Journl of Physics: Conference Series Solving the (3+-dimensionl potentil YTSF eqution with Exp-function method To cite this rticle: Y-P Wng 8 J. Phys.: Conf. Ser. 96 86 View the rticle online for updtes nd enhncements. Recent cittions - The (G/G-expnsion method for solving nonliner PDE descriing the nonliner low-pss electricl lines Ayd M. Shhoot et l - Solitons nd other solutions for higherorder NLS eqution nd quntum ZK eqution using the extended simplest eqution method Elsyed M.E. Zyed et l - Solitons nd other solutions to the resonnt nonliner Schrodinger eqution with oth sptio-temporl nd inter-modl dispersions using different techniques Elsyed M.E. Zyed nd Rehm M.A. Shohi This content ws downloded from IP ddress 8.5.3.83 on 8//8 t :3
7 Interntionl Symposium on Nonliner Dynmics (7 ISND IOP Pulishing Journl of Physics: Conference Series 96 (8 86 doi:.88/7-6596/96//86 Solving the (3+-dimensionl Potentil YTSF Eqution with Exp-function Method Yue-Peng Wng * Deprtment of Mthemtics Nnjing University of informtion Science nd Technology Nnjing P.R.Chin E-mil: eduwyp@63.com Astrct. In this pper the (3+-dimensionl potentil-ytsf eqution is solved using the exp-function method with the id of symolic computtion s result new generlized solitry solutions nd periodic solutions with free prmeters re therefore otined nd the free prmeters in the otined generlized solutions might imply some meningful results in physicl process. As specil cse some nown solutions cquired y the tnh-coth method in open literture re recovered due to the suitle choice of free prmeters. It is shown tht Expfunction method is stright concise relile nd promising mthemticl tool to solve nonliner evolution equtions rising in mthemticl physics.. Introduction Nonlinerity is fscinting element of nture tody mny scientists see nonliner science s the most importnt frontier for the fundmentl understnding of nture. Mny complex physicl phenomen re frequently descried nd modeled y nonliner evolution eqution so the exct or nlyticl solutions of the discussed nonliner evolution eqution ecome more nd more importnt which is considered not only vlule tool in checing the ccurcy of computtionl dynmics ut lso gives us good help to redily understnd the essentils of complex physicl phenomenon e.g. collision of two solitry solutions. For this purpose mny methods re therefore proposed such s the vritionl itertion method[] tnh method[3] nd others. A heuristic review on recently developed nlyticl methods is ville in Refs.[56]. Very recently strightforwrd nd concise method clled Exp-function method is proposed y He nd Wu [7] nd Some illustrtive exmples in Refs.[78] showed tht this method is very effective to serch for vrious solitry nd periodic solutions of nonliner equtions. Recently new (3 + -dimensionl nonliner evolution eqution clled the potentil- YTSF eqution ws derived y Yu Tod Ss nd Fuuym in [9] which reds: where u u( x y z t. * To whom ny correspondence should e ddressed. u + u + u u + u u + 3u xt xxxz x xz xx z yy ( c 8 IOP Pulishing Ltd
7 Interntionl Symposium on Nonliner Dynmics (7 ISND IOP Pulishing Journl of Physics: Conference Series 96 (8 86 doi:.88/7-6596/96//86 The Eq.( hs een lredy investigted y the uto-bclund trnsformtion[] nd tnh-coth method [] respectively. In the present pper we im to pply the Exp-function method to solve the potentil-ytsf eqution.. Procedure for solving the Potentil-YTSF eqution According to the Exp-function method first using wve trnsformtion: ξ x + ly + rz + ωt whereξ is complex vrile Eq.( cn e converted into n ordinry differentil eqution in vrileξ : 3 ( u + 6 ru u + ru ω where the prime denotes the derivtive with respect toξ. By virtue of the Exp-function method we ssume tht the solution of Eq.(3 is of the form cexp( cξ + + d exp( dξ u( ξ pexp( pξ + + qexp( qξ ( where the cpd nd q re positive integers which cn e freely chosen nnd m re unnown constnts to e determined lter. By lncing the highest order of liner term u with the highest order of nonliner term uu the vlues of p nd c cn e determined s follows: c exp[(5 p+ c ξ ] + u c exp(6 pξ + (5 nd c3exp[(c+ p ξ ] + c3exp[(c+ p ξ ] + uu cexp[(6 p ξ] + cexp[(6 p ξ ] + (6 Setting c+ 5p c+ p (7 we hve p c Similrly to determine vlues of q nd d we lnce the lowest order liner term of Expfunction in Eq.(3 + d exp[( d 5 q ξ ] u + d exp[( 6 q ξ ] (8 nd + d3exp[( d q ξ ] + d3exp[( d q ξ ] uu + dexp[( 6 q ξ] + dexp[( 6 q ξ] (9 This requires d 5q d q ( ( (3
