The Characteristic Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation ()
1. 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 the past few decades, many significant methods have been presented such as Bäcklund transformation [1], Darboux transformation [2], the extended tanh-function method [3], and the F-expansion method [4], Lie group analysis [5-8], homogeneous balance method [9], Jacobi elliptic function method [10] etc.
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.
2. 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 mathematical terms that these models possess. Omitting the higher order terms, one of these equations, Wu-Zhang (WZ) equation [14], can be written as:
(1)
(2)
(3)
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].
(4)
(5)
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 [19-21]. We will seek the solution of Equations (4) and (5) considering the following initial and boundary conditions:
(6)
where α(x) and β(x) are functions of x and f0(t) and f1(t) are functions of t to be specified.
2.1. The Group Systematic Formulation under the Following Transformations
(7)
Considering the infinitesimal transformations in (t, x, v, w) given by
(8)
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:
(9)
(10)
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:
(11)
And
(12)
where
(13)
where i, j and k stand for x and t.
2.2. The Invariance Analysis
Under the infinitesimal group of transformation, the system of differential equations (4) and (5) on the form Gi = 0, for i = 1, 2, will be invariant if
(14)
where D is the total derivative defined by:
(15)
The operator D can be rewritten as:
(16)
Invariance of equations (9) and (10) under Equation (16) gives:
(17)
(18)
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:
(19)
where A1, B1, B2 and B3 are real constants.
2.3. 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 (1.1), (1.2) and Case (2) respectively.
3. 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 A1 = B1 = B2 = 1.0 and B3 = 0, then we get the following system:
(20)
Solving these system leads to:
(21)
where η(x,t) is the similarity variable of the independent variables. And the similarity functions of the dependent variables are:
(22)
(23)
where F1(η) refer to the surface velocity and F2(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:
(24)
(25)
Equations (24) and (25) are solved numerically applying the shooting method coupled with Runge-kutta scheme using two groups of initial conditions illustrated in Cases (1.1) and (1.2) as follow.
Case (1.1)
Solving Equations (24) and (25), under the following boundary conditions:
(26)
The obtained results are presented diagrammatically in (figure 1).
Figure 1. Solution of (1 + 1)-dimensional dispersive long wave Equations (24) and (25) at A1 = B1 = B2 = 1.0. Where F1 refer to the surface velocity and F2 refer to the elevation of the water wave.
Case (1.2)
Also we solve Equations (24) and (25) under the following boundary conditions:
(27)
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 B1 = B2 = B3 = 0 and A1 ≠ 0, then we get the following system:
Figure 2. Solution of (1 + 1)-dimensional dispersive long wave Equations (24) and (25) at A1 = B1 = B2 = 1.0. Where F1 refer to the surface velocity and F2 refer to the elevation of the water wave.
(28)
In this case, similarity transformation is given as:
(29)
(30)
(31)
Substituting F1(η) and F2(η) from Equations (30), (31) into Equations (4) and (5) gives:
(32)
(33)
The numerical solutions under the following boundary conditions:
(34)
In addition to, the boundary conditions which illustrated in the figures. The obtained results are presented diagrammatically in (figure 3).
Figure 3. Solution of (1 + 1)-dimensional dispersive long wave Equations (32) and (33) at A1 = 1.0. Where F1 refer to the surface velocity and F2 refer to the elevation of the water wave.
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 
cannot be obtained by the generalized extended rational expansion method [22], and has not reported in the literature.
4. 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.