Generalized double sinh-gordon equation: Symmetry reductions, exact solutions and conservation laws

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1 IJS (05) 9A: Iranan Journal of Scence & echnology hp://ss.shrazu.ac.r Generalzed double snh-gordon equaon: Symmery reducons eac soluons and conservaon laws G. Magalawe B. Muaeea and C. M. Khalque School of Compuer Sascal and Mahemacal Scences Norh-Wes Unversy Pochefsroom Campus Prvae Bag 50 Republc of Souh Afrca Inernaonal Insue for Symmery Analyss and Mahemacal Modellng Deparmen of Mahemacal Scences Norh-Wes Unversy Mafeng Campus Prvae Bag 06 Mmabaho 75 Republc of Souh Afrca E-mals: & Absrac hs paper ams o sudy a generalzed double snh-gordon equaon whch appears n several physcal phenomena such as negrable quanum feld heory n dynamcs and flud dynamcs. Le symmery analyss ogeher wh he smples equaon mehod s used o oban eac soluons for hs equaon. Moreover we derve conservaon laws for he equaon by usng wo dfferen approaches namely he drec mehod and he new conservaon heorem due o Ibragmov. Keywords: Generalzed double snh-gordon equaon; Le symmery analyss; smples equaon mehod; conservaon laws. Inroducon Physcal phenomena n physcs and oher felds are ofen descrbed by nonlnear paral dfferenal equaons (NLPDEs). Fndng eac soluons of NLPDEs s one of he mos mporan as snce hey provde a beer undersandng of he physcal phenomena. Durng he pas several decades researchers have developed numerous mehods o fnd eac soluons of NLPDEs. Some of he mehods found n he leraure are he Hroa blnear mehod (Hroa 00; Ma e al. 0; Zhang and Khalque 0) he dynamcal sysem mehod (L 0: Zhang e al. 0; Zhang and Chen 009) he F -epanson mehod (Wang and L 005) he homogeneous balance mehod (Wang e al. 996) he ( G/ G) -epanson mehod (Wang e al. 008) he Weersrass ellpc funcon epanson mehod (Chen and Yan 006) he eponenal funcon mehod (He and Wu 006) he anh funcon mehod (Wazwaz 00) he eended anh funcon mehod (Wazwaz 007) he smples equaon mehod (Kudryashov 005; Vanov 00) and he Le group analyss mehod (Bluman and Kume 989; Olver 99; Ibragmov ). In hs paper we sudy he generalzed double snh-gordon equaon Correspondng auhor Receved: Sepember 0 / Acceped: March 05 u u snh( = 0 n () where and are non-zero real consans whch appear n several physcal phenomena such as negrable quanum feld heory n dynamcs and flud dynamcs. I should be noed ha when n = = = / and = 0 () reduces o he snh-gordon equaon (Wazwaz 005). Furhermore f = a = b/ and = 0 () becomes he generalzed snh-gordon equaon (Wazwaz 006). Varous mehods have been used o sudy (). In (Wazwaz 005) he anh mehod and varable separable ODE mehod were employed o fnd he eac soluons of (). he auhors of ang and Huang (007) suded he esence of perodc wave solary wave n and an-n wave and unbounded wave soluons of () by usng he mehod of bfurcaon heory of dynamcal sysems. he solary and perodc wave soluons of () were obaned n (Kher and Jabrar 00) by employng ( G/ G) -epanson mehod. In addon was shown ha he soluons obaned n (Kher and Jabrar 00) are more general han hose obaned n (Wazwaz 005). Recenly n (Magalawe and Khalque 0) new eac soluons of () were found by employng eponenal funcon mehod. he purpose of hs paper s wofold. Frsly we use he Le group analyss along wh he smples

