CONSERVATION LAWS OF COUPLED KLEIN-GORDON EQUATIONS WITH CUBIC AND POWER LAW NONLINEARITIES

Transcription

1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 5, Number /04, pp. 9 CONSERVATION LAWS OF COUPLED KLEIN-GORDON EQUATIONS WITH CUBIC AND POWER LAW NONLINEARITIES Aa BISWAS,, Abdul H. KARA, Lumiiţa MORARU 4, Ashfaque H. BOKHARI 5, F. D. ZAMAN 5 Delaware Sae Uiversiy, Deparme of Mahemaial Siees, Dover, DE , USA Kig Abdulaziz Uiversiy, Fauly of Siee, Deparme of Mahemais, Jeddah, Saudi Arabia Uiversiy of he Wiwaersrad, Shool of Mahemais, Cere for Differeial Equaios Coiuum Mehais ad Appliaios, Johaesburg, Wis 050, Souh Afria 4 Duărea de Jos Uiversiy, Deparme of Chemisry, Physis ad Evirome, Galaţi Domeasa Sree, 8000 Galaţi, Romaia 5 Kig Fahd Uiversiy of Peroleum ad Mierals, Deparme of Mahemais ad Saisis Dhahra-6, Saudi Arabia Correspodig auhor: Aa BISWAS, biswas.aa@gmail.om This paper deermies he oservaio laws of he oupled Klei-Gordo equaios ha arise i he quaum field heory. There are wo ypes of olieariy ha will be osidered, amely he ubi law ad he power law. The Lie symmery aalysis wih muliplier approah will be he mahemaial ool ha will be adoped o era he oserved desiies. The oservaio laws will be fially deermied from hese oserved desiies from he orrespodig -solio soluio. Key words: oservaio laws, solios, Lie symmery, iegrabiliy.. INTRODUCTION The heory of oliear evoluio equaios (NLEEs) is a very impora opi of researh i heoreial physis, applied mahemais ad egieerig siees. These NLEEs lay he basi foudaio for advaed sudies i his field. I fa NLEEs appear i fluid dyamis, plasma physis, oliear opis, ulear physis, ad several oher researh areas [-]. While he iegrabiliy aspes, perurbaio aalysis, umerial simulaios, sabiliy aalysis are he predomia fous areas i a lo of works, his paper is goig o address aoher impora feaure ha is less sudied globally. This is he oservaio law ha is also kow as he iegral of moio. These iegrals of moio are, o may oasios, ompued wih had alulaios ad herefore i does o lead o a ehausive ou for NLEEs. This paper is goig o use a adequae ool ha liss hese iegrals of moio ehausively. The oliear Klei-Gordo equaio (KGE) is he NLEE ha is goig o be osidered i his paper. KGE is a very impora equaio i he area of heoreial ad mahemaial physis. I pariular, i is sudied i he oe of relaivisi quaum mehais. KGE has bee sudied by several auhors ad has bee ehausively osidered. This paper is goig o ake a look a he oupled KGE where ubi ad power law olieariies are beig ake io aou [4, 5]. The eraio of oservaio laws is goig o be he fous here. The Lie symmery aalysis wih he muliplier approah will be he mahiery for eraio of he oservaio laws. Iiially, his approah will yield he oserved desiies. Subsequely, he -solio soluio will be used o fially obai he oserved quaiies. These -solio soluios were obaied earlier i he lieraure [4].. GOVERNING EQUATIONS The oupled KGE ha are goig o be sudied i his paper are firs lised i he followig wo subseios. The ubi law as well as he power law olieariy are goig o be he sudied. The followig

