Expanding the Tanh-Function Method for Solving Nonlinear Equations

In this paper, using the tanh-function method, we introduce a new approach to solitary wave solutions for solving nonlinear PDEs. The proposed method is based on adding integration constants to the resulting nonlinear ODEs from the nonlinear PDEs using the wave transformation. Also, we use a transformation related to those integration constants. Some examples are considered to find their exact solutions such as KdVBurgers class and Fisher, Boussinesq and Klein-Gordon equations. Moreover, we discuss the geometric interpretations of the resulting exact solutions.

Herman et al. [12] introduced a general physical approach to solitary wave construction from linear solutions and obtained many exact solution of nonlinear PDEs using the direct algebraic method [4].
In this paper, we introduce a similar technique to [12] using the tanh-function method to obtain exact solutions for nonlinear evolution and wave equations.The first step in the tanh-function method is using an independent variable to turn the nonlinear PDEs into other nonlinear ordinary differential equations (ODEs) which may or may not be integrable and neglecting the integration constants in case of integrable ODEs.Here, we add integration constants in the resulting integrable nonlinear ODEs.Also, we use a new transformation in which we express the solution function as a sum of another independent function and a constant which are determined later.By means of this modification, we get the exact solutions in which a free constant appears which for some values gives the solutions of the tanh-function method.When the resulting nonlinear ODEs is non integrable, we use the transformation only to get the same exact solutions of the tanh-function method.

Expanding the Tanh-Function Method for Solving Nonlinear Equations
The tanh-function method, pioneered by Malfliet [5,6], is a common powerful method for solving nonlinear equations.Here, we introduce a modification of the tanhfunction method through the following: Consider the nonlinear evolution and wave equations in the forms respectively.Introducing the wave transformation where is the wave number and > 0 k  is the travelling wave velocity.
Assuming (3) is integrated with respect to  as many times as possible without neglecting the integration constants.For the evolution equations the maximum number of integration is 1 and for the wave equations is 2. For reasons that will be explained below, we only leave the integration constant of the last integration.

. c
To obtain the exact solitary wave solution, possibly having a determined constant term , we introduce the transformation Substituting ( 4) into (3) and setting the constant part equals to zero in the resulting nonlinear ODE in  assuming that the function  and its derivatives have the following asymptotic values, and for where the superscripts denotes differentiation to the order , with respect to n  , also we assume that   satisfies the algebraic equation in  , then we get the values of .
and using the finite expansion where is a positive integer determined by the balancing procedure in the resulting nonlinear ODE in .Thus, we have an algebraic system of equations from which the constants are obtained and determine the function  , hence we get the exact solutions of (1).Now, we obtain exact solutions for some examples of nonlinear evolution and wave equations using the suggested method.

KdV-Burgers Class
Consider the KdV-Burgers class in the form = 0, where ,   and  are real constants.The class (9) gives the Burgers equation and the KdV equation at = 0  and = 0  respectively.

Burgers Equation
Consider the Burgers equation in the form = 0.
Using (2), for , to change (10) into the following nonlinear ODE Integrating (11) once to get a new nonlinear ODE in the form where is the integration constant. 1 Introducing ( 4) into ( 12), we have Using the conditions ( 5), (6) and that   satisfies the algebraic equation then the constant term in (13) equals to zero, Then we have the following two cases according to the values of .
Applying the tanh-function method by balancing the nonlinear term with the derivative term , we get , and using (8 Substituting ( 17) into (16), we obtain Setting zero all the coefficients of , we get the algebraic system of equations From which we have Using ( 4), we get the exact solutions of the tanh-function method in the form [13] values of .
In this case, we get the exact solutions of the tanhfunction method in the forms [14] = 0 : c these solutions represent 2-dimensional surfaces in the Monge form as shown in Figure 1 Using the same way as in Case (1), we obtain the exact solutions in the form Thus the solutions (22) represent a family of parabolic surfaces , and a family of planes at ; when , we get the solutions (21).

