The Characteristic Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation

Abstract

In this paper, the characteristic function method is applied to seek traveling wave solutions of nonlinear partial differential equations in a unified way. We consider the Wu-Zhang equation (which describes (1 + 1)-dimensional dispersive long wave). The equations governing the wave propagation consist of a pair of non linear partial differential equations. The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations which is solved via the shooting method, coupled with Runge-kutta scheme. The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions. As an illustrative example, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.

Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

“…Many authors proposed various methods to solve the Wu-Zhang system numerically. We summarize as follows: the first integral method [16], extended tanh-function method [17], characteristic function method [18], modified Conte's invariant Painlevé expansion method and truncation of the WTC's approach [19][20][21], elliptic function rational expansion method [22], generalized extended tanh-function method [23], generalized extended rational expansion method [24], and so on.…”

“…However, different from the aforementioned methods [16][17][18][19][20][21][22][23][24], in this paper, we apply the dynamical system method to study the bifurcation and exact solutions of NPDEs. Dynamical system method is quite different from the mentioned methods, and it has many successful applications [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].…”

Invariant subspaces, exact solutions and classification of conservation laws for a coupled (1+1)-dimensional nonlinear Wu-Zhang equation

“…On the other hand, conservation laws (Cls) [10][11][12] seem interesting and helpful to researchers in understanding the chemical, physical and biological behavior of nonlinear models. In the last few decades, lots of mathematicians and scientists have devoted their studies extensively to the dynamic behaviors of PDEs [13][14][15][16][17][18][19][20] using different ideas. The coupled nonlinear Wu-Zhang equation reads [13][14][15][16][17]:…”

“…In the last few decades, lots of mathematicians and scientists have devoted their studies extensively to the dynamic behaviors of PDEs [13][14][15][16][17][18][19][20] using different ideas. The coupled nonlinear Wu-Zhang equation reads [13][14][15][16][17]:…”

Non-Traveling Wave Solutions for the (1 + 1)-Dimensional Burgers System by Riccati Equation Mapping Approach

“…In the past few decades, many significant methods have been presented such as Bäklund transformation, Darboux transformation, the extended tanh-function method, and the F-expansion method, Lie group analysis, homogeneous balance method, Jacobi elliptic function method, and the mapping method, etc. [9][10][11][12][13][14][15]. The mapping approach is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations, the remarkable characteristic of which is that we can have many different ansatzs and therefore, a large number of solutions.…”