A supersymmetric Sawada-Kotera equation

A sup ersymmetric Saw ada-Kotera equation Kai Tian and Q. P . Liu ∗ Dep artment of Mathematics, China University of Mining and T e chnolo gy, Beijing 100083, P.R. China Abstract A new sup ersymmetric equation is prop osed for the Sa w ada-Kotera equation. The in tegrabilit y of this equation is shown b y the existence of Lax repr esentati on and infinite conserv ed quanti ties and a recursion op erator. P ACS : 02.30 .Ik; 0 5.45.Yv Key W ords : integrabilit y; Lax represen tation; recursion o p erator; supersymmetry 1 In t ro duction The follo wing fifth-order evolution equation u t + u xxxxx + 5 uu xxx + 5 u x u xx + 5 u 2 u x = 0 (1) is a w ell- known system in soliton theory . It w as pro p osed b y Sa w ada and Kotera, also b y Caudrey , Do dd and Gibb on indep enden tly , more than thirt y y ears ago [1][2], so it is referred as Sa w ada-Kotera (SK) equation or Caudrey-Do dd-Gibb on- Saw ada- Kotera equation in literature. No w there are a large n um b er of pap ers ab out it and th us its v ario us prop erties are established. F or example, its B¨ ac klund t r a nsformation and Lax represen tation w ere g iven in [3][4], its bi-Hamiltonian structure w as w ork ed out b y F uch ssteiner and Oev el [5], and a Darb oux transformation was deriv ed f o r this system [6][7], to men tion just a few ( see also [8][9]). Soliton equations o r in tegrable system s hav e sup ersymmetric analogues. Indeed, man y equations such as KdV, KP , and NLS equations w ere em b edded in to their sup ersymmetric coun terparts and it turns o ut tha t these sup ersymmetric systems ha v e also remark a ble prop- erties. Thus , it is interesting to w ork out sup ersymmetric exte nsions f o r a giv en integrable equation. ∗ Email: qpl@cumtb.edu.cn T el: 86 1 0 62 3390 15 1 The aim of the Not e is to prop ose a sup ersymmetric extension for the SK equation. In this regard, w e notice that Carstea [1 0], based on Hirota bilinear a pproac h, presen ted the follo wing equation φ t + φ xxxxx +  10( D φ ) φ xx + 5( D φ ) xx φ + 15 ( D φ ) 2 φ  x = 0 where φ = φ ( x, t, θ ) is a fermioic sup er v ariable dep ending on usual tempo r al v aria ble t and sup er spatial v ariables x and θ . D = ∂ θ + θ ∂ x is the super deriv ative . Rewriting the equation in comp onen ts, it is easy to see that this system do es reduce to the SK equation when the fermionic v ariable is absen t. Ho w eve r, apart from the fact that the system can b e put in to a Hirota’s bilinear form, not muc h is kno wn f o r its inte grability . W e will giv e an alternativ e sup ersymmetric extension for the SK equation and will show t he evidence fo r the in tegrabilit y of our system. The pap er is orga nized as f o llo ws. In section t w o , b y considering a Lax op erator and its factorization, we construct the sup ersymmetric SK (sSK) equation. In section three, we will show that our sSK equation has an in teresting prop ert y , namely , it do es no t hav e the usual b osonic conserv ed quan tities sinc e t ho se, resulted from the sup er residues of a frac- tional p o w er f o r La x op erator, a r e trivial. Ev ermore, there are infinite fermionic conserv ed quan tities. In the section four, w e construct a recursion op erator f or our sSK equation. Last section con ta ins a brief summary of our new findings and pres en ts some in teresting op en problems. 