Exact solitary-wave Special Solutions for the Nonlinear Dispersive K(m,n) Equations by Means of the Homotopy Analysis Method

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1 Available a hp://pva.ed/aa Appl. Appl. Mah. ISSN: Special Isse No. (Ags ) pp Applicaions Applied Maheaics: An Inernaional Jornal (AAM) Eac soliary-wave Special Solions for he Nonlinear Dispersive K(,n) Eqaions by Means of he Hooopy Analysis Mehod Ahe Yıldırı * Ege Universiy, Deparen of Maheaics, 35 Bornova İzir, Trkey ahe.yildiri@ege.ed.r, aheyildiri8@gail.co Canan Ünlü İsanbl Universiy, Deparen of Maheaics, 3434 Vezneciler İsanbl, Trkey Syed Taseef Mohyd-Din HITEC Universiy Taila Can, Pakisan Received: April 7, ; Acceped: May, Absrac In his paper, we sdy he nonlinear dispersive K(,n) eqaions which ehibi solions wih soliary paerns. New eac soliary solions are fond. The wo special cases, K(, ) K(3, 3), are chosen o illsrae he concree feares of he hooopy analysis ehod in K(,n) eqaions. The nonlinear eqaions K(,n) are sdied for wo differen cases, naely when = n being odd even inegers. General forlas for he solions of K(,n) eqaions are esablished. Keywords: Hooopy analysis ehod, nonlinear dispersive K(,n) eqaions MSC () #: 35C7, 35C9, 35C *Corresponding ahor: 8

2 AAM: Inern. J., Special Isse No. (Ags ) 83. Inrodcion Searching for soliary solions for nonlinear eqaion in aheaical physics is aracive in he soliary heory. For eaple, Wadai (97, 973) developed he solions for KdV MKdV eqaions. In 993, Rosena Hyan (993) presened a faily of flly nonlinear KdV eqaions K(,n) n,, n 3 () inrodced a class of soliary waves wih copac sppor ha are solions of a wo paraeer faily of flly nonlinear dispersive parial differen eqaions sch as K(, ) eqaion, () Recenly, Wazwaz (a) gave eac special solions wih soliary paerns for he nonlinear dispersive K(,n) eqaions n,, n (3) The new soliary-wave special solions wih copac sppor for he nonlinear dispersive K(,n) eqaions n,, n (4) are presened by Wazwaz (b). Of corse, oher soliary-wave solions of K(,n) eqaions were also fond by any ahors [Rosena (994, 997)]. In recen, Wazwaz (, a, b, 4) have sccessflly sed he Adoian decoposiion ehod o consrc soliary solions for any nonlinear eqaions. The ain goal of his paper is o invesigae he K(,n) eqaions of he for n,, n (5) we wold like o eend he hooopy analysis ehod (HAM) o seek eac special solions wih soliary paerns for (5). The HAM is developed in 99 by Liao (995, 999, 4). This ehod has been sccessflly applied o solve any ypes of nonlinear probles in science engineering by any ahors [Abbasby (7), Ayb e al. (3), Haya e al. (4)]. By he presen ehod, nerical resls can be obained wih sing a few ieraions. The HAM conains he ailiary paraeer, which provides s wih a siple way o adjs conrol he convergence region of solion

3 84 Yildiri e al. series for large vales of. Oher nerical ehods are given low degree of accracy for large vales of. Therefore, he HAM hles linear nonlinear probles wiho any asspion resricion.. The hooopy analysis ehod (HAM) We apply he HAM o he nonlinear dispersive K(,n) eqaions (5). We consider he following differenial eqaion N, (6) where N is a nonlinear operaor for his proble, denoe independen variables, is an nknown fncion. In he frae of HAM, we can consrc he following zeroh-order deforaion: q L U ; q q H N U ; q, (7) where q, is he ebedding paraeer, is an ailiary paraeer, H is an ailiary fncion, L is an ailiary linear operaor, is an iniial gess of U ; q is an nknown fncion on he independen variables q. Obviosly, when q q, i holds U ;, U ;. (8) Using he paraeer q, we ep U ; q in Taylor series as follows: where U ; q (9) q,! U ; q q q. () Asse ha he ailiary linear operaor, he iniial gess, he ailiary paraeer he ailiary fncion H are seleced sch ha he series (9) is convergen a q, hen de o (8) we have

4 AAM: Inern. J., Special Isse No. (Ags ) 85. () Le s define he vecor n,,..., n. () Differeniaing (7) ies wih respec o he ebedding paraeer q, hen seing q finally dividing he by!, we have he so-called h -order deforaion eqaion where L H R, (3) R! N U ; q q q, (4), (5). Finally, for he prpose of copaion, we will approiae he HAM solion () by he following rncaed series:. (6) k k 3. Applicaions In his secion, we wold like o choose wo special eqaions, naely K(, ) K(3, 3) wih specific iniial condiions, o illsrae he above-enioned schee. Eaple. Consider he K(, ) eqaion wih he iniial condiion, (7a) 4 v sinh, (7b) where v is an arbirary consan.

5 86 Yildiri e al. According o (7), he zeroh-order deforaion can be given by, ;,, q L U q q H U U U. (8) We can sar wih an iniial approiaion linear operaor 4 v sinh choose he ailiary L U ; q U ; q, wih he propery L C, where C is an inegral consan. We also choose he ailiary fncion o be H. Hence, he h -order deforaion can be given by where L H R, R 3 i i 3 i i i i. (9) Now he solion of he h -order deforaion eqaions (9) for becoes L R.. () Conseqenly, he firs few ers of he HAM series solion are as follows: 4 v v 3, sinh,, sinh, 3 v sinh v sinh v cosh, 3 3.