2 IJS (05) 9A: equaon mehod o oban eac soluons of he generalzed double snh-gordon equaon (). Secondly we derve conservaon laws for he equaon by usng wo dfferen echnques namely he drec mehod and he new conservaon heorem due o Ibragmov. he Le symmery mehod s based on symmery and nvarance prncples and s a sysemac mehod for solvng dfferenal equaons analycally. I was frs developed by Sophus Le (8-899) and snce hen has become an essenal mahemacal ool for anyone nvesgang mahemacal models of physcal engneerng and naural problems. he Le group analyss mehods are presened n many boos see for eample (Bluman and Kume 989; Olver 99; Ibragmov ). I s well nown ha conservaon laws play a val role n he soluon process of dfferenal equaons (DEs). here s no doub ha he esence of a large number of conservaon laws of a sysem of paral dfferenal equaons (PDEs) s an ndcaon of s negrably (Bluman and Kume 989). Comparson of dfferen approaches o conservaon laws for some paral dfferenal equaons n flud mechancs s gven n (Naz e al. 008). he organzaon of he paper s as follows. In Secon we oban symmery reducons of he generalzed double snh-gordon equaon () usng Le group analyss based on he opmal sysems of one-dmensonal subalgebras of (). Eac soluons are also obaned usng he smples equaon mehod. In Secon we frs recall some defnons and heorems on conservaon laws. We hen consruc conservaon laws for () usng wo approaches; he drec mehod and he new conservaon heorem due o Ibragmov. Fnally concludng remars are presened n Secon.. Symmery reducons and eac soluons of () We assume ha he vecor feld of he form = ( u) ( u) ( u) wll generae he symmery group of (). Applyng [] he second prolongaon o () we oban an overdeermned sysem of egh lnear paral dfferenal equaons namely = 0 = 0 u = 0 = 0 = 0 u = 0 u = 0 n n cosh( n cosh ( cosh( snh( cosh( snh( u Solvng he above equaons one obans he followng hree Le pon symmeres: = = =... One-dmensonal opmal sysem of subalgebras In hs subsecon we frs oban he opmal sysem of one-dmensonal subalgebras of (). hereafer he opmal sysem wll be used o oban he opmal sysem of group-nvaran soluons of (). For hs purpose we nvoe he mehod gven n (Olver 99). Recall ha he commuaor of and denoed by ] s gven by [ [ ] = and he adon ransformaons are gven by Ad ( ep( )) [ [ = ]]. [ he commuaor able of he Le pon symmeres of () and he adon represenaons of he symmery group of () on s Le algebra are presened n able and able respecvely. able. Commuaor able of he Le algebra of sysem () [ ] able. Adon able of he Le algebra of sysem () Ad ( ) ( ) ] hus from ables and and followng he mehod gven n (Olver 99) one can conclude ha an opmal sysem of one-dmensonal subalgebras of () s gven by { c } where c s a non-zero consan.

3 9 IJS (05) 9A: Symmery reducons of () Here he opmal sysem of one-dmensonal subalgebras consruced above wll be used o oban symmery reducons. hereafer we wll oban he eac soluons of (). Case. c he symmery generaor c gves rse o he group-nvaran soluon u = W( () where z = c s an nvaran of he symmery c and W s an arbrary funcon of z. he nseron of () no () yelds he ODE nw( snh nw( ) = 0. ( c ) W( snh z () Usng he ransformaon W ( = ln( H( ) () n on () we oban he nonlnear second-order ordnary dfferenal equaon ( c ) H( H( ( c nh( nh( n ) H( nh( (5) he negraon of he above equaon and reverng bac o orgnal varables yelds [ c (n ep( n ep( n nep() c ep(] = c c nep( du where c and c are consans of negraon. Case. he symmery operaor resuls n he groupnvaran soluon of he form u = W( (6) where z = s an nvaran of and W s an arbrary funcon sasfyng he ODE nw( snh nw( ) = 0. W( snh z (7) Agan usng he ransformaon () equaon (7) becomes H( H( H ( nh( nh( nh( n = 0 whose soluon s nep() c = c [ (n ep( n ep( n ep(] nep( du where c and c are consans of negraon and we oban a seady sae soluon of (). Case. (8) he symmery gves rse o he groupnvaran soluon u = W( (9) where z = s an nvaran of and W sasfes he ODE zw ( W ( snh nw( snh nw( = 0... Eac soluons of () usng smples equaon mehod In hs subsecon we nvoe he smples equaon mehod (Kudryashov 005; Vanov 00) o solve he hghly nonlnear ODE (5). hs wll hen gve us he eac soluon for he generalzed double snh-gordon equaon (). he smples equaons ha we wll use are he Bernoull and Rcca equaons. For deals see for eample (Adem and Khalque 0).... Soluons of () usng he Bernoull equaon as he smples equaon he balancng procedure (Vanov 00) yelds M = so he soluons of (5) ae he form H = A A (0) ( 0 G where G sasfes he Bernoull equaon (Adem and Khalque 0). Inserng (0) no (5) and usng he Bernoull equaon (Adem and Khalque 0) and hereafer equang he coeffcens of powers of G o zero we oban an algebrac sysem of fve equaons n erms of A 0 A namely

4 IJS (05) 9A: n n n n = 0 na A a A A na 6nA A A c = 0 0A a Ab n A A abc A b c A b = 0 A n Ab c A b na A abc A ab a na na = 0 Solvng he above sysem of algebrac equaons wh he ad of Maple one possble soluon s = 0 0 A b( A 0 ) ( A ) ( ) = 0 b c =. aa A na hus reverng bac o he orgnal varables a soluon of () s (Adem and Khalque 0) u( = ln( n cosh[ a( z C)] snh[ a( z C)] A a{ }) bcosh[ a( z C)] bsnh[ a( z C)] () where z = c and C s an arbrary consan of negraon.... Soluons of () usng he Rcca equaon as he smples equaon In hs case he balancng procedure yelds M =. So he soluons of (5) ae he form H = A A () ( 0 G where G sasfes he Rcca equaon (Adem and Khalque 0). Subsung () no (5) and mang use of he Rcca equaon (Adem and Khalque 0) we oban an algebrac sysem of equaons n erms of A 0 A by equang he coeffcens powers of G o zero. he resulng algebrac equaons are A A b na na A a a A c na = 0 aa A A bc A A b c ab a n A bc = 0 0 n A 0 A b 6n A na 0 = 0 n A abc na = 0 A c aa bc aa b a A n A n A A c A b aa A c na Solvng he above equaons we ge Ab A b a A a = a ( a a A ) = aa 0 a( c ) na = and consequenly he soluons of () are (Adem and Khalque 0) b u( = ln( A { anh[ ( z C)]}) n a a and b u( = ln( A { anh( n a a z sech }) z a z Ccosh snh () () where z = c and C s an arbrary consan of negraon.. Conservaon laws of () In hs secon conservaon laws wll be derved for (). However frs we brefly presen some noaons defnons and heorems ha wll be ulzed laer. For deals he reader s referred o (Ibragmov 007)... Prelmnares We consder a h-order sysem of PDEs E ( u u() u( ) ) = 0 = m (5) n of n ndependen varables = ( ) and m m dependen varables u = ( u u u ). Le u() u() u( ) denoe he collecons of all frs second h-order paral dervaves. hs means ha u = D ( u ) u = D D ( u ) respecvely where he oal dervave operaor wh respec o s gven by D = / / / = n. (6) Now consder he sysem of adon equaons o he sysem of h-order dfferenal equaons (5)

5 9 IJS (05) 9A: whch s defned by (Ibragmov 007) E ( u v u( ) v( ) ) = 0 = m (7) where ( v E ) E ( u v u( ) v( ) ) = (8) = m v = v( m and v = ( v v v ) are new dependen varables. he sysem (5) s sad o be self-adon f he subsuon of v = u no he sysem of adon equaons (7) yelds he same sysem (5). If he sysem (5) adms he symmery operaor = / / (9) hen he sysem of adon equaons (7) adms he operaor Y = / = [ v / v D ( )] / v (0) where he operaor (0) s an eenson of (9) o he varable v and are obanable from ( E ) = E. () heorem.. (Ibragmov 007) Every Le pon Le-Bäclund and nonlocal symmery (9) admed by he sysem (5) gves rse o a conservaon law for he sysem conssng of he equaon (5) and he adon equaon (7) where he componens of he conserved vecor = ( n ) are deermned by = L W = n L D D ( W ) L / s wh Lagrangan gven by L = v E ( u u ( )). s s he dfferenaon of L n () up o secondorder dervave yelds () () L L L L W [ D ( )] ( )( ). D W () =.. Consrucon of conservaon laws of () In hs secon conservaon laws wll be consruced for () by wo dfferen mehods namely he drec mehod and he new