Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Acta Mathematicae Applicatae Sinica, English Series Vol. 32, No. 2 (2016) 461–468 DOI: 10.1007/s10255-016-0572-y http://www.ApplMath.com.cn & www.SpringerLink.com Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 2016 Double Exp-Function Method for Multisoliton Solutions of The Tzitzeica-Dodd-Bullough Equation Alaattin ESEN, N. Murat YAGMURLU† , Orkun TASBOZAN Department of Mathematics, Faculty of Science and Art, Inönü University, Malatya 44280, Turkey († E-mail: [email protected]) Abstract In this work, it is aimed to find one- and two-soliton solutions to nonlinear Tzitzeica-Dodd-Bullough (TDB) equation. Since the double exp-function method has been widely used to solve several nonlinear evolution equations in mathematical physics, we have also used it with the help of symbolic computation for solving the present equation. The method seems to be easier and more accurate thanks to the recent developments in the field of symbolic computation. Keywords double exp-function method; one-soliton; two-soliton; Tzitzeica-Dodd-Bullough equation; solitary waves 2000 MR Subject Classification 1 35Q51; 74J35; 33F10 Introduction The travelling wave solutions of nonlinear partial differential equations and their investigation have an important place in the study of nonlinear physical phnomena. Numerical modellings of many phenomena in scientific and engineering fields such as fluid mechanics, plasma physcis, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, and geochemistry result in nonlinear phenomena. Thus, obtaining their exact and numerical solutions have become very important in recent years. Many authors have used many different approaches and techniques to develop nonlinear dispersive and dissipative problems. They include the inverse scattering transform method[2] , the Bäcklund transform[21] , Darboux transforms[6,16], the parameter-expansion method[17,22] , the tanh method[15] , extended tanh method[4,7] , sinecosine method[23] , exp-function method[5,11] , F -expansion method[14,24] and (G′ /G)-expansion method[13,18] . In the present study, double exp-function method has been proposed and succesfully applied to find the multisoliton solutions of nonlinear the TDB equation. 2 The Double-Exp Function Method The exp-function method first introduced by Wu and He in order to solve differential equations[19] , and it has been systematically investigated by many authors. Then multiple exp-function was first proposed by He[12] and used by many authors. Later, the double exp-function method and its application have been presented by Fu and Dai[8] . The method has been applied by many authors to solve different kinds of equations. In this study, the general nonlinear PDE given in the following form P (u, ut , ux , utt , utx , uxx , · · ·) = 0 Manuscript received October 19, 2011. Revised November 5, 2012. (1) 462 A. ESEN, N.M. YAGMURLU, O. TASBOZAN will be considered. It is assumed that the two-wave solution to Eq.(1) can be stated in terms of fractional form as f (x, t) , (2) u= g(x, t) where f (x, t) and g(x, t) are ansatz functions of the two-soliton forms f (x, t) = 1 + eξ + eη + A1 eξ+η , g(x, t) = 1 + eξ + eη + A2 eξ+η . (3) (4) If we put Eqs.(3) and (4) into Eq.