Conservation Laws and Hamiltonian Symmetries of Whitham-Broer-Kaup Equations

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1 Indian Jornal of Science and Technology Vol 8( Janary 05 ISSN (Prin : ISSN (Online : DOI : /ijs/05/8i/47809 Conseraion Laws and Hamilonian Symmeries of Whiham-Broer-Kap Eqaions Mehdi Nadjafikhah * and Parasoo Kabi-Nejad School of Mahemaics Iran Uniersiy of Science and Technology Narmak- Tehran I.R. Iran; m nadjafikhah@is.ac.ir parasoo kabinejad@is.ac.ir Absrac In he presen paper conseraion laws of he ri-hamilonain sysem of eqaions Whiham-Broer-Kap (WBK are inesigaed by applying he firs homoopy formla. Hamilonian symmeries of he sysem are consrced by sing he corresponding Hamilonian operaors and he consered densiies. Keywords: Eqaions Whiham-Broer-Kap Conseraion Laws Hamilonian Symmery. Inrodcion The sysem of eqaions Whiham-Broer-Kap (WBK ( ( is eqialen nder a change of ariables o a sysem of Bossinesq eqaions modelling he bi-direcional propagaion of long waes in shallow waer firs fond by Whiham Broer and Kap 4. Lie symmery analysis for he WBK eqaions was deried by Z. Zhang X. Yong and Y. Chen. In addiion hey fond conseraion laws of he sysem of eqaions WBK by means of scaling symmery. Kpershmid 5 showed ha he physical sysem is inegrable by inerse scaering and sbseqenly showed ha his sysem is ri-hamilonian. Or paper is organized as follows. In secion we analyze he hree Hamilonian srcres associaed wih he Hamilonian descripion of he sysem of eqaions WBK and probe Jacobi ideniy as well as compaibiliy of he hree srcres sing he mehod of fncional mli-ecors. In secion we consrc conseraion laws of he sysem by considering mlipliers of order hree. In secion 4 we dedce Hamilonian symmeries of he WBK eqaions from he Hamilonian operaors and consered densiies. In secion 5 we smmarize or resls.. Hamilonian Operaors In his secion we will proide he backgrond definiions and resls in Hamilonian operaors ha will be sed along his paper. Mch of i is saed as in 9. Le p ( denoe he spaial ariables and q ( he field ariables (dependen ariables so each α is a fncion of p and he ime. We will be concerned wih aonomos sysems of eolion eqaions K[ ] ( in which K[ ] ( K[ ] Kq[ ] is a q-ple of differenial fncions where he sqare brackes indicae ha each K α is a fncion of and finiely many parial deriaies of each α wih respec o p. A sysem of eolion eqaions is said o be Hamilonian if i can be wrien in he form D. E ( H ( Here H[ ] H[ d ] is he Hamilonian fncional and Hamilonian fncion H[] depends on and he deriaies of he s wih respec o he s; E ( E E q denoes he Eler operaor or ariaional deriaie wih respec o. The Hamilonian operaor D is a q q mari differenial operaor which may depend on boh and deriaies of (b no on and is reqired o be (formally skew-adjoin relaie o he L -inner prodc *Ahor for correspondence

