New exact travelling wave solutions of bidirectional wave equations

Transcription

1 Shirz University of Technology From the SelectedWorks of Hbiboll Ltifizdeh June, 0 New exct trvelling wve solutions of bidirectionl wve equtions Hbiboll Ltifizdeh, Shirz University of Technology Avilble t:

2 PRAMANA c Indin Acdemy of Sciences Vol. 76, No. 6 journl of June 0 physics pp New exct trvelling wve solutions of bidirectionl wve equtions JONU LEE nd RATHINASAMY SAKTHIVEL Deprtment of Mthemtics, Sungkyunkwn University, Suwon , Republic of Kore Corresponding uthor. E-mil: krskthivel@yhoo.com MS received 8 November 00; ccepted 3 Jnury 0 Abstrct. The surfce wter wves in wter tunnel cn be described by systems of the form [Bon nd Chen, Physic D6, 9 (998] vt u x (uv x u xxx bv xxt = 0, ( u t v x uu x cv xxx du xxt = 0,, b, c nd d re rel constnts. In generl, the exct trvelling wve solutions will be helpful in the theoreticl nd numericl study of the nonliner evolution systems. In this pper, we obtin exct trvelling wve solutions of system ( using the modified tnh coth function method with computerized symbolic computtion. Keywords. Trvelling wve solutions; tnh coth function method; Riccti equtions; symbolic computtion. PACS Nos 0.30.Jr; 0.30.Ik. Introduction The investigtion of exct trvelling wve solutions to nonliner evolution equtions plys n importnt role in the study of nonliner physicl phenomen. In this pper, we consider the following system which ws derived by Bon nd Chen [] hving the form vt u x (uv x u xxx bv xxt = 0, ( u t v x uu x cv xxx du xxt = 0,, b, c nd d re rel constnts. Here x represents the distnce long the chnnel, t is the elpsed time, the vrible v(x, t is the dimensionless devition of the wter surfce from its undisturbed position nd u(x, t is the dimensionless horizontl velocity. This set of equtions is used s model eqution for the propgtion of long wves on the surfce of wter with smll mplitude [,]. In the pst two decdes, severl methods such s Hirot s method [3], Jcobi elliptic function method [4], vritionl itertion method [5 0], exp-function method [ 7], 89

3 Jonu Lee nd Rthinsmy Skthivel homotopy perturbtion method [8 ] nd so on hve been developed nd extended for finding trvelling wve solutions to nonliner evolution equtions. However, prcticlly there is no unified method tht cn be used to hndle ll types of nonlinerity. The tnh-function method is n effective nd direct lgebric method for finding the exct solutions of nonliner evolution problems [,3]. The concept of tnh-function method ws first proposed in [] nd subsequently some generliztions of this method such s extended tnh function method [4,5], the modified extended tnh-function method [6] nd the modified tnh coth method [7] hve been proposed using different uxiliry ordinry differentil equtions nd pplied to mny nonliner problems [8,9]. The modified tnh coth expnsion method for finding solitry trvelling wve solutions to nonliner evolution equtions hs been used extensively in the literture. It is nturl extension to the bsic tnh-function expnsion method. Wzzn [7] used modified tnh coth method nd obtined new exct solutions for some importnt nonliner problems. More recently, Lee nd Skthivel [30] implemented modified tnh coth method to obtin single soliton solutions for the higher-dimensionl integrble equtions s the extended Jcobi elliptic function method is pplied to derive doubly periodic wve solutions. In this pper, we concentrte on finding trvelling wve solutions of system ( with the help of modified tnh coth method. The trvelling wve solutions my be useful in the theoreticl nd numericl studies of the model systems. The computer symbolic systems such s Mple nd Mthemtic llow us to perform complicted nd tedious clcultions.. Exct trvelling wve solutions The stndrd tnh method ws developed by Mlfliet [], the tnh ws introduced s new vrible, becuse ll derivtives of tnh re represented by tnh itself. In this section, we describe briefly the modified tnh coth function method in its systemtized form [6,8]. Suppose we re given nonliner evolution eqution in the form of prtil differentil eqution (PDE for function u(x, t. First, we seek trvelling wve solutions by tking u(x, t = u(η, η = kx ωt, k nd ω represent the wve number nd velocity of the trvelling wve respectively. Substitution into the PDE yields n ordinry differentil eqution (ODE for u(η. The ordinry differentil eqution is then integrted s long s ll terms contin derivtives, the integrtion constnts re considered s zero. The resulting ODE is then solved by the tnh coth method which dmits the use of finite series of functions of the form M M u(η = 0 m Y m (η b m Y m (η, m= m= (3 N N v(η = c 0 c n Y n (η d n Y n (η n= n= nd the Riccti eqution Y = A BY CY, (4 A, B nd C re constnts to be prescribed lter. Here M nd N re positive integers tht will be determined. The prmeters M nd N re usully obtined by blncing the 80 Prmn J. Phys., Vol. 76, No. 6, June 0

