Int. Journal of Math. Analysis, Vol. 7, 2013, no. 34, 1647-1666 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.3483 Traveling Wave Solutions for a Generalized Kawahara and Hunter-Saxton Equations Elham M. Al-Ali Mathematics Department, Faculty of Science Tabouk University, Tabouk, Saudi Arabia dr.elham.alali@gmail.com Copyright c 2013 Elham M. Al-Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, I use sine - cosine method to obtain a new traveling wave solutions of a generalized Kawahara equation and I obtain a new exact solution for Hunter - Saxton equation by direct integration. The traveling wave solutions for the above equation are presented to obtain novel exact solutions for the same equations. These solutions plays an important role in the modeling of many physical phenomena such as plasma waves, magento - acoustic wave and nematic liquid crystals. Mathematics Subject Classification: 35; 53C; 58J; 58Z05 Keywords: Nonlinear evolution equations; Kawahara equation; traveling wave solutions 1. Introduction Many partial differential equations which are of interest to study and investigate due to the role they play in various areas of mathematics and physics are included in this category [1-3].
1648 Elham M. Al-Ali The formulation of classical theory of surfaces in a form familiar to the soliton theory, which makes possible an application of the analytical methods of this theory to integrable cases [4-6]. When studying these nonlinear phenomena, which particularly plays a major role in many fields of physics, one can encounter with an equation of the form u t + ru n u x + pu 3x qu 5x =0. (1) where r, p and q are nonzero arbitrary constants and n =1, 2, 3,. When n = 1 and n =2, Eq. (1) is called Kawahara and modified Kawahara equations, respectively. In Eq. (1) the second term is convective part and the third term is dispersive part. The Kawahara and modified Kawahara equations have been the subject of extensive research work in recent decades in [7-9]. Nonlinear problems are more difficult to solve than linear ones. When studying these nonlinear phenomena, which particularly plays a major role in many fields of physics, such as fluid mechanics, solid state physics and plasma physics [10-13]. Several direct methods have been recently proposed for obtaining exact and approximate analytic solutions for nonlinear evolution equations, such as homogeneous balance method [14], the modified tanh-function method [15], the extended tanh-function method [16], the tanh-function method [17], the Jacobi elliptic function expansion method [18] and the sine-cosine method [19], and so on. The modified Kawahara equation also has wide applications in physics such as plasma waves, capillary-gravity water waves, water waves with surface tension, shallow water waves and so on [20-22]. The Hunter - Saxton equation describes the propagation of waves in a massive director field of a nematic liquid cristal u(x, t) being related to the deviation of the average orientation of the molecules from an equilibrium position [23]. The Hunter - Saxton equation first appeared in [24] as an asymptotic equation for rotators in liquid crystals. 2u x u 2x + u xxt + uu 3x =0, (2) where u(x, t) is a real - valued function, are integrable models for the propagation of nonlinear waves in 1 + 1 dimension. Rest of the paper is organized as follows. In section 2 sine-cosine method is produced. Section 3 is devoted to employed the method on the generalized Kawahara equation. In section 4 the new solution of The Hunter - Saxton equation by direct integration. The brief conclusion of this paper in section 5. Finally, some references are listed in the end.