2 Mehdi Nadjafikhah and Parasoo Kabi-Nejad < f > f. gd f. g d g a a so D D where denoes he formal L adjoin of a differenial operaor. In addiion D ms saisfy a nonlinear Jacobi condiion ha he corresponding poisson bracke { PQ } E [ P]. D E [ Q] d (4 P Pd [ ] Q Qd [ ] saisfies he Jacobi ideniy. In he spaial case ha D is a field-independen skew-adjoin differenial operaor meaning ha he coefficien of D do no depend on or is deriaies (b may depend on he Jacobi condiions are aomaically saisfied; For more general field-dependen operaors he complicaed Jacobi condiions can be considerably simplified by he fncional mli-ecor mehod which is described in deail in 9. Mli-ecors are he dal objecs of differenial forms. To presere he noaional disincion beween he wo we se he noaion q J a for he ni-ecor corresponding o he one-form d J a ; hs a erical mli-ecor is a finie sm of erms each of which is he prodc of a differenial fncion imes a wedge prodc of he basic ni-ecors. The space of fncional mli-ecors is he cokernel of he oal diergence so ha wo erical mli-ecors deermine he same fncional mli-ecor if and only if hey differ by a oal diergence. The fncional mliecor deermined by Θ is denoed sggesiely by an inegral sign: Θ Θd. In pariclar Θd 0 if and only if Θ Di Ψ for some erical mli-ecor Ψ. This implies ha we can inegrae fncional mli-ecors by pars: Θ ( DΨ d ( DΘ Ψ d. (5 i The principal eample of a fncional bi-ecor is ha deermined by a Hamilonian differenial operaor D which is Θ D i q D ( q d. ( Finally define he formal prolonged ecor field p rd q a J b b DJ ( Dabq a J (7 which acs on differenial fncions o prodce ni-ecors. We frher le pr D q ac on erical mliecors by wedging he resl of is acion on he coefficien differenial fncions wih he prodc of he θ s. Since pr D q commes wih he oal deriaie here is also a well-defined acion of pr D q on he space of fncional mli-ecors which essenially amons o bringing i nder he inegral sign. By ire of he following heorem one can deermine wheher or no a differenial operaor is geninely Hamilonian. Le D be a skew-adjoin differenial operaor wih corresponding bi-ecor Θ D as aboe. Then D is a Hamilonian operaor if and only if pr Dq 0. (8 The proof ha (8 is eqialen o he Jacobi ideniy for he poisson bracke deermined by D can be fond in (9. The sysem of eqaions WBK admis hree Hamilonian operaors D 0 D D D. D D 0 D D D 0 D 4D 4D D( D D 4D ( D D ( D (D (9 ( D ( D and so can be wrien in Hamilonian form in hree disinc ways 5. The skew symmery of hese Hamilonian srcres is manifes. The Proof of he Jacobi ideniy for his srcres as well heir compaibiliy can be shown hrogh he sandard mehod of fncional mli ecors. Since he coefficiens of he operaor D 0 do no depend on or is deriaies hen D 0 is aomaically a Hamilonian operaor. For he operaor D i is sfficien o proe ha pr D q 0 where Θ D is he corresponding fnc- ional bi-ecor and q ( qz s. θ and ζ are he basic ni-ecors corresponding o and respeciely. We can consrc he bi-ecor associaed wih he srcre D as Θ D D { q q } d (0 { q q q z q z q z z q z q z z z z} d So by sing inegraion by pars Θ D { q q z q q z z q z z } d ( Vol 8 ( Janary 05 Indian Jornal of Science and Technology 79

