In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics

Abdelrahman, Mahmoud A.E.

Khater, Mostafa

English

KHALSA PUBLICATIONS

ISSN 2347-3487                                                           3134 | P a g e                                                    D e c e m b e r  0 7 ,  2 0 1 5  Traveling Wave Solutions For The Couple Boiti-Leon-Pempinelli System By Using Extended Jacobian Elliptic Function Expansion Method Mahmoud A.E. Abdelrahman and Mostafa M.A. Khater* Mansoura University,Department of Mathematics, Faculty of Science, 35516 Mansoura, Egypt. Email*: mostafa.khater2024@yahoo.com  ABSTRACT 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 Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics. Keywords Extended Jacobian elliptic function expansion method; The Couple Boiti-Leon-Pempinelli System; Traveling wave solutions; hyperbolic solutions.                            Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN PHYSICS Vol. 11, No. 3 www.cirjap.com, japeditor@gmail.com  ISSN 2347-3487                                                           3135 | P a g e                                                    D e c e m b e r  0 7 ,  2 0 1 5  1- 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], modified simple equation method [11, 12], the exp expansion method [13]-[15], the extended exp  expansion method [16],  expansion method [17]-[19], Jacobi elliptic function method [20]- [23] 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 Couple Boiti-Leon-Pempinelli System  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      (2.1) Since, P is a polynomial in  and its partial derivatives. In the following, we give the main steps of this method  Step 1. We use the traveling wave solution in the form              (2.2) where c is a positive constant, to reduce Eq. (2.1) to the following ODE:   (2.3) where P is a polynomial in u (ξ) and its total derivatives, while duud  . Step 2. Making good use of ten Jacobian elliptic functions we assume that (2.3) have the solutions in these forms:      101( ) a , 1,2,3,...Nji j i jju f a f b g i          (2.4) With                    1 12 23 34 45 56 6, ,, ,, ,, ,, ,, ,f sn g cnf sn g dnf ns g csf ns g dsf sc g ncf sd g nd                             (2.5) where , , ,sn cn dn    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 ISSN 2347-3487                                                           3136 | P a g e                                                    D e c e m b e r  0 7 ,  2 0 1 5  1 1 1, , , ,, , ,cnns nc nd scsn cn dn snsn dn sncs ds sdcn sn dn                    (2.6) those have the relations 2 2 2 2 2 2 22 2 2 2 2 2 2 21, 1, 1 ,, 1 , 1 ,sn cn dn m sn ns csns m ds sc nc m sd nd                   (2.7) with the modulus m (0 < m < 1). In addition we know that 2, , .d d dsn cn dn cn sn dn dn m sn cnd d d                (2.8) The derivatives of other Jacobian elliptic functions are obtained by using Eq. (2.8). To balance the highest order linear term with nonlinear term we define the degree of u as D[u] = n which gives rise to the degrees of other expressions as  , .sq qpq qd u d uD n q D u np s n qd d                   (2.9) According the rules, we can balance the highest order linear term and nonlinear term in Eq. (2.3) so that n in Eq. (2.4) can be determined. In addition we see that when 1,m   ,sn cn and dn    degenerate as tanh ,sec ,sec ,h h   respectively, while when therefore Eq. (2.5) degenerate as the following forms                     101101101101( ) a tanh tanh sech ,( ) a coth coth csch ,( ) a tanh tan sec ,( ) a coth cot csc ,Njj jjNjj jjNjj jjNjj jju a bu a bu a bu a b                                  (2.10) 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- The Couple Boiti-Leon-Pempinelli System Consider the Couple Boiti-Leon-Pempinelli System [24] is in the form  (3.1) Using the transformation since and integrate the first equation of the system (3.1) three times with zero constant of integration and substitute it into the second equation of system (3.1) we obtained  (3.2) Balancing and So that, we assume the solution of Eq. (3.2) be in the form  (3.3) ISSN 2347-3487                                                           3137 | P a g e                                                    D e c e m b e r  0 7 ,  2 0 1 5  Where are arbitrary constants to be determined later. Substituting Eq. (3.3) and it's derivatives into Eq. (3.2) and equating all coefficients of  to zero, we obtain  (3.4) Solving above system by using Maple 16, we obtained  So that, we get the exact traveling wave solution of Eq. (3.3)  (3.5) When m = 1, we get hyperbolic solution of Eq. (3.3) 4- Conclusion We establish exact solutions for the Couple Boiti-Leon-Pempinelli System is the most fascinatin 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 Couple Boiti-Leon-Pempinelli System which has 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 are new and different from those obtained in [24]. It can be concluded that this method is reliable and propose a variety of exact solutions NPDEs. The performance of this method is effective and can be applied to many other nonlinear evolution equations. References [1] M. J. Ablowitz, H. Segur, Solitions and Inverse Scattering Transform, SIAM, Philadelphia 1981. [2] W. Maliet, Solitary wave solutions of nonlinear wave equation, Am. J. Phys., 60 (1992) 650-654. [3] W. Maliet, W. Hereman, The tanh method: Exact solutions of nonlinear evolution and wave equations, Phys.Scr., 54 (1996) 563-568. [4] A. M. 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Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

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Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

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