(2), then we easily obtain u= f (x, t) e−(ξ+η)/2 + e(ξ−η)/2 + e(η−ξ)/2 + A1 e(ξ+η)/2 = −(ξ+η)/2 . g(x, t) e + e(ξ−η)/2 + e(η−ξ)/2 + A1 e(ξ+η)/2 This newly obtained equation is a special case of the general ansatz of the exp-function method with N wave velocities and N frequencies, which was proposed by He[9] and then was used for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations[21] . Particularly, a two-soliton ansatz of the exp-function method can be defined as follows u= a−2 e−η + a−1 e−ξ + a0 + a1 eξ + a2 eη , b−2 e−η + b−1 e−ξ + b0 + b1 eξ + b2 eη where ξ = c1 x + c2 t, η = c3 x + c4 t and ai , bj (−2 ≤ i, j ≤ 2) are constants and will be determined. 3 The TBD Equation In this section, we consider the TBD equation[22] uxt = e−u + e−2u . (5) The equation plays an important role in many scientific applications such as solid state physics, nonlinear optics, dusty plasma, plasma physics, fluid dynamics, mathematical biology, dislocations in crystals, kink dynamics, chemical kinetics and quantum field theory[3] . The equation has been solved by many authors using different methods and techniques. Among others, Wazwaz[20] has applied tanh method to the Dod-Bullough-Mikhailov and the Tzitzeica-DodBullough equations to derive solitons and periodic solutions for these equations and Abazari[1] has applied (G′ /G)-expansion method for constructing more general exact solutions of the TDB equation. Using the transformation v(x, t) = e−u Eq.(5) becomes −vvxt + vx vt − v 3 − v 4 = 0. (6) Applying double exp-function method, we assume that v= a1 eξ + a2 e−ξ + a5 + a3 eη + a4 e−η , k1 eξ + k2 e−ξ + k5 + k3 eη + k4 e−η (7) where ξ = c1 x + c2 t and η = c3 x + c4 t. If we substitute (7) into (6) and set all of the coefficients of e(iξ+jη) to zero, then we result in a system of algebraic equations. If this system is solved with the help of any mathematical software, many abundant new solutions are obtained. After Double Exp-Function Method for Multisoliton Solutions of The Tzitzeica-Dodd-Bullough Equation 463 some simple computations, we finally obtained one- and two-soliton solutions for 22 different cases. (I) One-soliton Solutions: Case 1. 1 a3 = a4 = 0, a1 = −k1 , a2 = −k2 , a5 = −k5 , k3 = 0, c1 = c2 = 0, c3 = − , c4 k1 + k2 + k5 . u1 (x, t) = − ln − k1 + k2 + k5 + k4 e−(c3 x+c4 t) (8) If k1 + k2 + k5 = 1 and k4 = 1 are taken, using Eq.(8) results in the following solitary wave solution 1 1 1 u(x, t) = − ln tanh − (c3 x + c4 t) − . 2 2 2 If k1 + k2 + k5 = 1 and k4 = −1 are taken, using Eq.(8) results in the following solitary wave solution 1 1 1 u(x, t) = − ln coth − (c3 x + c4 t) − . 2 2 2 Case 2. a3 = a4 = 0, a1 = −k1 , a2 = −k2 , a5 = −k5 , k4 = 0, c1 = c2 = 0, c3 = − u2 (x, t) = − ln 1 , c4 −(k1 + k2 + k5 ) . k1 + k2 + k5 + k3 e(c3 x+c4 t) (9) If k1 + k2 + k5 = 1 and k3 = 1 are taken, using Eq.(9) results in the following solitary wave solution 1 1 1 u(x, t) = − ln tanh (c3 x + c4 t) − . 2 2 2 If k1 + k2 + k5 = 1 and k3 = −1 are taken, using Eq.(10) results in the following solitary wave solution 1 1 1 u(x, t) = − ln coth (c3 x + c4 t) − . 2 2 2 Case 3. a2 = a4 = a5 = 0, a1 = −k1 , a3 = −k3 , k5 = 0, c1 = c3 = − u3 (x, t) = − ln 1 , c2 = c 4 , 4c4 −k1 e(c1 x+c2 t) − k3 e(c3 x+c4 t) . k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) + k4 e−(c3 x+c4 t) (10) If k1 + k3 = 1 and k2 + k4 = 1 are taken, using Eq.(10) results in the following solitary wave solution 1 1 u(x, t) = − ln tanh(−(c3 x + c4 t)) − . 2 2 If k1 + k3 = 1 and k2 + k4 = −1 are taken, using Eq.(10) results in the following solitary wave solution 1 1 u(x, t) = − ln coth(−(c3 x + c4 t)) − . 2 2 Case 4. 