3 Conseraion Laws and Hamilonian Symmeries of Whiham-Broer-Kap Eqaions In coninaion we apply prolongaion relaions in order o proe he Jacobi ideniy. pr D q ( q z z z ( pr D q ( q q z z ( pr D q {( q z z z (4 z q ( q q z z z z } d again by sing inegraion by pars relaion (8 for he operaor D is obained. Ths D is a Hamilonian operaor. Similarly he operaor D is Hamilonian. Therefore he sysem of eqaions WBK is ri-hamilonian meaning ha i can be wrien as a Hamilonian sysem sing any one of he hree Hamilonian operaors. Concenraing on he simpler Hamilonian operaors D 0 and D he corresponding Hamilonian fncionals are respeciely d (5 H 0 H ( Frhermore any wo of hese operaors form a Hamilonian pair namely no only are D i and D j i j genine Hamilonian srcres any arbirary linear combinaion of hem is as well. In he following we show ha he Hamilonian operaors D 0 and D form a Hamilonian pair. So i is sfficien o proe ha pr ( Θ pr ( Θ 0 (7 D q D D q D 0 0 where Θ D0 and Θ D are he biecors corresponding o he Hamilonian srcres D 0 and D respeciely. Since D 0 has consan coefficiens pr D q 0. 0 So we only need o erify pr D q 0 where 0 pr D0 q ( z pr D0 q ( q. Hence pr D q { z q z q z z } 0 0 d (8 In a similar way Hamilonian operaors D 0 D and D D are compaible. As a resl we can sae he following proposiion: The sysem of eqaions WBK admis hree Hamilonian operaors D 0 D D and so i can be wrien as a Hamilonian sysem sing any one of he hree Hamilonian operaors. In addiion any wo of hese operaors form a Hamilonian pair.. Conseraion Laws Consider a sysem of N parial differenial eqaions of order n wih p independen ariables ( p and q dependen ariables ( q gien by ( n 0 n N. (9 n ( A conseraion law of a PDE sysem (9 is a diergence epression D P D p P p 0 (0 holding for all solions f( of he gien sysem. ( r In (0 P i ( i p are called he fles of he conseraion law and he highes-order deriaie r presen in he fles is called he order of he conseraion law. If one of he independen ariables of PDE sysem (9 is ime he conseraion law (0 akes he form DT DiX 0 ( where Di is he spaial diergence of X wih respec o he spaial ariables ( p. Here T is referred o as a densiy and X ( X X p as spaial fles of he conseraion law (0. The consered densiy T and he associaed fl X ( X X p are fncions of and he deriaies of wih respec o boh and. In pariclar eery admied conseraion law arises from mlipliers l n ( ( l sch ha l n ( l ( n ( r n i i ( ( DP ( ( holds idenically where he smmaion conenion is sed wheneer appropriae. Throgh his approach he deermining of conseraion laws for a gien PDE sysem (9 redces o finding ses of mlipliers. By direc calclaion one can show ha he Eler operaors annihilae any diergence epression ( r D i P( i. Ths he following ideniies hold for arbirary fncion : ( r E D P j ( i i ( 0 j q. ( The conerse also holds. Specifically he only scalar epressions annihilaed by Eler operaors are diergence epressions. In coninaion he following heorem is applied which connecing mlipliers and conseraion ( laws. A se of mlipliers { ( l } N yields a l n n conseraion law for he PDE sysem (9 if and only if he se of ideniies 80 Vol 8 ( Janary 05 Indian Jornal of Science and Technology

4 Mehdi Nadjafikhah and Parasoo Kabi-Nejad E j ( l ( n n ( ( ( 0 j q. (4 l n holds idenically. See for more deails. The se of eqaions (4 yields he se of linear deermining eqaions o find all ses of conseraion law mlipliers of he PDE sysem (9 by considering mlipliers of all orders. In his secion we consrc conseraion laws for he sysem of eqaions (. Consider he mlipliers of he form l ( l ( (5 for he sysem of eqaions (. The deermining eqaions for mlipliers is l ] 0 ( E [ ( l ] 0 E [ ( ( E [ l ( ] 0 E [ l ( ( ] 0 Therefore afer sraighforward b edios calclaion we conclde ha ( c c (( c c c c c l l 5 4 ( c c ( 4 c c (7 4 c c (( c c4 ( c c( c c 7 where ci i 7 are consans. In coninaion we apply he firs homoopy formla which is de o blman and Anco o consrc conseraion laws of he sysem (. I is described in deail in. In he following consered ecors are represened by wo componens T and T which are consered densiy and fl respeciely. So we obain he following consered densiies and fles. Case : l l T ( T Case : l l 4 T 4 ( c c5 ( c c c ( 4 ( T 5 Vol 8 ( Janary 05 Indian Jornal of Science and Technology 8

5 Conseraion Laws and Hamilonian Symmeries of Whiham-Broer-Kap Eqaions 4 9 Case 4: l l T 9 Case 5: T l l T Case : T l l T 4 T Case : Case 7: l l 0 T T l 0 l T T 4. Hamilonian Symmeries The correspondence beween Hamilonian symmery grops and conseraion laws for sysems of eolion eqaions in Hamilonian form is known as Noher s 8 Vol 8 ( Janary 05 Indian Jornal of Science and Technology

