Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov–Kuznetsov equation using the modified simple equation method

Abstract

Akbar (2014) Exact solutions of the (2+1)dimensional cubic Klein-Gordon equation and the (3+1)-dimensional Zakharov-Kuznetsov equation using the modified simple equation method,

New Hyperbolic Function Solutions for Some Nonlinear Partial Differential Equation Arising in Mathematical Physics

“…in which α is constant and the (2 + 1)-dimensional nonlinear cubic Klein-Gordon equation (cKGE) [15] defined by: …”

“…in which , α β coefficients are constants and not zero [15]. When it comes to convert Equation (38) into NLODE, we can perform the travelling wave transformation: 2 2 2 2 2 22 2 2 2 , ,…”

Elastic collision of mobile solitons of a (3 + 1)-dimensional soliton equation

“…From the physical point of view, the propagation of nonlinear localized waves in multidimensional system is an exciting and important task that received much attention during the last decades. The typical example of higher-dimensional NLPDEs is the (2 + 1)-dimensional cubic Klein-Gordon equation [1], (3 + 1)-dimensional Zakharov-Kuznetsov equation [2], the (2 + 1)-dimensional Kadomtsev-Petviashvili (KP) equations [3], (3 + 1)-dimensional Boussinesq equations [4], (3 + 1)-dimensional Burgers equation [5], (2 + 1)-and (3 + 1)-dimensional nonlinear Schrödinger equations [6,7], etc. Solving such nonlinear equations may inculcate to know the profound dynamical process, and sometimes it leads to know some facts that are not simply understood through common observations.…”

New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures

“…Especially various type travelling wavesolutions such as dark, complex, elliptic, Jacobi elliptic, exponential, rational, hyperbolic and trigonometric function solutions means that they have new properties of physical problems. In the process many powerful methods such as sumudu transform method, Riccati-Bernoulli sub-ODE method, GG  -expansion method, Exp-function method, Fitted finite difference method, extended jacobi elliptic function expansion method, modified simple equation method and Generalized Bernoulli Sub-ODE method, functional variable method, variational iteration method, improved Bernoulli sub-equation function method, Laplacevariationaliteration method, finite difference method, generalized Kudryashov method and so on have been used to find new solutions of nonlinear evolution equations [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In the rest of this paper, we present the general properties of the sine-Gordon expansion method(SGEM) in comprehensive manner in section 2.…”