1 1 , c2 = −c4 , c3 = − , 4c4 4c4 −k1 e(c1 x+c2 t) − k4 e−(c3 x+c4 t) u4 (x, t) = − ln . k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) + k4 e−(c3 x+c4 t) a2 = a3 = a5 = 0, a1 = −k1 , a4 = −k4 , k5 = 0, c1 = (11) 464 A. ESEN, N.M. YAGMURLU, O. TASBOZAN If k1 + k4 = 1 and k2 + k3 = 1 are taken, using Eq.(11) results in the following solitary wave solution 1 1 tanh(c3 x + c4 t) − . u(x, t) = − ln 2 2 If k1 + k4 = 1 and k2 + k3 = −1 are taken, using Eq.(11) results in the following solitary wave solution 1 1 u(x, t) = − ln coth(c3 x + c4 t) − . 2 2 Case 5. c1 c4 − 1 k1 k2 , c3 = 0, c2 = , k4 c1 −k1 e(c1 x+c2 t) − k4 e−c4 t u5 (x, t) = − ln . (12) k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + kk1 k4 2 ec4 t + k4 e−c4 t a2 = a3 = a5 = 0, a1 = −k1 , a4 = −k4 , k5 = 0, k3 = If k4 = 1 and k2 = 1 are taken, using Eq.(12) results in the following solitary wave solution u(x, t) = − ln 1 1 1 tanh − (c1 x + (c2 − c4 )t) − . 2 2 2 If k4 = 1 and k2 = −1 are taken, using Eq.(12) results in the following solitary wave solution u(x, t) = − ln 1 1 1 coth − (c1 x + (c2 − c4 )t) − . 2 2 2 Case 6. 1 + c1 c4 k1 k2 , c3 = 0, c2 = − , k4 c1 −k2 e−(c1 x+c2 t) − k4 e−c4 t u6 (x, t) = − ln . (13) k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k3 ec4 t + k4 e−c4 t a1 = a3 = a5 = 0, a2 = −k2 , a4 = −k4 , k5 = 0, k3 = If k4 = 1 and k1 = 1 are taken, using Eq.(13) results in the following solitary wave solution u(x, t) = − ln 1 2 tanh 1 1 (c1 x + (c2 + c4 )t) − . 2 2 If k4 = 1 and k1 = −1 are taken, using Eq.(13) results in the following solitary wave solution u(x, t) = − ln 1 2 coth 1 1 (c1 x + (c2 + c4 )t) − . 2 2 Case 7. a2 = a3 = a5 = 0, a1 = −k1 , a4 = −k4 , k5 = 0, k3 = u7 (x, t) = − ln (c1 x+c2 t) −c3 x k1 k2 c2 c3 − 1 , c4 = 0, c1 = , k4 c2 −k1 e − k4 e k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + kk1 k4 2 ec3 x + k4 e−c3 . x (14) (15) If k4 = 1 and k2 = 1 are taken, using Eq.(15) results in the following solitary wave solution u(x, t) = − ln 1 2 tanh 1 1 ((c3 − c1 )x − c2 t) − . 2 2 Double Exp-Function Method for Multisoliton Solutions of The Tzitzeica-Dodd-Bullough Equation 465 If k4 = 1 and k2 = −1 are taken, using Eq.(15) results in the following solitary wave solution u(x, t) = − ln 1 2 coth 1 1 ((c3 − c1 )x − c2 t) − . 2 2 Case 8. a1 = a2 = a3 = 0, u8 (x, t) = − ln a4 = −k4 , k1 = k2 = 0, k5 = − k4 k3 + a25 , a5 c3 = − 1 , c4 a5 − k4 e−(c3 x+c4 t) . (c x+c t) −(c x+c t) k5 + k3 e 3 4 + k4 e 3 4 (16) If a5 = −1 and k1 = 1 are taken, using Eq.(16) results in the following solitary wave solution u(x, t) = − ln 1 2 tanh 1 1 (c3 x + c4 t) − . 2 2 If a5 = −1 and k1 = −1 are taken, using Eq.(16) results in the following solitary wave solution 1 1 1 coth (c3 x + c4 t) − . u(x, t) = − ln 2 2 2 Case 9. a2 = a3 = a4 = 0, u9 (x, t) = − ln a1 = −k1 , k3 = k4 = 0, k5 = − k1 k2 + a25 , a5 c1 = − 1 , c2 −k1 e(c1 x+c2 t) + a5 . k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k5 (17) If a5 = −1 and k2 = 1 are taken, using Eq.(17) results in the following solitary wave solution u(x, t) = − ln 1 1 1 tanh − (c1 x + c2 t) − . 2 2 2 If a5 = −1 and k2 = −1 are taken, using Eq.(17) results in solution 1 1 u(x, t) = − ln coth − (c1 x + c2 t) − 2 2 the following solitary wave 1 . 