In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the system of shallow water wave equations and modified Liouville equation which play an important role in mathematical physics.
Extended Jacobian Elliptic Function Expansion Method The System of Shallow Water Wave Equations Modified Liouville Equation Traveling Wave Solutions Solitary Wave Solutions1. Introduction
The nonlinear partial differential equations of mathematical physics are major subjects in physical science [1] . Exact solutions for these equations play an important role in many phenomena in physics such as fluid mechanics, hydrodynamics, optics, plasma physics and so on. Recently many new approaches for finding these solutions have been proposed, for example, tanh-sech method [2] -[4] , extended tanh-method [5] -[7] , sine-cosine method [8] -[10] , homogeneous balance method [11] [12] , F-expansion method [13] -[15] , exp-function method
[16] , the modified simple equation method [17] , the -expansion method [18] , -expansion
method [19] -[22] , Jacobi elliptic function method [23] -[26] and so on.
The objective of this article is to apply the extended Jacobian elliptic function expansion method for finding the exact traveling wave solution the system of shallow water wave equations and modified Liouville equation which play an important role in mathematical physics.
The rest of this paper is organized as follows: In Section 2, we give the description of the extended Jacobi elliptic function expansion method. In Section 3, we use this method to find the exact solutions of the nonlinear evolution equations pointed out above. In Section 4, conclusions are given.
2. Description of Method
Consider the following nonlinear evolution equation
where F is polynomial in and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following, we give the main steps of this method [23] -[26] .
Step 1. Using the transformation
where c is wave speed, to reduce Equation (1) to the following ODE:
where P is a polynomial in and its total derivatives, while.
Step 2. Making good use of ten Jacobian elliptic functions, we assume that (3) has the solutions in these forms:
with
where, , , are the Jacobian elliptic sine function, the jacobian elliptic cosine function and the Jacobian elliptic function of the third kind and other Jacobian functions which is denoted by Glaisher’s symbols and are generated by these three kinds of functions, namely
that have the relations
with the modulus m In addition we know that
The derivatives of other Jacobian elliptic functions are obtained by using Equation (8). To balance the highest order linear term with nonlinear term we define the degree of u as which gives rise to the degrees of other expressions as
According the rules, we can balance the highest order linear term and nonlinear term in Equation (3) so that n in Equation (4) can be determined.
Noticed that, , when the modulus and, , when the modulus, we can obtain the corresponding solitary wave solutions and triangle function solutions, respectively, while when therefore Equation (5) degenerate as the following forms
Therefore the extended Jacobian elliptic function expansion method is more general than sine-cosine method, the tan-function method and Jacobian elliptic function expansion method.
3. Application3.1. Example 1: The System of Shallow Water Wave Equations
We first consider the system of the shallow water wave equation [27] in order to demonstrate the - expansion method
We use the wave transformation, to reduce Equations (14) to the following nonlinear system of ordinary differential equations:
where by integrating once the second equation with zero constant of integration, we find
substituting Equation (16) into the first equation of Equation (15) we obtain
Integrating Equation (17) with zero constant of integration, we find
Balancing and in Equation (18) yields,. This suggests the choice of in Equation (18) as
where, and are constant such that or. From (19), it is easy to see that
Substituting Equations (19) and (21) into Equation (18) and equating all coefficients of, , , , , , respectively to zero, we obtain:
Solving the above system with the aid of Maple or Mathematica, we have the following solution:
Case 1.
So that the solution of Equation (18) can be written as
when, the solution can be in the form
Case 2.
So that the solution of Equation (18) can be written as
when, the solution can be in the form
Case 3.