Traveling wave solutions 1649 2. The sine - cosine method Now we describe the sine - cosine method [19], for a given nonlinear evolution equation, say, in two variables [25-30], f(u, u x,u t,u xx,u xt,u tt, )=0, (3) where u(x, t) is traveling wave solution. We first consider it traveling wave solutions u(x, t) =u(ξ), ξ =(x ct + k), where u t = cu ξ, u x = u ξ, u 2x = u 2ξ, (4) where c is constant parameter to be determined, and k is an arbitrary constant, then equation (3) becomes an ordinary differential equation, which is integrated as long as all terms contain derivatives. The associated integration constants can be taken as zero. The next crucial step is that the solution we are looking for is expressed in the form or u(ξ) =λ sin β (μξ), and μξ mπ m =0, ±1, ±2, (5) u(ξ) =λ cos β (2m +1)π (μξ), and μξ m =0, ±1, ±2, (6) 2 where λ and μ are constants to be determined, μ and c are the wave number and the wave speed, respectively [20-23]. We use u(ξ) =λ sin β (μξ), u n+1 = λ n+1 sin (n+1)β (μξ), u ξ = λβμ cos(μξ) sin β 1 (μξ) u 2ξ = μ 2 λβ(β 1) sin β 2 (μξ) μ 2 λβ 2 sin β (μξ), u 3ξ = μ 3 λβ(β 1)(β 2) cos(μξ) sin β 3 (μξ) μ 3 λβ 3 cos(μξ) sin β 1 (μξ), (7) u 4ξ = μ 4 λβ 4 sin β (μξ) 2μ 4 λβ(β 1)(β 2 2β + 2) sin β 2 (μξ)+μ 4 λβ(β 1)(β 2)(β 3) sin β 4 (μξ), u 5ξ = μ 5 λβ 5 cos(μξ) sin β 1 (μξ) 2μ 5 λβ(β 1)(β 2)(β 2 2β+2) cos(μξ) sin β 3 (μξ) +μ 5 λβ(β 1)(β 2)(β 3)(β 4) cos(μξ) sin β 5 (μξ). The parameter β will be found by balancing the highest-order nonlinear terms with the highest-order partial derivative term in the given equation and then give the formal solution. Substituting the formal solution (5) or (6)
1650 Elham M. Al-Ali into the ordinary differential equation obtained above and the change it into hyperbolic polynomial identities for the intermediate variable ξ. Collect all terms with the same power in sin k (μξ) and cos(μξ) sin k (μξ) or cos j (μξ) and sin(μξ) cos j (μξ) and set to zero their coefficients to get algebraic relations among the unknowns c, λ, μ, β. With the aid of Mathematica and using the Wu s elimination method [31], we solve for the above unknowns of the equations to finally obtain the traveling wave solutions of the given nonlinear evolution equation. 3. Traveling wave solutions for a generalized Kawahara equation For the generalized Kawahara equation (1) which contains some particular important equations such as Kawahara and modified Kawahara equations. In order to get a new traveling wave solutions of equation (1). Substituting (4) into (1), we obtain an ordinary differential equation cu ξ + ru n u ξ + pu 3ξ qu 5ξ =0. (8) Integrating Eq. (8) and integration constant can be taken as zero, we find cu + r n +1 un+1 + pu 2ξ qu 4ξ =0. (9) Substituting Eq. (7) into Eq. (9), we obtain cλ sin β (μξ)+ r n +1 λn+1 sin (n+1)β (μξ)+pμ 2 λβ(β 1) sin β 2 (μξ) pμ 2 λβ 2 sin β (μξ) qμ 4 λβ 4 sin β (μξ)+2qμ 4 λβ(β 1)(β 2 2β + 2) sin β 2 (μξ) qμ 4 λβ(β 1)(β 2)(β 3) sin β 4 (μξ) =0. (10) Setting the coefficients of sin β μξ, sin β 2 μξ to zero and equating the exponents of sin (n+1)β μξ, and sin β 4 μξ, we find the following equations: cλ pμ 2 λβ 2 qμ 4 λβ 4 =0, pμ 2 λβ(β 1) + 2qμ 4 λβ(β 1)(β 2 2β +2)=0, β 4=nβ + β (11) r n +1 λn+1 qμ 4 λβ(β 1)(β 2)(β 3) = 0.
Traveling wave solutions 1651 We now solve the above set of equations (11) by using the Wu s elimination method, and obtain the following solutions: β = 4 n, c = 4p2 (2 + n) 2 q(8 + 4n + n 2 ), 2 μ = n p 2 q(8 + 4n + n 2 ), (12) { } 1 c(n + 1)(4 + n)(4+3n) n λ =. 8r(n +2) We find the following four types of traveling wave solutions for the generalized Kawahara equation (1): Type 1. For pq > 0 { } 1 c(n + 1)(4 + n)(4+3n) u 1 (x, t) =( i) 4 n 4 n csch n 8r(n +2) Type 2. For pq > 0 { } 1 c(n + 1)(4 + n)(4+3n) n 4 u 2 (x, t) = sech n 8r(n +2) Type 3. For pq < 0 { } 1 c(n + 1)(4 + n)(4+3n) n 4 u 3 (x, t) = sec n 8r(n +2) Type 4. For pq < 0 { } 1 c(n + 1)(4 + n)(4+3n) n 4 u 4 (x, t) = csc n 8r(n +2) n [ 2 n [ 2 n [ 2 n [ 2 p (x ct + k)]. q(8 + 4n + n 2 ) (13) p (x ct + k)]. q(8 + 4n + n 2 ) (14) p (x ct + k)]. q(8 + 4n + n 2 ) (15) p (x ct + k)]. q(8 + 4n + n 2 ) (16) From the above solution of generalized Kawahara equation, I can find the solutions of Kawahara and modified Kawahara equations as follows:
1652 Elham M. Al-Ali (1) For n = 1, we obtain the Kawahara equation in the form u t + ruu x + pu 3x qu 5x =0, (17) and the solitary wave solutions of Kawahara equation are: Type 1. For pq > 0 Type 2. For pq > 0 Type 3. For pq < 0 Type 4. For pq < 0 u 5 (x, t) = 35c 12r csch4 [ 1 2 u 6 (x, t) = 35c 12r sech4 [ 1 2 u 7 (x, t) = 35c 12r csc4 [ 1 2 u 8 (x, t) = 35c 12r sec4 [ 1 2 p 13q p 13q p 13q p 13q 36p2 (x t + k)]. (18) 169q 36p2 (x t + k)]. (19) 169q 36p2 (x t + k)]. (20) 169q 36p2 (x t + k)]. (21) 169q (2) For n = 2, we obtain the modified Kawahara equation in the form u t + ru 2 u x + pu 3x qu 5x =0, (22) we obtain the following four types of solitary wave solutions Type 1. For pq > 0 45c p 4p2 u 9 (x, t) = 8r csch2 [ (x t + k)]. (23) 20q 25q Type 2. For pq > 0 Type 3. For pq < 0 u 10 (x, t) = u 11 (x, t) = 45c p 4p2 8r sech2 [ (x t + k)]. (24) 20q 25q 45c p 4p2 8r csc2 [ (x t + k)]. (25) 20q 25q
Traveling wave solutions 1653 Type 4. For pq < 0 u 12 (x, t) = 45c p 4p2 8r sec2 [ (x t + k)]. (26) 20q 25q 4. Exact solution for Hunter - Saxton equation I now consider the Hunter - Saxton equation (2), by using the traveling wave solutions u(x, t) = u(ξ), ξ =(x ct + k) in Eq. (2) and I obtained nonlinear ordinary differential equation in the form [32-38] 2u u cu + uu =0, (27) by integrating Eq. obtain (27) and integration constant can be taken as zero, we u 2 2 +(u c)u =0, (28) 1 multiply Eq. (28) by u c, I obtain I can write Eq. (29) in the form 1 2 u c u 2 + u c u =0, (29) [ u c u ] =0, (30) by integrating Eq. (30), I obtain the solution of Hunter - Saxton equation as the form u 13 =(d 1 (x ct + k)+d 2 ) 2 3 + c, where d1 and d 2 are constants. (31) 5. Conclusions In this paper, I obtain a new traveling wave solutions for the generalized Kawahara and Hunter - Saxton equations. With the aid of a symbolic computation system, three types of more general traveling wave solutions ( including hyperbolic functions, trigonometric functions and radical functions) with free
1654 Elham M. Al-Ali parameters are constructed. Solutions concerning solitary and periodic waves are also given by setting the three arbitrary parameters, involved in the traveling waves, as special values. The obtained results show that sine - cosine method is very powerful and convenient mathematical tool for nonlinear evolution equations in science and engineering. References. 1. Hirota, R., The direct method in soliton theory, Cambridge University Press, Cambridge 2004. 2. Crampin, M., Solitons and SL(2,R), Phys. Lett. 66A(1978)170-172. 3. Khater, A. H.; Callebaut, D. K.; Sayed, S. M., Conservation laws for some nonlinear evolution equations which describe pseudospherical surfaces, J. of Geometry and Phys. 51(2004) 332-352. 4. Chadan, K.; Sabatier, P. C., Inverse problem in quantum scattering theory (springer, New York, 1977). 5. Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear Media, Soviet Phys. JETP 34(1972) 62-69. 6. Chern, S. S.; Tenenblat, K., Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74(1986)55-83. 7. Kudryashov, N. A., A note on new exact solutions for the Kawahara equation using Exp. function method, Journal of Computational and applied mathematics, 234(2012)3511-3512. 8. Wazwaz, A. M., Compacton solutions of the Kawahara type nonlinear dispersive equation, Appl. Math. Comput. 25 (2005) 1155-1160. 9. Wazwaz, A. M., New solitary wave solutions to the modified Kawahara equation, Phys. Lett. A8 (2007) 588-592. 10. Scott, A. C.; Chu, F.Y.F.; Mclaughlin, D.W., The soliton. A new concept in applied Science, Proc. IEEE 61(1973)1443-1483. 11. Reyes, E. G., On Geometrically integrable equations and Hierarchies of pseudo-spherical type, Contemporary Mathematics, 285(2001)145-155.