2 Case 10. a2 (1 + c1 c2 ) a1 = a2 = a3 = a4 = 0, k3 = k4 = 0, k1 = 5 2 2 , 4c1 c2 k2 a5 u10 (x, t) = − ln . (c x+c t) −(c x+c t) k1 e 1 2 + k2 e 1 2 + k5 k5 = a5 , c1 c2 (II) Two-soliton Solutions Case 11. a1 = a3 = a4 = a5 = 0, u11 (x, t) = − ln a2 = −k2 , k1 = k4 = k5 = 0, −k2 e−(c1 x+c2 t) . k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) c1 = − 1 + c2 c3 + c3 c4 , c2 + c4 466 A. ESEN, N.M. YAGMURLU, O. TASBOZAN Case 12. a1 = a3 = a4 = a5 = 0, u12 (x, t) = − ln a2 = −k2 , k1 = k3 = k5 = 0, c1 = −1 + c2 c3 − c3 c4 , c2 − c4 c1 = −1 + c2 c3 − c3 c4 , c2 − c4 −k2 e−(c1 x+c2 t) . k2 e−(c1 x+c2 t) + k4 e−(c3 x+c4 t) Case 13. a1 = a2 = a4 = a5 = 0, u13 (x, t) = − ln a3 = −k3 , k2 = k4 = k5 = 0, −k3 e(c3 x+c4 t) . k1 e(c1 x+c2 t) + k3 e(c3 x+c4 t) Case 14. a2 = a3 = a4 = a5 = 0, u14 (x, t) = − ln a1 = −k1 , k2 = k3 = k5 = 0, c1 = − 1 + c2 c3 + c3 c4 , c2 + c4 −k1 e(c1 x+c2 t) . k1 e(c1 x+c2 t) + k4 e−(c3 x+c4 t) Case 15. a2 = a3 = a4 = a5 = 0, u15 (x, t) = − ln a1 = −k1 , k2 = k4 = k5 = 0, c1 = −1 + c2 c3 − c3 c4 , c2 − c4 −k1 e(c1 x+c2 t) . k1 e(c1 x+c2 t) + k3 e(c3 x+c4 t) Case 16. k3 k4 k3 k4 1 + c1 c2 − c1 c4 , a4 = −k4 , k5 = 0, k1 = , c3 = , k2 k2 c2 − c4 a1 e(c1 x+c2 t) − k4 e−(c3 x+c4 t) u16 (x, t) = − ln . k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) + k4 e−(c3 x+c4 t) a2 = a3 = a5 = 0, a1 = − Case 17. a1 = a2 = a4 = a5 = 0, u17 (x, t) = − ln a3 = −k3 , k1 = k4 = k5 = 0, c1 = − 1 + c2 c3 + c3 c4 , c2 + c4 −k3 e(c3 x+c4 t) . k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) Case 18. k3 k4 1 + c1 c2 − c1 c4 k3 k4 , a3 = −k3 , k5 = 0, k2 = , c3 = , k1 k1 c2 − c4 a2 e−(c1 x+c2 t) − k3 e(c3 x+c4 t) u18 (x, t) = − ln . k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) + k4 e−(c3 x+c4 t) a1 = a4 = a5 = 0, a2 = − Case 19. k3 k4 k3 k4 1 + c1 c2 + c1 c4 , a4 = −k4 , k5 = 0, k2 = , c3 = − , k1 k1 c2 + c4 a2 e−(c1 x+c2 t) − k4 e−(c3 x+c4 t) u19 (x, t) = − ln . k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) + k4 e−(c3 x+c4 t) a1 = a3 = a5 = 0, a2 = − 467 Double Exp-Function Method for Multisoliton Solutions of The Tzitzeica-Dodd-Bullough Equation Case 20. a1 = a2 = a3 = a5 = 0, u20 (x, t) = − ln a4 = −k4 , k2 = k3 = k5 = 0, c1 = − 1 + c2 c3 + c3 c4 , c2 + c4 −k4 e−(c3 x+c4 t) . k1 e(c1 x+c2 t) + k4 e−(c3 x+c4 t) Case 21. k3 k4 k3 k4 1 + c1 c2 + c1 c4 , a3 = −k3 , k5 = 0, k1 = , c3 = − , k2 k2 c2 + c4 a1 e(c1 x+c2 t) − k3 e(c3 x+c4 t) u21 (x, t) = − ln . k1 e(c1 x+c2 t) + k2 e−(c1 x+c2 t) + k3 e(c3 x+c4 t) + k4 e−(c3 x+c4 t) a2 = a4 = a5 = 0, a1 = − Case 22. a1 = a2 = a3 = a5 = 0, u22 (x, t) = − ln a4 = −k4 , k1 = k3 = k5 = 0, c1 = c2 , c4 (c4 − c2 ) c3 = − 1 , c4 −k4 e−(c3 x+c4 t) . k2 e−(c1 x+c2 t) + k4 e−(c3 x+c4 t) As seen from the conclusions, the method seems to be easier and faster with the usage of advanced symbolic computation systems. Our solutions are new soliton solutions of the Tzitzeica-Dod-Bullough equation. In this study, we have obtained a wide range of solutions of the problem, thus those solutions will be more meaningful and helpful in the examination of physical problems related to the equation. 4 Conclusions In this study, analytical solutions of the TDB equation has been successfully obtained and presented using symbolic computation systems. In the solution process, the double exp-function method has been preferred due to its widespread usage in handling well-known evolution equations, discrete nonlinear equations and nonlinear equation systems. The solutions obtained in the present study are more extensive compared with those available in the literature. Thus, we are in the opinion that the presented results will be helpful for the applications in mathematical physics and applied