-wave and three-wave solutions.

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

“…where 1,2 (x, t) are the two components of the seed solution satisfying Equations (14) and (15), upon substituting 1 for 1 and 2 for 2 .…”

“…4 This was possible due to cyclic identities satisfied by the Jacobi elliptic functions where the nonlinear cross terms reduce to, and combine with, other linear terms. 5 Other examples include time modes of nonlinear systems, 6 Einstein nonlinear electrodynamics (NLE) equations, 7 nonlinear gas governing equations, 8 the Novikov-Veselov equation, 9 Maxwell-Schrödinger equations, 10 the (2 + 1)-dimensional KdV equations, 11 Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation and the (2 + 1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation, 12 the (2 + 1)-dimensional Sawada-Kotera (SK) equation, 13 the (3 + 1)-dimensional Jimbo-Miwa (JM) equations, 14 the (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation, 15 higher order NLSE, 16 coupled nonlinear Klein-Gordon and Schrödinger equations, 17 the (2 + 1)-dimensional modified Zakharov-Kuznetsov (ZK) equation and the (3 + 1)-dimensional KP equation, 18 the (2 + 1)-dimensional ZK equation and the Davey-Stewartson (DS) equation, 19 generalized KdV equation, the Oliver water wave equation, the k(n; n) equation, 20 the fifth-order KdV equation, 21 the cubic-quintic NLSE, 22 Hirota bilinear equations, 23,24 and the (2 + 1)-dimensional Boussinesq equation. 25 The main property that allows for the application of superposition principle to all of the above-mentioned nonlinear systems is the reduction of the nonlinear cross terms into linear ones, which then combine with other linear terms.…”

Multiple rogue wave solutions and the linear superposition principle for a (3 + 1)‐dimensional Kadomtsev–Petviashvili–Boussinesq‐like equation arising in energy distributions

“…-expansion method, 7 Hirota's bilinear method, [8][9][10][11][12][13][14][15] He's variational principle, 16,17 binary Darboux transformation, 18 Lie group analysis, 19,20 Bäcklund transformation method, 21 and the multiple exp-function method. [22][23][24][25][26] Moreover, many powerful methods have been employed to investigate the new properties of mathematical models which are symbolizing serious real-world problems. [27][28][29] Originally, the Kadomtsev-Petviashvili (KP) equation introduced by Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili to describe the evolution of the nonlinear and long waves with small and slow dependence on the transverse coordinate [30][31][32] as follows:…”

Solitons and Other Solutions for Two Higher-Order Nonlinear Wave Equations of KdV Type Using the Unified Auxiliary Equation Method

“…In recent years, quite a few methods for constructing explicit and solitary wave solutions of these nonlinear evolution equations have been presented. A variety of powerful methods, such as the semi-inverse variational method [1][2][3][4][5], the exp-function method [6][7][8], the Jacobi elliptic function method [9][10][11][12], the (G /G)expansion method [13,14], the Kudryashov method [15][16][17], the multiple exp-function method [18,19], the modified simple equation method [20][21][22], the auxiliary equation method [23,24], the extended auxiliary equation method [25][26][27][28], the soliton ansatz method [29][30][31][32][33][34][35], the traveling wave hypothesis [36], the unified auxiliary equation method [37,38], the conformable fractional derivatives [39][40][41], the Collocation finite element method [42] and so on.…”