N-dimensional Auto-Bäcklund Transformation and Exact Solutions to n-dimensional Burgers System

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1 N-dmesoal Ato-Bäckld Trasformato ad Eact Soltos to -dmesoal Brgers System Mglag Wag Jlag Zhag * & Xagzheg L. School of Mathematcs & Statstcs Hea Uversty of Scece & Techology Loyag 4703 PR Cha. School of Mathematcs & Statstcs Lazho Uversty Lazho PR Cha * Correspodg athor. E-mal: zhag660@63.com Abstract: N-dmesoal Bäckld trasformato (BT) Cole-Hopf trasformato ad Ato-Bäckld trasformato (Ato-BT) of -dmesoal Brgers system are derved by sg smplfed homogeeos balace (SHB). By the Ato-BT aother solto of the -dmesoal Brgers system ca be obtaed provded that a partclar solto of the Brgers system s gve. Sce the partclar solto of -dmesoal Brgers system ca be gve easly by the Cole-Hopf trasformato the sg the Ato-BT repeatedly more soltos of -dmesoal Brgers system ca be obtaed sccessvely. Keywords: -dmesoal Brgers system; -dmesoal BT; -dmesoal Cole-Hopf trasformato; -dmesoal Ato-BT; SHB; eact solto AMS(000) Sbect Classfcato: 35Q0; 35Q53. Itrodcto The classcal Brgers eqato s the smplest olear model fld dyamcs [] ad has bee wdely sed the srface pertrbatos acostcal waves electromagetc waves desty waves poplato growth magetohydrodyamc waves [-4] etc. The -dmesoal Brgers system the form [5-7] t μ Δ 0 Δ () wth a rrotatoal codto: s a mportat geeralzato of the Brgers eqato ad was vestgated recetly by Yag Che etc. I paper [5] where the athors have show that the -dmesoal Brgers system () ca be trasformed to -dmesoal lear heat eqato by a -dmesoal Cole-Hopf trasformato. I the preset paper we wll stdy frther -dmesoal Brgers system () by sg the dea of SHB [8] whch meas that a determed fcto f f ( ) ad ts dervatves f f () that appearg the orgal homogeeos balace(hb) [9-] are replaced by a

2 logarthmc fcto l( ) A A A ad ts dervatves ( l ) respectvely where costat A ad the fcto ( are to be determed. The am of ths paper s to derve the -dmesoal BT for -dmesoal Brgers system () ad from the BT to reaso ot the -dmesoal Cole-Hopf trasformato ad the -dmesoal Ato-BT for the -dmesoal Brgers system (). By the Ato-BTaother eact solto of the Brgers system ca be obtaed provded a partclar solto of t s gve. Hece oce the partclar solto s gve sg the Ato-BT repeatedly more eact soltos of the Brgers system ca be obtaed sccessvely.. Dervato of -dmesoal Ato-BT Cosderg the homogeeos balace betwee ad system () ( m m ) accordg to SHB we ca sppose that the solto of the -dmesoal Brgers system () s of the form A(l ) A () ( ) where t be a gve partclar solto of -dmesoal Brgers system ().e. t μ Δ 0 (3) wth the rrotatoal codto costat A ad the fcto ( are to be determed later. Sbstttg () to the left had sde of system () otcg that t t A t A Δ Δ ad sg (3) we have A Δ

3 t t μδ μ Δ A A μ. (4) I order to fd A settg the coeffcet of to zeroyelds A μ 0 whch mples that A -μ. (5) Usg (5) the epresso () becomes μ (6) ad the epresso(4) becomes t t μδ -μ provded the fcto ( satfes the eqato μδ 0... (7) t μδ 0. (8) Based o (6) (7) ad (8) we come to the mportat coclso that f ( ) t be a partclar solto of the -dmesoal Brgers system () ad the fcto ( be a ozero solto of Eq.(8) the the olear trasformato (6) satstes eactly the -dmesoal Brgers system (). The trasformato (6) wth Eq.(8) s called a BT of the -dmesoal Brgers system (). I order to fd eact solto of the -dmesoal Brgers system () we cosder two partclar cases of the -dmesoal BT: The frst case s that Whe 0 the (6) (7) ad (8) become μ ( 6 ) t -μ μδ t μδ 0 ( 7 ) 3

4 t μδ 0 ( 8 ) respectvely. Based o ( 6 ) ( 7 ) ad ( ) we ca say that f 8 ( s a solto of the -dmesoal lear heat eqato (8) the the -dmesoal olear trasformato (6) satfes eactly the -dmesoal Brgers system (). Ths (6) wth (8) have made p the -dmesoal Cole-Hopf trasformato for the -dmesoal Brgers system () whch cocdes wth that obtaed paper [5]. The secod case s that Whe the (6) (7) ad (8) become ( ) μ. ( 6) t - t μδ μ μδ 0 (. 7 ) t μ Δ 0. ( 8 ) respectvely. Based o ( 6 ) ( 7 ) ad ( 8 ) we ca say that f t ( ) s a gve partclar solto of -dmesoal Brgers system () the the -dmesoal olear trasformato ( 6 ) s aother solto of t. Ths ( 6 ) wth ( 8 ) have made p the -dmesoal Ato-BT for the -dmesoal Brgers system (). I the et secto we wll se the Ato-BT to get more soltos of the -dmesoal Brgers system () whe a partclar solto of the system s gve. 3. Applcato of Ato-BT It s well-kow that the -dmesoal lear heat eqato (8) has a solto ( ep (9) ( πμt ) 4μt whch s deoted by 4