mathematics, particularly numerical simulations. In conclusion, we assert that the exp-function method is plain and can be applied to a wider range of several nonlinear evolution equations. References [1] Abazari, R. The (G′ /G)-expansion method for Tzitzeica type nonlinear evolution equations. Mathematical and Computer Modelling, 52: 1834–1845 (2010) [2] Ablowitz, M.J., Clarkson, P.A. Solitons: Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, 1991 [3] Bhrawy, A.H., Obaid, M. New exact solutions for the Zhiber-Shabat equaton using the extended F expansion method. Life Science Joıurnal, 9: 1154–1162 (2012) [4] El-Wakil, S.A., Abdou, M.A. New Exact Travelling Wave Solutions Using Modified Extended TanhFunction Method. Chaos Solitons Fractals, 31: 840–852 (2007) [5] Esen A., Kutluay, S. Application of the Exp-function method to the two dimensional sine-Gordon equation. Int. J. Nonlinear Sci., 10: 1355–1359 (2009) [6] Esteevez, P.G. Darboux transformation and solutions for an equation in 2+1 dimensions. J. Math. Phys., 40: 1406–1419 (1999) 468 A. ESEN, N.M. YAGMURLU, O. TASBOZAN [7] Fan, E. Exended Tanh-function Method and Its Applications to Nonlinear Equations. Phys. Lett. A, 277: 212–218 (2000) [8] Fu, H.M., Dai, Z.D. Double Exp-function Method and Application. Int. J. Nonlin. Sci. Num., 10: 927–933 (2009) [9] He, J.H. Some Asymptotic Methods for Strongly Nonlinear Equations. Int. J. Mod. Phys. B, 20: 1141–1199 (2006) [10] He, J.H. Some Asymptotic Methods for Solitary Solutions and Compactons. Abstract and Applied Analysis, Volume 2012, Article ID 916793, 130 pages, doi:10.1155/2012/916793 [11] He, J.H., Wu, X.H. Exp-Function Method for Nonlinear Wave Equations. Chaos Solitons Fractals, 30: 700–708 (2006) [12] He, J.H. An Elementary Introduction to Recently Developed Asymptotic Methods and Nanomechanics in Textile Engineering. Int. J. Mod. Phys. B, 22: 3487–3578 (2008) [13] Kutluay S., Esen A., Tasbozan O. The (G’/G)-expansion method for some nonlinear evolution equations. Appl. Math. and Comput., 217: 384–391 (2010) [14] Liu, J.B., Yang, K.Q. The extended F-expansion method and exact solutions of nonlinear PDEs. Chaos Solitons Fractals, 22: 111—121 (2004) [15] Malfliet, W., Hereman, W. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scripta, 54: 563–568 (1996) [16] Matveev, V.B., Salle, M.A. Darboux Transformation and Solitons. Springer, Berlin, 1991 [17] Shou, D.H., He, J.H. Application of parameter-expanding method to strongly nonlinear oscillators. Int. J. Nonlinear Sci., 8: 121–124 (2007) [18] Wang, M.L., Li, X., Zhang, J. The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A, 372: 417–423 (2008) [19] Wu, X.H., He J.H. Solutary Solutions, Periodic Solutions and Compacton Like Solutions Using the ExpFunction Method. Comput. Math. Appl., 54: 966–986 (2007) [20] Wazwaz, A.M. The Tanh Method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations. Chaos Solitons Fractals, 25: 55–63 (2005) [21] Wadati, W., Sanuki, H., Konno, K. Relationships among inverse method, Backlund transformation and infinite number of conservation laws. Prog. Theor. Phys., 53: 419–436 (1975) [22] Xu, L. He’s parameter-expanding methods for strongly nonlinear oscillators. J. Comput. Appl. Math., 207: 148–154 (2007) [23] Yan, C. A simple transformation for nonlinear waves. Phys. Lett. A, 224: 77—84 (1996) [24] Zhou, Y.B., Wang, M.L., Wang, Y.M. Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys. Lett. A, 308: 31–36 (2003)