= 0 C
These relations represent surfaces whose Gaussian curvature and mean curvature K H are given by (29)

KdV Equation
Thus the solutions (28) represent a family of parabolic , and a family of planes Consider the KdV equation in the form as shown in Figure 4 for   C ; when , we get the solutions (27).= 0 = 0.

KdV-Burgers Equation
Consider the KdV-Burgers equation in the form: where is the integration constant. 1 Introducing (4) into (25), we get c C = 0.
Using (2) to change (30) into a nonlinear ODE, then integrating once ) Using the conditions (5), ( 6) and ( 14), we get (15), hence we obtain the following two cases according to the     where is the integration constant.
In this case, we get the exact solutions of the tanhfunction method in the forms [13] = 0 : c Introducing ( 4) into (37), we get Using conditions ( 5), (6) and that   satisfies the algebraic equation then the constant term in (38) equals to zero    

Case (1). c
In this case, we have the exact solutions in the forms In this case, we get the exact solutions of the tanhfunction method in the form [14] = 0 : c  these solutions are plotted as shown in Figure 7 for = =1.

Case (2). 1
In this case, we get the same exact solutions (41).
, for , to change (42) into the nonlinear ODE, we get as shown in Figure 6 for       Integrating twice and leave the integration constant of the last integration, we have Introducing ( 4) into (44), we obtain (45) Using the conditions ( 5), (6) and that   satisfies the algebraic equation then the constant term in (45) equals to zero Then we have the following two cases according to the values of . 1

Case (1).
: In this case, we get the exact solutions of the tanhfunction method in the forms [15] = 0 c

Case (2).
  In this case, we get the exact solutions in the forms These relations represent surfaces whose Gaussian curvature K and mean curvature H are given by (50)(51).then the solutions (49) represent a family of parabolic surfaces , and a family of planes at as shown in Figure 9 for and = 0,1 and when , we get the solutions (48).= 0 C

Klein-Gordon Equation
Consider the Klein-Gordon equation in the form 3 = 0.
Introducing ( 4) into (53) Using the conditions ( 5), (6) and that   satisfies the algebraic equation then the constant term in (54) equals to zero Then we have the following two cases according to the values of . 1

Case (1).
: In this case, we obtain the solutions of the tanhfunction method in the form [7] = In this case, we get the same exact solutions (57).

Conclusions
In this paper, we introduced a new technique, by adding an integration constant and a new transformation (4) then using the tanh-function method, to obtain exact solitary wave solutions in case of the nonlinear evolution and wave equations that turn into nonlinear integrable ODEs using the wave transformation (2).
By this technique, we obtained exact solutions of the Burgers equation in (22), the KdV equation in (28), the     KdV-Burgers equation in (34) and the Boussinesq equation in (49) which all give the exact solutions obtained before by the tanh-function method as a special cases [13][14][15].Moreover, we discussed the geometric interpretations of the resulting exact solutions.Also, we get the same exact solutions by using (4) then using the tanh-function method.In case of the nonlinear evolution and wave equations that turn into nonlinear non integrable ODEs using (2), Fisher and Klein-Gordon equations are considered to illustrate our technique.
The presented technique can be applied to obtain exact solutions for many nonlinear evolution and wave equa-tions.

1
Applying the tanh-function method by introducing the new independent variable c = tanh Y  which leads to the change of derivatives in the forms


) these solutions represent 2-dimensional surfaces in the Monge form as shown in Figure 3 for = =1.These relations represent surfaces whose Gaussian curvature and mean curvature K H are given by (23) In this Case, we have the exact solutions in the forms

1 1 1
= 0. c c  (40) these solutions represent 2-dimensional surfaces in the Monge form as shown in Figure 5 for = = =1    Then we have the following two cases according to the values of .Case (2).


are given in (34), thus the solutions (34) represent a family of parabolic surfaces , and a family of planes
) these solutions represent 2-dimensional surfaces in the Copyright © 2011 SciRes.AM Monge form as shown in Figure 8 for = 1, = 1.
57)these solutions are plotted as shown in Figure10for =