5 E ep. ( πμt ) 4μt ( Sbstttg (9) to the -dmesoal Cole-Hopf trasformato ( 6 ) we have a eact solto of -dmesoal Brgers system () as follows E μ E t (0) where E ep. ( πμt ) μt 4μt Sbstttg (0) whch s called a seed solto to the -dmesoal ato-bt ( 6) we have aother solto of -dmesoal Brgers system () as follows ( ) t μ μ. () t ( ) Sbstttg () stead of ( 6 ) to ( 6 ) we have aother ew solto of -dmesoal Brgers system ()as follows () () ( ) 3 () 6μ t μ (). () t( μ Sbstttg () stead of ( 6 ) to ( 6 ) we have aother ew solto of -dmesoal Brgers system ()as follows (3) () ( ) 4 () μ t μ t μ. (3) t 6μ t () ad so o. The soltos ( k ) 3 (... k 0... ) are all ratoal fctos. I the above stace t seems that the compoet of solto for -dmesoal Brgers system () depeds po the correspodg varable ad t oly. For smplcty the soltos () () ad (3) are cosdered as the fctos of varables t ad ad are show Fg.(). However we have other eamples that the compoet ( ) of solto for -dmesoal Brgers system depeds po all varables t. For eample the 5

6 -dmesoal heat eqato (8) has a solto t e cosη ω -μ α η α α -arbtrary costat. (4) Sbstttg (4) to -dmesoal Cole-Hopf trasformato (6) we have a solto of -dmesoal Brgers system () e sη μα ω -μ e cosη α η α. (5) Sbstttg (5) as a seed solto to -dmesoal Ato-BT (6) yelds aother solto of -dmesoal Brgers system () () e s η - cosη e μα ( e cosη) sη (6) ω -μ α η α α -arbtrary costats. The compoet (6) depeds po all varables... ad t ; ad t s a () perodc fcto varableη whch s a lear combato of space varables (... ). For smplcty the soltos (5) ad (6) are cosdered as the fctos of varables ηt ad ad are show Fg.(). (a) (b) (c) Fg.. Plots of the soltos () () ad (3) wth μ (a) Fg.. Plots of the soltos (5) ad (6) wth μ ω α Smlarly we have a solto of -dmesoal Brgers system (b) 6

7 e shη (7) μα e coshη where ω μ. α η α Sbstttg (7) as a seed solto to -dmesoal Ato-BT ( 6 ) yelds a ew solto of -dmeso Brgers system () () e sh η coshη e μα (8) e cosh sh ( η ) η ω μ α η α α -arbtrary costats. The compoet (8) depeds po all varables... ad t as well for () smplcty the soltos (5) ad (6) are cosdered as the fctos of varables are show Fg.(3). ηt ad ad (a) Fg. 3. Plots of the soltos (7) ad (8) wth μ ω α. (b) 4. Coclso Usg the SHB we have derved ot the -dmesoal BT for the -dmesoal Brgers system.the BT volves a gve partclar solto (... ) of the -dmesoal Brgers system as well as the solto φ φ(... of Eq.(8). If (... ) the Eq.(8) becomes classcal -dmesoal heat eqato the BT becomes the well kow Cole-Hopf trasformato; f 0 ad takg φ the Eq.(8) becomes rght the -dmeso Brgers system for. Ths the BT becomes the Ato-BT of the -dmeso Brgers system. Usg Cole-Hopf trasformatot s easy to get a partclar solto of the -dmeso Brgers system. Takg the partclar solto as a seed solto of -dmeso Brgers system ad sg Ato-BT repeatedly more soltos of the -dmeso Brgers system ca be obtaed sccessvely. 7

8 Ackowledgmets The athors epress ther scere thaks to the referee for valable sggestos. Ths work were spported part by the Basc Scece ad the Frot Techology Research Fodato of Hea Provce of Cha (Grat o ) ad the Scetfc Research Iovato Ablty Cltvato Fodato of Hea Uversty of Scece ad Techology (Grat o.0cx0). Refereces [] J. M. Brgers Mathematcal eamples llstratg relatos occrrg the theory of trblet fld moto Verhad. Ko. Neder. Akad. Weteschappe Afd. Natrkde Eerste Sect. 7 (939) -53. [] R. A. Kraekel J. G. Perera M.A. Maa Nolear srface-wave ectatos the Berard Marago system Phys. Rev. A 46 (99) [3] M. J. Ablowtz S. De Lllo The Brgers eqato der determstc ad stochastc forcg Physca D 9 (996) [4] V. Grare A. Mgdal Istatos the Brgers eqato Phys. Rev. E 54(996) [5] Y. Che E. G. Fa M. W. Ye The Hopf-Cole trasformato topologcal soltos ad mltple fso soltos for the -dmesoal Brgers system Physcs Letters A 380(06)9-4. [6] S. F. She J. Zhag ad Z. L. Pa New Eact Solto of (N)-Dmesoal Brgers System Comm. Theor. Phys. 43 (005) [7] C. H. Cha M. Czbak Reglarty of soltos for the crtcal N-dmesoal Brgers eqato A. I. H. Pocaré AN 7 (00) [8] M. L. Wag ad X. Z. L Smplfed Homogeeos balace method ad ts applcato to the Whtham-Broer-Kap model eqatos Joral of Appled Mathematcs ad Physcs (04) [9] M. L. Wag Soltary wave soltos for varat Bossesq eqatos Physcs Letters A 99(995)69-7. [0] M. L. Wag Eact soltos for a compod KdV-Brgers eqato Physcs Letters A 3 (996) [] M. L. WagY. B. Zho Z. B. L Applcatos of a homogeeos balace method to eact soltos of olear eqatos methematcl physcs Physcs Letters A 6( 996) [] J. L. Zhag Y. M. Wag M. L. Wag Z. D. Fag New Applcato of the Homogeeos Balace Prcple. Chese Physcs (003)