conservaon heorem. We recall ha he equaon () adms he followng hree Le pon symmery generaors: = = =.... Applcaon of he drec mehod I s well-nown ha here ess a fundamenal relaonshp beween he pon symmeres admed by a gven equaon and he conservaon laws of ha equaon. Followng (Khalque and Mahomed 009) we see ha he conservaon law D D = 0 (5) whch mus be evaluaed on he paral dfferenal equaon can be consdered ogeher wh he followng requremens: [ n] ( ) D ( ) D ( ) = 0 (6) [ n] ( ) D ( ) D ( ) = 0 (7) [n] n whch s he nh prolongaon of a pon symmery of he orgnal equaon. he order of he eenson s equal o he order of he hghes dervave n and. Consequenly for he gven (5)-(7) can be solved o oban he conserved vecors or uple = ( ). he condon (5) on he equaon () gves u u ( u snh() u u u Snce and are ndependen of he second dervaves of u mples ha he coeffcens of u u and u mus be zero. Hence u u = 0 = 0 (8) (9) u ( snh () (0) u

6 IJS (05) 9A: We now consruc he conservaon laws for () usng he hree admed Le pon symmeres. We sar wh he ranslaon symmery = / whch s already n s eended form. he symmery condons (6)-(7) yeld = 0 = 0 () respecvely. herefore from (8)-(0) and () he componens of he conserved vecor of () assocaed wh he symmery are gven by cu c = cu c5u cosh( ] ( c6 n c u u c u c = 5 7 cosh( [ n where c c5 c6 and c 7 are arbrary consans and ( s an arbrary funcon of. Connung n he same manner usng and we oban he componens of he conserved vecor for equaon () as and u c c cosh( = cu c5u [ n cosh( ] c8 n c u u c u p( ) = 5 = 5 6 = c5u c6 c u c respecvely where c c5 c6 and c 8 are consans and p () s an arbrary funcon of. However we noe ha he symmery gves a rval conserved vecor.... Applcaon of he new conservaon heorem In hs subsecon we use he new conservaon heorem gven n (Ibragmov 007) and consruc conservaon laws for (). For applcaons of hs heorem see for eample (racna e al. 0; Gandaras and Khalque 0; da Slva and Frere 0). he adon equaon of () by nvong (8) s E (uv u v) = [v(u u snh( snh(]= 0 () where v = v( s a new dependen varable. hus from () we have v v nv[ cosh( cosh(] () I s clear from he adon equaon () ha equaon () s no self-adon. By recallng () we oban he Lagrangan for he sysem of equaons () and () as L = v[ u u snh(]. () () We frs consder he Le pon symmery generaor = /. I can easly be seen from (0) ha he operaor Y s he same as and ha he Le characersc funconw =. hus by usng () he componens = of he conserved vecor = ( ) are gven by = v( u = uv snh() u v vu. Remar: he conserved vecor conans he arbrary soluon v of he adon equaon () and hence gves an nfne number of conservaon laws. hs remar also apples o he wo cases gven below. () For he symmery = / we have W = u. hus by usng () he symmery generaor gves rse o he followng componens of he conserved vecor: = v u vu = v( u snh() vu. () he symmery = / / has he Le characersc funconw = u u. hus nvong () we oban he conserved vecor gven by = v( u snh() vu vu vu vu = v( u v u snh() v u vu vu.. Concludng remars We have suded he generalzed double snh- Gordon equaon () usng he Le symmery analyss. Symmery reducons based on he opmal

7 95 IJS (05) 9A: sysems of one-dmensonal subalgebras of () and eac soluons wh he help of smples equaon mehod were obaned. hese eac soluons obaned here are dfferen from he ones obaned n (Wazwaz 005; Wazwaz 006; Wazwaz 005; ang and Huang 007; Kher and Jabrar 00; Magalawe and Khalque 0). Also he correcness of he soluons obaned here has been verfed by subsung hem bac no (). Fnally conservaon laws for () were derved by employng wo dfferen mehods; he drec mehod and he new conservaon heorem. he usefulness of conservaon laws was dscussed n he nroducon. Acnowledgemens