6 AAM: Inern. J., Special Isse No. (Ags ) 87 so on. Hence, he HAM series solion (for ) is, vsinh v sinh v cosh v sinhh... () 3 7 Using Taylor series ino (), we find he closed for solion 4 v v sinh. () In addiion, we can develop anoher eac solion for he K(, ) eqaion. Now we consider anoher iniial vale proble of K(, ) eqaion, (3a) 4 v cosh, (3b) Using he anner as discssed above, we obain anoher eac solion given by 4 v v cosh. (4) Eaple. Consider he iniial vale proble K(3, 3) 3 3, (5a) 6v, sinh, (5b) 3 where v is an arbirary consan. According o (7), he zeroh-order deforaion can be given by, ;,, q L U q q H U U 3 U 3 (6) We can sar wih an iniial approiaion 6v sinh 3 we choose he ailiary linear operaor

7 88 Yildiri e al. L U ; q U ; q, wih he propery L C, where C is an inegral consan. We also choose he ailiary fncion o be H. Hence, he h-order deforaion can be given by where L H R, R i 3 i i k i k 3 i k i k i k i k (7) Now he solion of he h-order deforaion eqaions (7) for becoe L R. (8) Conseqenly, he firs few ers of he HAM series solion are as follows: 6v sinh, 3 6 v 6 3 3/, cosh, v v v / 3/ 5/, cosh cosh sinh,. so on. Hence, he HAM series solion (for ) is, v 6 3/ 6 5/ 6 7 / 3 sinh v cosh v sinh v cosh... (9)

8 AAM: Inern. J., Special Isse No. (Ags ) 89 Using Taylor series ino (9), we find he closed for solion 6v v sinh (3) 3 To obain anoher eac solion for K(3, 3), we consider he iniial vale proble of K(3, 3) eqaion 3 3, (3a) 6v, sinh, (3b) 3 According o he siilar seps as discssed above, we have anoher eac solion given by 6v v sinh 3 4. More eac solions 4.. The K(,) ype In Secion 3, wo eac soliary paerns solions were developed in he for 4 v v sinh (3) 4 v v cosh (33) By cobining he wo resls, we will find ha 4 v 4 v Mv sinh Nv cosh, (34) saisfies he K(, ) eqaion, where M N are consans if M=N or M = -N (35) (a) When M = N, we can obain he rivial solion

9 9 Yildiri e al. 4 Nv (36) 3 (b) When M = -N, we can obain he new eac solion 4 v 4 v N vsinh Nv cosh, (37) Moreover, adding a consan o he argens in (3) (33) will ehibi ore eac solions. In oher words, we inrodce he eac solions 4 v vsinh c (38) 4 v v cosh c (39) where c is a consan. 4.. The K(3,3) ype As discssed before, he eac solions for he K(3, 3) eqaion are given in he for 6v v sinh (4) 3 6v v sinh (4) 3 We can obain a new eac solion by cobining he wo resls (4) (4) we find ha 6v v 6v v M sinh N sinh (4) 3 3 saisfies he K(3, 3) eqaion if M = N, M = +N, M = -+N, (43) (a) When M = N, we can obain he rivial solion

10 AAM: Inern. J., Special Isse No. (Ags ) 9 (44) (b) When M = +N, we can obain he new eac solion 6v v 6v v N sinh N sinh (45) 3 3 (c) When M = -+N, we can obain he new eac solion 6v v 6v v N sinh N sinh (46) 3 3 Moreover, adding a consan o he argens in (4) (4) will ehibi ore eac solions 6v v sinh c (47) 3 6v v sinh c (48) 3 where c is a consan. 5. General solions for K(,n) 5.. The K(,n), = n Being Even Ineger When = n being even ineger, we can es several iniial vale probles by sing hooopy analysis ehod. By properly observing several resls, we inrodce he general forlas vp n sinh v c p n n (49) vp n cosh v c p n n, (5)

11 9 Yildiri e al. where c is a consan. 5.. The K(,n), = n Being Odd Ineger For = n being odd ineger n >, several probles were esed by sing hooopy analysis ehod by careflly observing several resls, we inrodce he general forlas vp n sinh v c p n n (5) vp n sinh v c p n n, (5) where c is a consan. And by cobining he wo resls (5) (5), we will find ha n n vp n vp n M sinh v c N sinh v c, p n p n saisfies he K(n,n) eqaion wih n being odd ineger >, where M N are consans if M = N, M = +N, M = -+N (54) Ths his coplees or goal by esablishing general forlas for solions of he nonlinear dispersive K(n,n) ha work for all vales of n >. (53) 6. Conclsion In sary, we have presened he applicaion of hooopy analysis ehod o he nonlinear dispersive K(,n) eqaions. We chose wo special cases, K(, ) K(3, 3) eqaions o illsrae he schee sch ha new eac solions wih soliary paerns are of iporan significance. We developed he new eac solions which are generaed by cobining wo disinc solions of he K(, ) K(3, 3) eqaions. A las, we esablish he general forlas for eac solions of eqaions K(,n) when = n being even odd inegers for n >. I is worh noing ha oher soliary solions of K(,n) eqaion ay be also consrced by sing he hooopy analysis ehod.

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