So that the solution of Equation (18) can be written as
when, the solution can be in the form
3.2. Example 2: Modified Liouville Equation
Now, let us consider the modified Liouville equation [28] .
respectively, where, and are non zero and arbitrary coefficients. Using the wave transformation, , , to reduce Equation (35) to be in the form:
Balancing and in Equation (36) yields,. Consequently, we have the formal solution:
where, , are constants to be determined, such that or. It is easy to see that
Substituting (37) and (39) into Equation (36) and equating all the coefficients of, , , , , , , , , , , , to zero, we deduce respectively
Solving the above system with the aid of Maple or Mathematica, we have the following solution:
So that the solve of Equation (36) can be written in the form
When, the solution can be in the form
4. Conclusions
We establish exact solutions for the system of shallow water wave equations and modified Liouville equation which are two of the most fascinating problems of modern mathematical physics.
The extended Jacobian elliptic function expansion method has been successfully used to find the exact traveling wave solutions of some nonlinear evolution equations. As an application, the traveling wave solutions for the system of shallow water wave equations and modified Liouville equation, have been constructed using the extended Jacobian elliptic function expansion method. Let us compare between our results obtained in the present article with the well-known results obtained by other authors using different methods as follows: our results of the system of shallow water wave equations and modified Liouville equation are new and different from those obtained in [27] and [28] and Figure 1 and Figure 2 show the solitary wave solution of Equations
Solitary wave solution of Equation (30)
Solitary wave solution of Equation (56)
(30) and (56). It can be concluded that this method is reliable and proposes a variety of exact solutions NPDEs. The performance of this method is effective and can be applied to many other nonlinear evolution equations.
NOTESReferencesAblowitz, M.J. and Segur, H. (1981) Solitions and Inverse Scattering Transform. SIAM, Philadelphia.
http://dx.doi.org/10.1137/1.9781611970883Malfliet, W. (1992) Solitary Wave Solutions of Nonlinear Wave Equation. American Journal of Physics, 60, 650-654.
http://dx.doi.org/10.1119/1.17120Malfliet, W. and Hereman, W. (1996) The tanh Method: Exact Solutions of Nonlinear Evolution and Wave Equations. Physica Scripta, 54, 563-568. http://dx.doi.org/10.1088/0031-8949/54/6/003Wazwaz, A.M. (2004) The tanh Method for Travelling Wave Solutions of Nonlinear Equations. Applied Mathematics and Computation, 154, 714-723. http://dx.doi.org/10.1016/S0096-3003(03)00745-8El-Wakil, S.A. and Abdou, M.A. (2007) New Exact Travelling Wave Solutions Using Modified Extented tanh-Function Method. Chaos, Solitons & Fractals, 31, 840-852. http://dx.doi.org/10.1016/j.chaos.2005.10.032Fan, E. (2000) Extended tanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 277, 212-218. http://dx.doi.org/10.1016/S0375-9601(00)00725-8Wazwaz, A.M. (2007) The Extended tanh Method for Abundant Solitary Wave Solutions of Nonlinear Wave Equations. Applied Mathematics and Computation, 187, 1131-1142. http://dx.doi.org/10.1016/j.amc.2006.09.013Wazwaz, A.M. (2005) Exact Solutions to the Double sinh-Gordon Equation by the tanh Method and a Variable Separated ODE Method. Computers & Mathematics with Applications, 50, 1685-1696.
http://dx.doi.org/10.1016/j.camwa.2005.05.010Wazwaz, A.M. (2004) A sine-cosine Method for Handling Nonlinear Wave Equations. Mathematical and Computer Modelling, 40, 499-508. http://dx.doi.org/10.1016/j.mcm.2003.12.010Yan, C. (1996) A Simple Transformation for Nonlinear Waves. Physics Letters A, 224, 77-84.
http://dx.doi.org/10.1016/S0375-9601(96)00770-0Fan, E. and Zhang, H. (1998) A Note on the Homogeneous Balance Method. Physics Letters A, 246, 403-406.
http://dx.doi.org/10.1016/S0375-9601(98)00547-7Wang, M.L. (1996) Exact Solutions for a Compound KdV-Burgers Equation. Physics Letters A, 213, 279-287.