4 New exct trvelling wve solutions liner terms of highest order in the resulting eqution with the highest order nonliner terms. Substituting (3 in the ODE nd using (4 results in n lgebric system of equtions in powers of Y tht will led to the determintion of the prmeters m, b m, c n, d n, k nd ω. Hving determined these prmeters we obtin n nlytic solution u(x, t in closed form. Note.. In this pper, we shll consider the following specil solutions of the Riccti eqution (4: (i A =, B = 0, C =, eq. (4 hs solutions Y = tnh η ± i sech η nd Y = coth η ± csch η. (ii A =, B = 0, C =4, eq. (4 hs solutions Y = tnh η nd Y = 4 (tnh η coth η. (iii A =, B = 0, C = 4, eq. (4 hs solutions Y = tn η nd Y = (tn η cot η. 4 To look for the trvelling wve solutions of eq. (, we mke the trnsformtions u(x, t = u(η, v(x, t = v(η, η = kx ωt. Now eq. ( cn be written s ωv ku ku v kuv k 3 u bk ωv = 0, ωu kv kuu ck 3 v dk ωu (5 = 0, the prime denotes derivtive with respect to η. To determine prmeters M nd N, we blnce the liner terms of highest order in eq. (5 with the highest order nonliner terms. This in turn gives M = nd N =. As result, the modified tnh coth method (3 dmits the use of the finite expnsion u(η = 0 Y Y b Y b Y, v(η = c 0 c Y c Y d Y d Y. Substituting eq. (6 in the reduced ODE (5 nd using eq. (4 collecting the coefficients of Y, yields system of lgebric equtions for 0,,, b, b, c 0, c, c, d, d, k nd ω. If we set A =, B = 0, C = in eq. (4, nd solving the system of lgebric equtions using Mple, we obtin the following three sets of nontrivil solutions: 0 = (b 8c 4d E, =±6 E, =±9(3b d E, b = 0, b = 0, (b 46d(3b d c 0 = c(b 6d(3b d, c = (3b d E E, 9(3b d c = c(3b d 3(3b d 3(3b d d = 0, d = 0, k =±,ω = 8c E, (7 c(b d c(b d (6 Prmn J. Phys., Vol. 76, No. 6, June 0 8

5 Jonu Lee nd Rthinsmy Skthivel E = c(b 6d(3b d, (b d(3b d E = c(b 6d(3b d. 0 = E 3, = 0, = b, b = 0, b = b, c 0 = E 4, c = 0, k b ((d bb 36bcdk 4 (b d b c =, d = 0, d = c, 6bc 3(3b d k = c(b d, ω = (b db ± 36bcdk 4 (b d b 6bdk, (8 (d(c c d b(c d 4cdk b (c d 36bcdk 4 (b d b E 3 = 6bdk, E 4 = c k 4 8bc(d bck 4 (b d bck b ((b db ± 36bcdk 4 (b d b 8b. 0 = (3b4c d E, = 3 E 5, 9(3b d =± E, b =±3 E 5, c(b 6d(3b d (3b d b =, c 0 =, c(b 6d(3b d c = 6 (3b d E E, 9(3b d c = 4c(3b d, d 3(3b d =c, d = c, k =± c(b d, 3(3b d ω =±8c E, (9 c(b d d(b d E 5 = c(b 6d(3b d. (0 8 Prmn J. Phys., Vol. 76, No. 6, June 0