Traveling wave solutions 1655 12. K. Tenenblat, Transformations of manifolds and applications to differential equations, Pitman Monographs and Surveys in Pure and Applied Mathematics 93(1998). Addison Wesley Longman, England. 13. Kawahara, T., Oscillatory solitary waves in dispersive media J. Phys. Soc. Jpn. 33(1972) 260-271. 14. Fan, E.; Zhang, H., A note on the homogeneous balance method, Phys. Lett. A246 (1998) 403-406. 15. Sirendaoreji, Auxiliary equation method and new solutions of kleingordon equations, Chaos, Solitons and Fractals 31 (2007) 943-950. 16. Fan, E., Extended tanh- function method and its applications to nonlinear equations, Phys. Lett. A277 (2000) 212-219. 17. Malfliet, W.; Hereman, W., The tanh method:i. Exact solutions of nonlinear wave equations, Phys. Scripta 54 (1996) 569-575. 18. Liu, S. K.; Fu, Z. T.; Liu, S. D., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A298 (2001) 69-74. 19. Yusufoglu, E.; Bekir, A.; Alp, M., Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using sine - cosine method, Chaos, Solitons and Fractals 37 (2008) 1193-1197. 20. Shuangping, T.; Shuangbin, C., Existence and uniqueness of solutions of nonlinear Kawahara equations, Chinese Annals. of Mathematics, Series 23A (2002) 221-228. 21. Sirendaoreji, New exact traveling wave solutions for the Kawahara and modified Kawahara equations, Chaos, Solitons and Fractals 19 (2004) 147-150. 22. Kaya, D.; Al - Khaled, K., A numuerical comparison of a Kawahara equation, Phys. Lett. A363 (2007) 433-439. 23. Gleason, K., On the periodic Hunter - Saxton equation, Senior Thesis, University of Notre Dame (2005). 24. Hunter, J. K.; Saxton, R., Dynamics of director fields, SIAM J. Appl. Math. 51 (1991) 1498-1521.
1656 Elham M. Al-Ali 25. Khater, A. H.; Callebaut, D. K.; Sayed, S. M.; Exact solutions for some nonlinear evolution equations which describe pseudospherical surfaces, J. Comp. Appl. Math. 189(2006) 387-411. 26. Sayed, S. M.; Elkholy, A. M.; Gharib, G. M., Exact solutions and conservation laws for Ibragimov- Shabat equation which describe pseudospherical surface, Computational & Applied Mathematics, 27(2008)305-318. 27. Sayed, S. M.; Elhamahmy, O. O.; Gharib, G. M.; Travelling wave solutions for the KdV-Burgers-Kuramoto and nonlinear Schrodinger equations which describe pseudo-spherical surfaces, Journal of Applied Mathematics 2008 (2008)10-17. 28. Sayed, S.M.; Gharib, G. M., Canonical reduction of Self-Dual Yang- Mills equations to Fitzhugh-Nagumo equation and exact solutions, Chaos Solitons & Fractals 39 (2009)492-498. 29. Bobenko, A. I., Surfaces in Terms of 2 by 2 Matrices. Old and New Integrable Cases, SFB288 Preprint 66 (1993) In: A.P. Fordy, J.C. Wood (eds.) Harmonic Maps and Integrable Systems, Vieweg, Braunschweig/Wiesbaden 1994, pp. 81-127 30. Terng, C.L., Soliton equations and differential geometry, J. of Diff. Geometry 45(1997) 407-412. 31. Wu, W.T., Polynomial equations-solving and its applications, Algorithms and Computation, (Beijing 1994), 1-9, Lecture Notes in Comput. Sci. 834, Springer-Verlag, Berlin, (1994). 32. Wang, M.L.; Li, Z.B., Application of homogeneous balances method to exact solution of nonlinear equation in mathematical physics, Phys. Lett. A216(1996)67-72. 33. Fokas, A. S.; Gelfand, I. M., Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transform. Lett. Math. Phys. 32 (1994) 189-210. 34. Fokas, A. S.,A unified transform method for solving linear and certain nonlinear PDE s. Proc. R. Soc. A. 453 (1997) 1411-1443. 35. Fokas, A. S., Two dimensional linear PDE s in a convex polygon. Proc. R. Soc. A. 457 (2001) 371-393.
Traveling wave solutions 1657 36. Fokas, A. S., Nonlinear evolution equations on the half line. Commun. Math. Phys. 230 (2002) 1-39. 37. Xia, T.C.; Zhang, H.Q.; Yan, Z.Y., New explicit exact travelling wave solution for a compound KdV-Burgers equation, Chinese Phys. 8, (2001)694-697. 38. Yan, C.T., A simple transformation for nonlinear waves, Phys. Lett. A224(1996)77-82. Received: April 10, 2013
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