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

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 disper-sive 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 Rungekutta 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.


Introduction
Exact traveling wave solutions of nonlinear partial differential equations (NLPDEs) have been of a major concern for both mathematicians and physicists.Many efforts have been made on the study of NLPDEs.Long wave in shallow water is a subject of broad interests and has a long colorful history.Physically, it has a rich variety of phenomenological manifestation, especially the existence of waves permanent in form and robust in maintaining their entities through mutual interaction and collision as well as the remarkable property of exhibiting recurrences of initial data when circumstances should prevail.These characteristics are due to the intimate interplay between the roles of nonlinearity and dispersion.Mathematically, it has been noted that validity of theoretical models critically epends on the domain of underlying key parameters which characterize the specific motions to be modelled.
In 1985 Seshadri et al. [11] put the base-stone of the characteristic function method which was after then applied to solve a variety of nonlinear boundary value problems.
Lastly, Abd-el-malek et al. [12,13] applied the characteristic function method to solve some types of nonlinear partial differential equations which proved to be a powerful tool to be used.
In this present work, we shall derive the solutions of (1 + 1)-dimensional dispersive long wave equation via the characteristic function method, which depends on the application of a one-parameter group transformation to the system of the partial differential equations.These transformations reduce the number of independent variables in the system of partial differential equations by one.Our aim is to make use of the characteristic function method to represent the system of non-linear partial differential equations in the form of a system of ordinary differential equations which will be solved numerically using the shooting method, coupled with Runge-Kutta scheme.

Formulation of the Problem
Mathematically, it has been noted that validity of theoretical models critically depends on the domain of underlying key parameters which characterize the specific motions to be modelled.In [14], Wu and Zhang derived three sets of model equations for modelling nonlinear and dispersive long gravity waves travelling in two horizontal directions on shallow waters of uniform depth.Their comparative study of these models is directed to explore the intrinsic properties in physical and mathe-matical terms that these models possess.Omitting the higher order terms, one of these equations, Wu-Zhang (WZ) equation [14], can be written as: where w is the elevation of the water wave, u is the surface velocity of water along x-direction and v is the surface velocity of water along y-direction.The system (1)-( 3) can be reduced to (1 + 1)-dimensional dispersive long wave equation [14].
  where w is as previously defined, v is now is the surface velocity of water along x-direction.The system ( 4) and ( 5) can be traced back to the works of Broer [15], Kaup [16], Martinez [17], Kupershmidt [18], etc.A good understanding of all solutions of Equations ( 4) and ( 5) is very helpful for coastal and civil engineers to apply the nonlinear water wave model in a harbor and coastal design.Therefore, finding more types of exact solutions of Equations (4) and ( 5) is of fundamental interest in fluid dynamics.There is an amount of papers devoted to these equations [19][20][21].We will seek the solution of Equations ( 4) and ( 5) considering the following initial and boundary conditions: where α(x) and β(x) are functions of x and f 0 (t) and f 1 (t) are functions of t to be specified.

The Group Systematic Formulation under the Following Transformations
Considering the infinitesimal transformations in (t, x, v, w) given by , , , q q Q t x v w q q q q q q q q q r r O r r R t x v w r r r r r r r r r , , , , , , , , , q q q q q q q r r O r r R Q t x v w q q q q q q q q O    where i, j and k stand for t, x and A, B, M, N, are the infinitesimals of the group of transformations.Hence Equations ( 4) and (5) becomes: , , , , , , To calculate the prolongation of a given transformation, we need to differentiate Equations ( 9) and (10) with respect to each of the variables t, x.To do this we introduce the following total derivatives: , And where where i, j and k stand for x and t.

The Invariance Analysis
Under the infinitesimal group of transformation, the system of differential Equations ( 4) and (5) on the form G i = 0, for i = 1, 2, will be invariant if where D is the total derivative defined by: The operator D can be rewritten as: Invariance of Equations ( 9) and (10) under Equation ( 16) gives: We conclude that the system of Equations ( 4) and ( 5) will be invariant under an infinitesimal group of transformations.Therefore, it is now possible to reduce the number of independent variables by one.
The characteristic system of Equations ( 4) and ( 5) is given by: where A 1 , B 1 , B 2 and B 3 are real constants.

The Reduction to Ordinary Differential Equations
Our aim is to make use of CFM to represent the problem in the form of ordinary differential equations.Then we have to proceed our analysis to complete the transformation in the next section in three Cases ( and Case (2) respectively.

Experiments and Results
In this section similarity transformation can be obtained by solving the characteristic system (19) which gives the similarity transformation for the dependent and independent variables.

Case (1)
In Equation (19), assuming A 1 = B 1 = B 2 = 1.0 and B 3 = 0, then we get the following system: Solving these system leads to: where η(x,t) is the similarity variable of the independent variables.And the similarity functions of the dependent variables are:   where F 1 (η) refer to the surface velocity and F 2 (η) refer to the elevation of the water wave and they are arbitrary functions of η.Substituting the self-similar variables v and w from the Equations ( 22) and (23) into Equations ( 4) and (5) yields a system of ordinary differential equations in the similarity variable η: in Cases (1.1) and (1.2) as follow.

Case (1.1)
Solving Equations ( 24) and (25), under the following boundary conditions: Equations ( 24) and ( 25) are solved numerically applying the shooting method coupled with Runge-kutta scheme using two groups of initial conditions illustrated The obtained results are presented diagrammatically in (Figure 1).
In addition to, the boundary conditions which illustrated in the figures.The obtained results are presented diagrammatically in (Figure 2).

Case (2)
In Equation ( 19), assuming B 1 = B 2 = B 3 = 0 and A 1 ≠ 0, then we get the following system: In this case, similarity transformation is given as: Substituting F 1 (η) and F 2 (η) from Equations ( 30), (31) into Equations ( 4) and ( 5) gives: The numerical solutions under the following boundary conditions: In addition to, the boundary conditions which illustrated in the figures.The obtained results are presented diagrammatically in (Figure 3).Remark: Using MAPLE, we have verified all solutions we obtained by putting them back into the original NPDEs, WZ Equation ( 4) and (5).And it is worthy to point out that the new forms of solutions in Case (1), where the similarity variable 2 0.5 x t     t cannot be obtained by the generalized extended rational expansion method [22], and has not reported in the literature.

Conclusion
In this work, we have used the characteristic function technique to obtain similarity reductions of nonlinear partial differential Equations ( 4) and ( 5), for (1 + 1)-dimensional dispersive long wave equation (DLWE) in shallow water with uniform depth.The resulting system of non-linear ordinary differential with the associated boundary conditions is solved numerically using shooting method, coupled with Runge-Kutta scheme.The numerical solution was carried out for various values of the parameters of the equations to study different cases and illustrate the effect of critical and special values on the performance of velocity and elevation of water wave.

Figure 1 .
Figure 1.Solution of (1 + 1)-dimensional dispersive long wave Equations (24) and (25) at A 1 = B 1 = B 2 = 1.0.Where F 1 refer to he surface velocity and F 2 refer to the elevation of the water wave.t

Figure 2 .
Figure 2. Solution of (1 + 1)-dimensional dispersive long wave Equations (24) and (25) at A 1 = B 1 = B 2 = 1.0.Where F 1 refer to he surface velocity and F 2 refer to the elevation of the water wave.t

Figure 3 .
Figure 3. Solution of (1 + 1)-dimensional dispersive long wave Equations (32) and (33) at A 1 = 1.0.Where F 1 refer to the surface velocity and F 2 refer to the elevation of the water wave. 