http://dx.doi.org/10.1016/0375-9601(96)00103-XAbdou, M.A. (2007) The Extended F-Expansion Method and Its Application for a Class of Nonlinear Evolution Equations. Chaos, Solitons & Fractals, 31, 95-104. http://dx.doi.org/10.1016/j.chaos.2005.09.030Ren, Y.J. and Zhang, H.Q. (2006) A Generalized F-Expansion Method to Find Abundant Families of Jacobi Elliptic Function Solutions of the (2 + 1)-Dimensional Nizhnik-Novikov-Veselov Equation. Chaos, Solitons & Fractals, 27, 959-979. http://dx.doi.org/10.1016/j.chaos.2005.04.063Zhang, J.L., Wang, M.L., Wang, Y.M. and Fang, Z.D. (2006) The Improved F-Expansion Method and Its Applications. Physics Letters A, 350, 103-109. http://dx.doi.org/10.1016/j.physleta.2005.10.099He, J.H. and Wu, X.H. (2006) Exp-Function Method for Nonlinear Wave Equations. Chaos, Solitons & Fractals, 30, 700-708. http://dx.doi.org/10.1016/j.chaos.2006.03.020Zahran, E.H.M. and Khater, M.M.A. (2014) The Modified Simple Equation Method and Its Applications for Solving Some Nonlinear Evolutions Equations in Mathematical Physics. Jokull Journal, 64.Abdelrahman, M.A.E., Zahran, E.H.M. and Khater, M.M.A. (2014) Exact Traveling Wave Solutions for Power Law and Kerr Law Non Linearity Using the -Expansion Method. Global Journal of Science Frontier Research, 14-F.Wang, M.L., Zhang, J.L. and Li, X.Z. (2008) The -Expansion Method and Travelling Wave Solutions of Nonlinear Evolutions Equations in Mathematical Physics. Physics Letters A, 372, 417-423.
http://dx.doi.org/10.1016/j.physleta.2007.07.051Zhang, S., Tong, J.L. and Wang, W. (2008) A Generalized -Expansion Method for the mKdv Equation with Variable Coefficients. Physics Letters A, 372, 2254-2257. http://dx.doi.org/10.1016/j.physleta.2007.11.026Zayed, E.M.E. and Gepreel, K.A. (2009) The -Expansion Method for Finding Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics. Journal of Mathematical Physics, 50, Article ID: 013502. http://dx.doi.org/10.1063/1.3033750Zahran, E.H.M. and Khater, M.M.A. (2014) Exact Solution to Some Nonlinear Evolution Equations by the -Expansion Method. Jokull Journal, 64.Zaki, S.I. (2000) Solitary Wave Interactions for the Modified Equal width Wave Equation. Computer Physics Communications, 126, 219-213. http://dx.doi.org/10.1016/S0010-4655(99)00471-3Fan, E. and Zhang, J. (2002) Applications of the Jacobi Elliptic Function Method to Special-Type Nonlinear Equations. Physics Letters A, 305, 383-392. http://dx.doi.org/10.1016/S0375-9601(02)01516-5Liu, S., Fu, Z., Liu, S. and Zhao, Q. (2001) Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289, 69-74. http://dx.doi.org/10.1016/S0375-9601(01)00580-1Zhao, X.Q., Zhi, H.Y. and Zhang, H.Q. (2006) Improved Jacobi-Function Method with Symbolic Computation to Construct New Double-Periodic Solutions for the Generalized Ito System. Chaos, Solitons & Fractals, 28, 112-126.
http://dx.doi.org/10.1016/j.chaos.2005.05.016Dolapci, ?.T. and Yildirim, A. (2013) Some Exact Solutions to the Generalized Korteweg-de Vries Equation and the System of Shallow Water Wave Equation. Nonlinear Analysis, Modeling and Control, 18, 27-36. Salam, M.A. (2012) Traveling-Wave Solution of Modified Liouville Equation by Means of Modified Simple Equation Method. International Scholarly Research Network ISRN Applied Mathematics, 2012, Article ID: 565247.