6 Mehdi Nadjafikhah and Parasoo Kabi-Nejad heorem afer he prooype [8]. This relaionship has been analyzed eensiely by Oler 9 Wilson ] Gel fand and Dikii. As we remarked in he las secion Any conseraion law of a sysem of eolion eqaions akes he form DT DiX 0 in which Di denoes spaial diergence. Noe ha if ( T( n is any sch differenial fncion and is a solion o he eolionary sysem K[ ] hen D T pr ( T. (9 where / denoes he parial -deriaie. Ths T is he densiy for a conseraion law of he sysem if and only if is associaed fncional T saisfies K T/ pr ( T 0. (0 In he case or sysem is of Hamilonian form he following proposiion 9 is sed. Le D be a Hamilonian operaor wih poisson bracke (4. To each fncional H Hd here is an eolionary ecor field H called he Hamilonain ecor field associaed wih H which saisfies K pr H ( P { P H} ( for all fncionals P. Indeed H has characerisic Dd H DE( H. Hence he bracke relaion ( immediaely leads o he Noeher relaion beween Hamilonian symmeries and conseraion laws. So for he sysem of eqaions WBK generalized symmeries which are Hamilonian can be dedced from consered densiies by sing he Hamilonian operaors. The sysem of eqaions Whiham-Broer-Kap admis Hamilonian symmeries wih he following characerisics in he case of he Hamilonian operaor D 0 Q 5 Also generalized symmeries corresponding o he Hamilonian operaors D can be dedced from he conseraion laws. Ths he Hamilonian symmeries relaie o D are Q Q Q Q Q Q 4 4 ( Q 4 4. Q Q Q Q Q Q 0 ( Q4 Q 4 Q Q Vol 8 ( Janary 05 Indian Jornal of Science and Technology 8

7 Conseraion Laws and Hamilonian Symmeries of Whiham-Broer-Kap Eqaions 5. Conclsion In his paper he ri-hamilonian sysem of eqaions Whiham-Broer-Kap is sdied. conseraion laws of he sysem by considering mlipliers of order hree are consrced. Moreoer generalized symmeries of he sysem of eqaions WBK which are Hamilonian are obained by sing he Hamilonian operaors and consered densiies.. References. Blman GW Cheiako AF Anco SC. Applicaions of Symmery Mehods o Parial Differenial Eqaions. Applied Mahemaical Sciences. Springer; 00.. Broer LJF. Approimae eqaions for long waer waes. Appl Sci Res. 975; : Whiham GB. Variaional mehods and applicaions o waer waes. Proc Roy Soc. 97; 99A: Kap DJA. A higher order waer wae eqaion and he mehod for soling i. Proc Theor Phys. 975; 54: Kpershmid BA. Mahemaics of dispersie waes. Comm Mah Phys. 985; 99:5 7.. Gel fand IM Dikii LA. A Lie algebra srcre in a formal ariaional calclaion. Fncional Anal Appl. 97; 0: Gel fand IM Dorfman IYA. Hamilonain operaors and algebraic srcres relaed o hem. Fncional Anal Appl. 979; : Noeher E. Inariane ariaions-probleme. Kgl Ger Wiss Nachr.Mah Phys Kl. 98; Oler PJ. Applicaion of Lie Grops o Differenial Eqaions. nd ed. New York: Springer-Verlag; Oler PJ. Rosena P. Tri-Hamiloinan daliy beween solions and soliary-wae solions haing compac sppor. Phys Re E. 99; 5:900.. Oler PJ. On he Hamilonian srcre of eolion eqaions. Mah Proc Cambridge Philos Soc. 980: 88: Zhang Z Yong X Chen Y. Symmery analysis for Whiham-Broer-Kap eqaions. Jornal of Nonlinear Mahemaical Physics. 008; 5: Vol 8 ( Janary 05 Indian Jornal of Science and Technology

, u denotes uxt (,) and u. mean first partial derivatives of u with respect to x and t, respectively. Equation (1.1) can be simply written as

, u denotes uxt (,) and u. mean first partial derivatives of u with respect to x and t, respectively. Equation (1.1) can be simply written as Proceedings of he rd IMT-GT Regional Conference on Mahemaics Saisics and Applicaions Universii Sains Malaysia ANALYSIS ON () + () () = G( ( ) ()) Jessada Tanhanch School of Mahemaics Insie of Science Sranaree