6 New exct trvelling wve solutions Substituting Y = tnh η ± i sech η nd Y = coth η ± csch η in eq. (6, the first two sets (7 nd (8 gives the trvelling wve solutions in the following form: u, (x, t = (b 8c 4d E ± 6 E (tnh η ± i sech η v, (x, t = ± 9(3b d E (tnh η ± i sech η, ( (b 46d(3b d c(b 6d(3b d (3b d E E (tnh η ± i sech η 9(3b d c(3b d (tnh η ± i sech η, ( u, (x, t = (b 8c 4d E ± 6 E (coth η ± csch η v, (x, t = η =± ± 9(3b d E (coth η ± csch η, (3 (b 46d(3b d c(b 6d(3b d (3b d E E (coth η ± csch η 3(3b d c(b d 9(3b d c(3b d (coth η ± csch η, (4 ( x 8c ( c d(c d b(c d 4cdk u,3 (x, t = c(b 6d(3b d t. b (c d 36bcdk 4 (b d b 6bdk b (tnh η ± i sech η (tnh η ± i sech η, (5 c k 4 8bc(d bck 4 (b d bck b ((b db ± 36bcdk 4 (b d b v,3 (x, t = 8b k b ((d bb 36bcdk 4 (b d b 6bc (tnh η ± i sech η, (6 (tnh η ± i sech η ( c d(c d b(c d 4cdk b (c d 36bcdk 4 (b d b u,4 (x, t = 6bdk b (coth η ± csch η (coth η ± csch η, (7 Prmn J. Phys., Vol. 76, No. 6, June 0 83

7 Jonu Lee nd Rthinsmy Skthivel c k 4 8bc(d bck 4 (b d bck b ((b db ± 36bcdk 4 (b d b v,4 (x, t = 8b k b ((d bb 36bcdk 4 (b d b 6bc (coth η ± csch η (coth η ± csch η, (8 3(3b d η = c(b d x (b db ± 36bcdk 4 (b d b t. 6bdk Finlly, the third set gives the trvelling wve solutions s u,5 (x, t = (3b 4c d E 3 E 5 (tnh η ± i sech η ± 9(3b d E c(b 6d(3b d (3b d v,5 (x, t = c(b 6d(3b d 6 (3b d E E (tnh η ± i sech η (tnh η ± i sech η u,6 (x, t = (3b 4c d E 3 E 5 (coth η ± csch η ± 9(3b d E (tnh η ± i sech η (tnh η ± i sech η, (tnh η±i sech η (9 (tnh η ± i sech η (tnh η ± i sech η (coth η ± csch η 9(3b d 4c(3b d, (0 (coth η±csch η (coth η ± csch η c(b 6d(3b d (3b d v,6 (x, t = c(b 6d(3b d 6 (3b d E E (coth η ± csch η (coth η ± csch η 9(3b d (coth η ± csch η 4c(3b d (coth η ± csch η, (, ( 84 Prmn J. Phys., Vol. 76, No. 6, June 0

8 New exct trvelling wve solutions 3(3b d η =± c(b d ( x 8c c(b 6d(3b d t. Note.. The modified tnh coth expnsion method is nturl extension to the bsic tnh-function expnsion method. It gives three types of solutions, nmely tnh function expnsion, coth function expnsion, nd tnh coth expnsion. For every tnh function expnsion solution, there is corresponding coth function expnsion solution. It should be mentioned tht by mistke in mny ppers, such tnh coth solutions re climed to be new. However, tnh coth solutions my be delivered tht re new in the sense tht they would not be delivered vi the bsic tnh-function method. Remrk.3. If we set A =, B = 0, C =4ineq. (4 nd by repeting the sme clcultion s bove, we obtin the following trvelling wve solutions for eq. (: u, (x, t = (b 8c 4d E ± 6 E tnh η v, (x, t = ± 9(3b d E tnh η, (3 (b 46d(3b d c(b 6d(3b d (3b d E E tnh η 9(3b d c(3b d tnh η, (4 u, (x, t = (b 8c 4d E ± 3 E (tnh η coth η ± v, (x, t = 9(3b d E (tnh η coth η, (5 4 (b 46d(3b d c(b 6d(3b d 6(3b d E E (tnh η coth η 9(3b d 8c(3b d (tnh η coth η, (6 η =± ( 3(3b d x 8c 4 c(b d c(b 6d(3b d t. u,3 (x, t = (b 8c 4d E ± 6 E coth η v,3 (x, t = ± 9(3b d E coth η, (7 (b 46d(3b d c(b 6d(3b d (3b d E E coth η 9(3b d c(3b d coth η, (8 Prmn J. Phys., Vol. 76, No. 6, June 0 85