2 4 Aa Biswas, Abdul H. Kara, Lumiiţa Moraru, Ashfaque H. Bokhari, F. D. Zama wo subseios will lis he dimesioless form of hese equaios alog wih heir respeive -solio soluios. The osrai odiios or he domai resriios for he eisee of he solios will also be give... Cubi olieariy The oupled KGE wih ubi law olieariy, i dimesioless form is give by [4, 5]: q k q + aq + bq + qr 0, () r k r + a r + b r + q r 0. () Here i () ad (), q (, ) ad r (, ) are he depede variables ad represe he wave profiles. The idepede variables are ad ha respeively, represe he spaial ad emporal variable. The o-zero real-valued osas are k, a, b ad for,. The ouplig erms are give by he oeffiies of. I pariular, equaios () ad () ogeher form he oupled Φ 4 equaio. The ea -solio soluios o () ad (), are [4]: q(, ) A seh[ B( v)] () ad where he ampliudes A for, are r(, ) A seh[ B( v)], (4) A a ( b b b ) (5) ad, ad. The widh of he solio is give by: a a B. (6) v k v k Addiioally, he parameer v represes he solio veloiy i () ad (4). These relaios osequely irodue he osrai odiios a a, (7) a ( v k ) < 0, (8) for,. Also, he ampliude relaios diae he osrai odiio give by a ( b )( b b ) > 0. (9).. Power law olieariy For power law olieariy, he dimesioless form of he oupled KGE is give by [4, 5] m q k q + a q + b q + q r 0, (0) m r k r + a r + b r + q r 0, ()

3 Coservaio laws of oupled Klei-Gordo equaios wih ubi ad power law olieariies 5 where. While he oeffiies have he same ierpreaio as i he previous subseio, he addiioal parameers i his ase are he power law olieariy parameers m ad ad here m > 0 as well as > 0. I his ase, he -solio soluios o (0) ad () are [9]: ad q (, ) A seh [ B ( v)] () where he ampliudes r (, ) A seh [ B ( v)] () A are oeed o eah oher by he oupled relaios: m b A + A A + ( m + ) a 0, (4) where, ad. The widh of he solios is give by: m + a m + a B, (5) v k v k where v is he veloiy of he solio. Equaio (5) is valid wih he odiio a ( v k ) < 0. Boh he ampliude ad widh relaios lead o he osrai odiios give by (7) ad (8).. SYMMETRIES AND CONSERVATION LAWS The Lie symmery approah o differeial equaios is well kow; for deails see e.g., [4, ]. I order o deermie oserved desiies ad flues, we resor o he ivariae ad muliplier approah based o he well kow resul ha he Euler-Lagrage operaor aihilaes a oal divergee (see [7]). Firsly, if ( T, T ) is a oserved veor orrespodig o a oservaio law, he D T + D T 0 (6) alog he soluios of he differeial equaio (say, de 0 ). Moreover, if here eiss a orivial differeial fuio Q, alled a muliplier', suh ha he Q(de) is a oal divergee, i.e., E u [ Q( de)] 0, Q ( de) D T + D T, for some (oserved) veor ( T, T ) ad E u is he respeive Euler-Lagrage operaor. Thus, a kowledge of eah muliplier Q leads o a oserved veor deermied by, ier alia, a homoopy operaor; see more deails ad referees i [7, 8]. For a sysem de 0ad de 0, Q ( Q, Q ) say, so ha ad Q ( de) + Q ( de) D T + D T, I eah ase, T is he oserved desiy. E ( u, v)[ D T + D T ] 0 (7)

4 6 Aa Biswas, Abdul H. Kara, Lumiiţa Moraru, Ashfaque H. Bokhari, F. D. Zama 4.. Cubi olieariy For oupled KGE wih ubi olieariy, hese equaios are: r k q k q r + a q + b q + a r + b r + qr + rq A oe parameer Lie group of Lie poi rasformaios (as veor fields) ha leave ivaria (8) are: X 0, 0.,, k X, X + The oserved veors (flues ad desiies) are lised below: (i) ( Q, Q ) ( k q + q, k r + r ), (8) T k [ b q + a r + b r + q ( a + r ) ( q q + k q r ( r + k r ) + q( q + q + q ) + r( r + r + r )], T [ bq + ar + br + q ( a + r ) + ( q + k qq r ( r + k r ) k q( q + q + q ) k r( r + r + r )]; Q Q q r (ii) (, ) (, ); T [ k ( q q r r + qq + rr )], T [ b q + a r + b r + q ( a + v ) + q +, r k qq k rr ]; Q Q q r (iii) (, ) (, ); T T 4 [ b q + qq ( q q 4 + a r + rr + b r k + r r qq 4 ( q + q rr + r )]. ( a )], + v ) + q +.. Power law olieariy I he ase of power law olieariy, he oupled KGE a be wrie: q k q r k r + a q + b q + a r + b r + r m m + q r q 0, 0. The oserved desiies are, however, obaiable oly for m, ad are give by (9)