7 Interntionl Symposium on Nonliner Dynmics (7 ISND IOP Pulishing Journl of Physics: Conference Series 96 (8 86 doi:.88/7-6596/96//86 which leds to the result d q... Cse I : p c q d In this cse Eq.( ecomes exp( ξ + + exp( ξ u( ξ exp( ξ + + exp( ξ Sustituting Eq.( into Eq.(3 we hve [ C exp( ξ + C exp(3 exp( exp( exp( 3 ξ + C ξ + C ξ + C + C ξ A + C exp( ξ + C exp( 3 ξ + C exp( ξ ] 3 Where A [ exp( ξ + + exp( ξ ] Equting the coefficients of exp( nξ to e zero we hve C C3 C C C C 3 C C C Solving this system of lgeric equtions y the help of Mple softwre we cn identify the four sets of reltions: 3 l l r r + r l l r r 3 + r l l r r + r 3 ( ( (3 ( ( + ± ( + ( + l l r r (5 + r 3 Sustituting Eqs.(3-6 into Eq.( four sets of generlized solutions with some free prmeters cn e therefore otined for exmple y Eq.(3 the following solution yields: (6 3
7 Interntionl Symposium on Nonliner Dynmics (7 ISND IOP Pulishing Journl of Physics: Conference Series 96 (8 86 doi:.88/7-6596/96//86 u + exp( ξ exp( ξ 3 + r where ξ x + ly + rz + t ; r lre free prmeters. Prticulrly for Eq.(7 when c < 3 we ssume tht l r c 3 α + α then the following solution cn e derived: u α + (c+ 3 tnh[ (c+ 3( x+ y+ z ct] Which is just the solution Eq.(58 rising in open literture[]; if l r c 3 α α nother solution Eq.(6 in literture [3 ] is lso otined s follows: u α + (c+ 3 coth[ (c+ 3( x+ y+ z ct]. (9 In ddition when c > 3 we cn set tht l r I c+ 3 I α + α or l r I c+ 3 I α α correspondingly the two solutions elow shown in open literture[] re derived gin: u α c+ 3tn[ c+ 3( x+ y+ z ct ] u α + c+ 3cot[ c+ 3( x+ y+ z ct ].. Cse II: p c q d For the se of simplicity we set so Eq.( reduces to exp( ξ + exp( ξ + + exp( ξ u( ξ exp( ξ + exp( ξ + + exp( ξ Inserting Eq.( into Eq.(3 y the sme mnipultion s cse I leds to the following results: ( + ( + r r (7 (8 ( ( ( r+ 3 l l (3
7 Interntionl Symposium on Nonliner Dynmics (7 ISND IOP Pulishing Journl of Physics: Conference Series 96 (8 86 doi:.88/7-6596/96//86 3 + 3 r 6+ l l r r ( ( ( + + 3 + r ( + ( ( + + l l r r (5 + r 3 l l r r (6 + 3 r five solutions of Eq.( cn e so derived respectively..3. Cse III: p c q d For simplicity we ssume tht then we hve l l r r (7 exp( ξ + expξ + + exp( ξ + exp( ξ u( ξ exp( ξ + + exp( ξ Sustituting Eq.(8 into Eq.(3 nd y smple clcultion with Mple the following reltions re otined: ( + ( (8 r+ l 3 3 l l r r (9 ( 8 3 + 6 r l l r r (3 ( + ( ± 5
7 Interntionl Symposium on Nonliner Dynmics (7 ISND IOP Pulishing Journl of Physics: Conference Series 96 (8 86 doi:.88/7-6596/96//86 3 + r l l r r (3 6 + ( 6 + 3 r+ 3 l l l r r (3 ( r+ l 3 3 l l r r (33 ( r+ l 3 3 l l r r (3 Inserting Eqs.(9-3 into the Eq.(8 respectively the corresponding six solutions cn e cquired. Aiming t Eq.(9 which is sustituted into the Eq.(8 then there is solution elow: exp( ξ ( + ( + exp( ξ u exp( ξ + + exp( ξ 3 r+ Where ξ x + ly + rz + t ; nd l r re free prmeters. In cse then the Eq.(35 cn e converted into the following form: u h ξ + ξ ξ sec ( [ cosh( sinh( ] 3 r+ whereξ x + ly + rz + t ; l r re free prmeters. To our nowledge these solitry solutions with some free prmeters otined ove re first ppered in literture. 3. Conclusions The Exp-function method leds to generlized solitry solutions with some free prmeters. The free prmeters might imply some physiclly meningful results of the (3+-dimensionl (35 (36 6
7 Interntionl Symposium on Nonliner Dynmics (7 ISND IOP Pulishing Journl of Physics: Conference Series 96 (8 86 doi:.88/7-6596/96//86 potentil-ytsf equtionnd due to the suitle choice of free prmeters some nown solutions cquired y the tnh-coth method in open literture[] re recovered. The Expfunction method is proved to e powerful mthemticl tool to the serch for generlized solitry solutions nd generlized periodic solutions of nonliner wve equtions it is simple strightforwrd nd effective. Acnowledgment The wor ws supported y the Post-Doctorl Science Foundtion of Jingsu Province of Chin(Grnt No. 6C; KLME of Jingsu Province of Chin (Grnt No. KLME6 References [] Moghimi M. nd Hejzi F S A. 7 Chos. Soliton. Frct. 33 756 [] Aulwf E M Adou M A nd MhmoudA A 7 Chos. Soliton. Frct. 3 38 [3] Wzwz A M Appl. Mth. Comput.5 73 [] El-Wil S A nd Adou M A. 7 Chos. Soliton. Frct. 3.8 [5] He J H 6 Non-perturtive method for strongly nonliner prolems (Disserttion Berlin: De-Verlg im internet GmH. [6] He J H 6 Int. J. Mod. Phys. B [7] He J H nd Wu X H 6 Chos. Soliton. Frct. 3 7 [8] Zhng S 7 Appliction of Exp-function method to KdV eqution with vrile coefficients. Phys. Lett. A [9] Yu S J Tod K Ss N nd Fuuym T. 998 J. Phys. A 3 3337 [] Yn Z. 3 Phys. Lett. A 38 78 [] Wzwz A M. 7 New solutions of distinct physicl structures to high-dimensionl nonliner evolution equtions Appl. Mth. Comput. 7