9 Jonu Lee nd Rthinsmy Skthivel u,4 (x, t = (b 8c 4d E ± E (tnh η coth η ± 36(3b d E (tnh η coth η, (9 (b 46d(3b d v,4 (x, t = c(b 6d(3b d 8(3b d c(3b d 4(3b d E E (tnh η coth η (tnh η coth η, (30 η =± ( 3(3b d x 8c 4 c(b d c(b 6d(3b d t. u,5 (x, t = (3b4c d E ± 3 E (tnh η coth η 9(3b d ± E (tnh η coth η, (3 c(b 6d(3b d (3b d v,5 (x, t = c(b 6d(3b d ± 6 E E (tnh η coth η 9(3b d 4c(3b d (tnh η coth η, (3 u,6 (x, t = (3b4c d E ± 3 tnh ηcoth η E ± 9(3bd tnh ηcoth η (tnh η coth η ( E, (33 tnh η coth η c(b 6d(3b d (3b d v,6 (x, t = c(b 6d(3b d ± 6 (3b d E E tnh η coth η tnh η coth η (tnh 9(3b d η coth η (, 4c(3b d tnh η coth η (34 η =± ( 4 3(3b d x 8c c(b d c(b 6d(3b d t. 86 Prmn J. Phys., Vol. 76, No. 6, June 0

10 New exct trvelling wve solutions Remrk.4. If we set A =, B = 0, C = 4 in eq. (4, nd solving the system of lgebric equtions using Mple by the sme clcultion s bove, we obtin the following trvelling wve solutions of eq. (: u 3, (x, t =±(b 8c 4d E ± 6 E tn η ± 9(3b d E tn η, (35 (b 46d(3b d v 3, (x, t = c(b 6d(3b d ± (3b d E E tn η 9(3b d c(3b d tn η, (36 u 3, (x, t =±(b 8c 4d E ± 3 E (tn η cot η 9(3b d ± E (tn η cot η, (37 4 (b 46d(3b d v 3, (x, t = c(b 6d(3b d ± 6(3b d E E (tn η cot η 9(3b d 8c(3b d (tn η cot η, (38 η =± ( 3(3b d x 8c 4 c(b d c(b 6d(3b d t. u 3,3 (x, t =±(b 8c 4d E ± 6 E cot η ± 9(3b d E cot η, (39 (b 46d(3b d v 3,3 (x, t = c(b 6d(3b d ± (3b d E E cot η 9(3b d c(3b d cot η, (40 u 3,4 (x, t =±(b 8c 4d E ± E (tn η cot η ± 36(3b d E (tn η cot η, (4 (b 46d(3b d v 3,4 (x, t = c(b 6d(3b d 8(3b d c(3b d ± 4(3b d E E (tn η cot η (tn η coth η, (4 Prmn J. Phys., Vol. 76, No. 6, June 0 87

11 Jonu Lee nd Rthinsmy Skthivel η =± ( 3(3b d x 8c 4 c(b d c(b 6d(3b d t. u 3,5 (x, t =±(3b4c d E ± 3 E (tn η cot η 9(3b d ± E (tn η cot η, (43 c(b 6d(3b d (3b d v 3,5 (x, t = c(b 6d(3b d 6 (3b d E E (tn η cot η 9(3b d 4c(3b d (tn η cot η, (44 u 3,6 (x, t =±(3b4c d E ± 3 tn η cot η E tn η cot η 9(3b d (tn η cot η ( ± E, (45 tn η cot η c(b 6d(3b d (3b d v 3,6 (x, t = c(b 6d(3b d 6 (3b d tn η cot η E E 9(3b d 4c(3b d (tn η cot η ( tn η cot η, (46 tn η cot η η =± ( 4 3(3b d x 8c c(b d c(b 6d(3b d t. Remrk.5. It should be mentioned tht we hve verified ll the obtined solutions by putting them bck into the originl eqution. To the uthors knowledge this is the first ttempt to solve the bidirectionl wve equtions with tnh function method, ll solutions re new nd cnnot be found in the literture. 3. Conclusion In this pper, using the solution of the uxiliry eqution (4 in the modified tnh coth function method, we hve found some new exct trvelling wve solutions for bidirectionl wve equtions. This method lso suggests tht one cn get different exct solutions by 88 Prmn J. Phys., Vol. 76, No. 6, June 0