5 5 Coservaio laws of oupled Klei-Gordo equaios wih ubi ad power law olieariies 7 (i) a λq b λq λq k λqq + + T + + a λ r + λ q r + bλ r + +λr k λrr dλ 0 a q b q q k qq + + a r + q r + br + + r k rr, (ii) T [ qq + r r qq rr ], (iii) + a k λq b k λ q + + T + + a k λ r + k λ q r + bk λ r + 0 k λqq k λqq k λq q k λq + + λ + + λ λ λ k rr k rr k rr k r k λqq + + k λrr ]dλ a k q b k q k qq + + a k r + k q r + b k r + + k rr + k qq k q q k q k qq k rr k rr k r k rr. 4. CONSERVED QUANTITIES I his seio we will obai he oserved quaiies from he oserved desiies ha are disussed i he previous seio afer usig he - solio soluios ha are disussed i Seio. The sudy will be agai spli io he followig wo subseios depedig o he ype of olieariy i quesio. 4.. Cubi olieariy For ubi law olieariy, he respeive oserved quaiies are 4 4 I Td [ b q r b r q ( a r ) { q + k q q + r ( r + k r )} k q( q + q + q ) k rr ( r r)]d 0, + + I T d { b q + a r + b r + q ( a + r ) q + r k qq k rr }d 4 4 { b A + ba + a A + a A + A A + ( v + k )( A + A ) B } B ad I T d [ q q r r qq rr ]d + vb ( A + A ). (0) () ()

6 8 Aa Biswas, Abdul H. Kara, Lumiiţa Moraru, Ashfaque H. Bokhari, F. D. Zama 6 I order o evaluae hese oserved quaiies, he -solio soluios give by () ad (4) are used i he oserved desiies. 4.. Power law olieariy I ase of KGE wih power law olieariy, he oserved quaiies are give by: 4 I Td [ a q + a r + q + r + k ( q + r ) + ( ) Γ( ) ( ) ( + b q + br )]d B ( v + k )( A + A) ( b A + ba ) ( a A + aa ) + + B ( )( + ) ( + ) Γ Γ +, I T qq rr qq rr vb( A + A ) Γ ( ) Γ ( ) ( ) d [ ]d + ( )( + ) Γ + k I T d {( a + a )( q + r ) + ( b + b + ) ( q + q r + r ) + + ( qq + rr ) + ( qq + rr ) ( q q + r r ) k ( q + r ) + ( qq + rr )}d 0. (), (4) Oe agai, he values of he oserved quaiies are obaied from he -solio soluio ha was disussed i he previous seio. Also, i was osidered he fa ha m. (5) 5. CONCLUSIONS This paper obaied he oserved quaiies for he oupled KGE. Boh he ubi law ad he power law are osidered. There are hree differe ypes of oservaio laws for eah ype of olieariy sudied i his paper. The oserved quaiies were obaied from heir respeive oserved desiies from he - solio soluios ha were published earlier [4]. The Lie symmery aalysis was employed o ahieve his goal. I was observed ha for power law olieariy he oserved quaiies eis oly if he epoes are oeed by a simple relaio. This paper oly addressed he -oupled KGE, however. I fuure, his mahemaial mehod will be applied o oher forms of he oupled KGE suh as he -oupled as well as he N-oupled KGE []. Those resuls will be repored i fuure publiaios. ACKNOWLEDGMENTS The firs ad seod auhors (AB & AHK) are graeful o Kig Fahd Uiversiy of Peroleum ad Mierals for sposorig heir aademi visi o he uiversiy durig Deember 0. REFERENCES. B. AHMED, A. BISWAS, Solios, kiks ad sigular solios of oupled Koreweg-de Vries equaio, Pro. Romaia Aad. A, 4, pp. 0, 0.. T. ALAGESAN, Y. CHUNG, K. NAKKERAN, Solio soluios of oupled oliear Klei-Gordo equaios, Chaos, Solios & Fraals,, pp , A. BISWAS, A. YILDIRIM, O. M. ALDOSSARY, R. SASSAMAN, Solio perurbaio heory for he geeralized Klei- Gordo equaio wih full olieariy, Pro. Romaia Aad. A,, pp. 4, 0.

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