G. M. would le o han SANHARP NRF and Norh-Wes Unversy Mafeng Campus Souh Afrca for her fnancal suppor. References Adem K. R. & Khalque C. M. (0). Eac Soluons and Conservaon Laws of a (+)-Dmensonal Nonlnear KP-BBM Equaon. Absrac and Appled Analyss Arcle ID Bluman G. W. & Kume S. (989). Symmeres and Dfferenal Equaons. New Yor: Sprnger-Verlag. Chen Y. & Yan Z. (006). he Weersrass ellpc funcon epanson mehod and s applcaons n nonlnear wave equaons. Chaos Solons and Fracals Da Slva P. L. & Frere I. L. (0). Src selfadonness and ceran shallow waer models. arv:.99v [mah-ph] 5 Mar 0. Gandaras M. L. & Khalque C. M. (0). Nonlnearly Self-Adon Conservaon Laws and Soluons for a Forced BBM Equaon. Absrac and Appled Analyss Arcle ID 608. He J. H. & Wu. H. (006). Ep-funcon mehod for nonlnear wave equaons. Chaos Solons and Fracals Hroa R. (00). he Drec Mehod n Solon heory. Cambrdge: Cambrdge Unversy Press. Ibragmov N. H. (Ed.) (99 996). CRC Handboo of Le Group Analyss of Dfferenal Equaons. Vols. Boca Raon FL: CRC Press. Ibragmov N. H. (007). A new conservaon heorem. Journal of Mahemacal Analyss and Applcaons 8. Khalque C. M. & Mahomed F. M. (009). Sol waer redsrbuon and eracon flow models: Conservaon laws. Nonlnear Analyss: Real World Applcaons Kher H. & Jabrar A. (00). Eac soluons for he double snh-gordon and generalzed form of he double snh-gordon equaons by usng G/ G - epanson mehod. ursh Journal of Physcs 7 8. Kudryashov N. A. (005). Smples equaon mehod o loo for eac soluons of nonlnear dfferenal equaons. Chaos Solons and Fracals 7. L J. (0). Sngular ravelng Wave Equaons: Bfurcaons and Eac Soluons. Beng Chna: Scence Press. Ma W.-. Zhang Y. ang Y. & u J. (0). Hroa blnear equaons wh lnear subspaces of soluons. Appled Mahemacs and Compuaon Magalawe G. & Khalque C. M. (0). New eac soluons for a generalzed double snh-gordon equaon. Absrac and Appled Analyss Arcle ID Naz R. Mahomed F. M. & Mason D. P. (008). Comparson of dfferen approaches o conservaon laws for some paral dfferenal equaons n flud mechancs. Appled Mahemacs and Compuaon Olver P. J. (99). Applcaons of Le Groups o Dfferenal Equaons Graduae es n Mahemacs. Berln: Sprnger-Verlag. nd edon. ang S. & Huang W. (007). Bfurcaon of ravellng wave soluons for he generalzed double snh-gordon equaon. Appled Mahemacs and Compuaon racnà R. Bruzón M. S. Gandaras M. L. & orrs M. (0). Nonlnear self-adonness conservaon laws eac soluons of a sysem of dspersve evoluon equaons. Communcaons n Nonlnear Scence Numercal Smulaon Vanov N. K. (00). Applcaon of smples equaons of Bernoull and Rcca nd for obanng eac ravelng-wave soluons for a class of PDEs wh polynomal nonlneary. Absrac and Appled Analyss Wang M. & L. (005). Applcaons of F - epanson o perodc wave soluons for a new Hamlonan amplude equaon. Chaos Solons Fracals Wang M. angzheng L.. & Jnlang Z. J. (008). he ( G/G )-epanson mehod and ravellng wave soluons of nonlnear evoluon equaons n mahemacal physcs. Physcs Leer A 7 7. Wang M. Zhou Y. & L Z. (996). Applcaon of a homogeneous balance mehod o eac soluons of nonlnear equaons n mahemacal physcs. Physcs Leer A Wazwaz A. M. (006). he anh funcon mehod for ravelng wave soluons of nonlnear equaons. Appled Mahemacs and Compuaon Wazwaz A. M. (005). Eac soluons o he double snh-gordon equaon by he anh mehod and a varable separaed ODE mehod. Compuers and Mahemacs wh Applcaon Wazwaz A. M. (005). he anh mehod: eac soluons of he sne-gordon and he snh-gordon equaons. Appled Mahemacs and Compuaon Wazwaz A. M. (006). Eac soluons for he generalzed sne-gordon and generalzed snh-gordon equaons. Chaos Solons and Fracals

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