12 New exct trvelling wve solutions choosing different uxiliry equtions in the tnh-function. It should be noted tht the method used here cn generte not only regulr solutions but lso singulr ones involving csch nd coth functions. Acknowledgements The work of Lee ws supported by the Ntionl Reserch Foundtion Grnt funded by the Koren Government (Ministry of Eduction, Science nd Technology with grnt number NFR C References [] J Bon nd M Chen, Physic D6, 9 (998 [] M Chen, Int. J. Theor. Phys. 37, 547 (998 [3] A M Wzwz, Appl. Mth. Comput. 04, 94 (008 [4] E J Prkes, B R Duffy nd P C Abbott, Phys. Lett. A95, 80 (00 [5] S T Mohyud-Din, M A Noor nd K I Noor, Int. J. Nonliner Sci. Numer., 87 (00 [6] E Hesmeddini nd H Ltifizdeh, Int. J. Nonliner Sci. Numer. 0, 377 (009 [7] M Dehghn nd F Shkeri, J. Comput. Appl. Mth. 4, 435 (008 [8] J H He, Int. J. Non-Liner Mech. 34, 699 (999 [9] JHHendXHWu,Comput. Mth. Appl. 54, 88 (007 [0] R Mokhtri, Int. J. Nonliner Sci. Numer. Simul. 9, 9 (008 [] M A Abdou, A A Solimn nd S T Bsyony, Phys. Lett. A369, 469 (007 [] J H He nd X H Wu, Chos, Solitons nd Frctls 30, 700 (006 [3] R Skthivel nd C Chun, Rep. Mth. Phys. 6, 389 (008 [4] R Skthivel nd C Chun, Z. Nturforsch. A (J. Phys. Sci. 65, 97 (00 [5] R Skthivel, C Chun nd J Lee, Z. Nturforsch. A (J. Phys. Sci. 65, 633 (00 [6] H Hosseini, M M Kbir nd A Khjeh, Int. J. Nonliner Sci. Numer. Simul. 0, 307 (009 [7] S Zhng nd H Zhng, Phys. Lett. A373(30, 50 (009 [8] M Dehghn nd F Shkeri, J. Porous Medi, 765 (008 [9] M Dehghn nd J Mnfin, Z. Nturforsch. A 64, 4 (009 [0] J H He, Comput. Meth. Appl. Mech. Eng. 78, 57 (999 [] A Yildirim nd D Agirseven, Int. J. Nonliner Sci. Numer. Simul. 0, 35 (009 [] W Mlfliet, Am. J. Phys. 60(7, 650 (99 [3] W Mlfliet nd W Heremn, Phys Scr. 54, 563 (996 [4] E J Prkes nd B R Duffy, Comput. Phys. Commun. 98, 88 (996 [5] E Fn, Phys. Lett. A77, (000 [6] S A Elwkil, S K El-lbny, M A Zhrn nd R Sbry, Phys. Lett. A99, 79 (00 [7] L Wzzn, Commun. Nonliner Sci. Numer. Simul. 4, 64 (009 [8] S A El-Wkil, S K El-Lbny, M A Zhrn nd R Sbry, Appl. Mth. Comput. 6, 403 (005 [9] S Zhng nd H Zhng, Phys. Lett. A373(33, 905 (009 [30] J Lee nd R Skthivel, Mod. Phys. Lett. B4, 0 (00 Prmn J. Phys., Vol. 76, No. 6, June 0 89