2 Sup ersymmetric Sa wa da-Kotera Equation The main purp o se of this section is to construct a sup ersymmetric a na logy fo r the SK equation. T o this end, w e will w ork with the algebra o f sup er-pseudo differen tial op erators on a (1 | 1) sup erspace with coo r dina t es ( x, θ ). W e start with the follow ing general Lax op erator L = ∂ 3 x + Ψ ∂ x D + U ∂ x + Φ D + V . (2) By the standard fractional p o w er metho d [12], we ha v e an integrable hierarch y of equations giv en b y ∂ L ∂ t n = [( L n 3 ) + , L ] (3) where w e are using the standard notations: [ A, B ] = AB − ( − 1) | A || B | B A is the sup ercom- m utator a nd the subscript + means ta king the pro jec tion to the differen tial part for a given sup er-pseudo differen tial op erator. It is remark ed that the sys tem (3) is a kind of ev en order generalized SKdV hierarc hies considered in [11 ]. In the follo wing, w e will consider the particular t 5 flo w. Our in terest here is t o find a minimal sup ersymmetric extension for the SK equation, so we hav e to do reductions for the general Lax o p erator (2). T o this end, w e imp ose L + L ∗ = 0 2 where ∗ means taking formal adj o in t. Then w e find Ψ = 0 , V = 1 2 ( U x − ( D Φ)) that is L = ∂ 3 x + U ∂ x + Φ D + 1 2 ( U x − ( D Φ)) a Lax op erator with t w o field v ariables. In this case, w e ta ke B = 9( L 5 3 ) + , namely B = 9 ∂ 5 x + 15 U ∂ 3 x + 15Φ D ∂ 2 x + 15( U x + V ) ∂ 2 x +15Φ x D ∂ x + (10 U xx + 15 V x + 5 U 2 ) ∂ x +10(Φ xx + Φ U ) D + 10 V xx + 10 U V + 5Φ( D U ) for con ven ience. Then, the flow of equations, resulted from ∂ L ∂ t = [ B , L ] reads as U t + U xxxxx + 5  U U xx + 3 4 U 2 x + 1 3 U 3 + Φ x ( D U ) + 1 2 Φ( D U x ) + 1 2 ΦΦ x − 3 4 ( D Φ) 2  x = 0 (4a) Φ t + Φ xxxxx + 5  U Φ xx + 1 2 U xx Φ + 1 2 U x Φ x + U 2 Φ + 1 2 Φ( D Φ x ) − 1 2 ( D Φ)Φ x  x = 0 (4b) where w e identify t 5 with t for simplicit y . Remarks : 1. It is interesting to note that the ab o v e system has an ob vious reduction. Indeed, setting Φ = 0, w e will hav e the standard Kaup- Kup ershimdt (KK) equation. Therefore, w e ma y consider it as a supersymmetric extension of the KK equation. 2. The coupled system (4a-4b) admits the following simple Hamiltonian structure  U t Φ t  =  0 ∂ x ∂ x 0  δ H where the Hamilto nia n is given by H = Z  5 4 Φ( D φ ) 2 − ( D U x )( D Φ xx ) − 5 3 Φ U 3 − 5 4 ( D U x )( D U )Φ + 5 4 ( D U ) U x ( D Φ) + 5( D U ) U ( D Φ x ) + 5 2 ( D U ) Φ x Φ  d x d θ. 3 A t this p oin t, it is not clear ho w this system (4a-4b) is related to the SK equation. T o find a sup ersymmetric SK equation from it, w e no w conside r the factorization of the Lax op erator in the f o llo wing w ay L = ∂ 3 x + U ∂ x + Φ D + 1 2 ( U x − ( D Φ)) = ( D 3 + W D + Υ)( D 3 − D W + Υ) , (5) whic h giv es us a Miura-type transformation U = − 2 W x − W 2 + ( D Υ) , Φ = − Υ x − 2Υ W , and the mo dified syste m corresp o nding to this fa cto r izat io n is giv en b y W t + W xxxxx + 5 W xxx ( D Υ) − 5 W xxx W x − 5 W xxx W 2 − 5 W 2 xx + 10 W xx ( D Υ x ) − 20 W xx W x W − 5 W xx W ( D Υ) − 5 W 3 x − 5 W 2 x ( D Υ) + 5 W x ( D Υ xx ) + 5 W x ( D Υ) 2 + 5 W x W 4 − 5 W x W ( D Υ x ) + 10 W ( D Υ x )( D Υ) − 10Υ x Υ W x + 5( D W xx )Υ x + 5( D W x )Υ xx + 5( D W x )Υ W x + 10( D W x )Υ W 2 − 5( D W )Υ x W x + 10( D W )Υ x W 2 + 10( D W )Υ( D Υ x ) − 5 ( D W ) Υ W xx + 30( D W )Υ W x W = 0 , Υ t + Υ xxxxx + 5Υ xxx ( D Υ) − 5Υ xxx W 2 + 5Υ xx ( D Υ x ) + 5Υ xx W xx − 25 Υ xx W x W + 5Υ xx W ( D Υ) + 5Υ x ( D Υ) 2 + 5Υ x W xxx − 25Υ x W xx W − 2 5Υ x W 2 x + 5Υ x W x ( D Υ) + 10Υ x W x W 2 + 5Υ x W 4 − 10 Υ x W 2 ( D Υ) + 5 Υ x W ( D Υ x ) − 10Υ W xxx W − 20Υ W xx W x + 10Υ W xx W 2 + 30Υ W 2 x W + 20Υ W x W 3 − 30 Υ W x W ( D Υ) − 10Υ W 2 ( D Υ x ) − 5( D W x )Υ x Υ − 5 ( D W ) Υ xx Υ − 1 0( D W )Υ x Υ W = 0 . Although this mo dification do es indeed ha v e a complicated form, the remark able fact is that it allow s a simple r eduction. What we need to do is simply putting W to zero, namely W = 0 , Υ = φ . In this case, w e ha v e φ t + φ xxxxx + 5 φ xxx ( D φ ) + 5 φ xx ( D φ x ) + 5 φ x ( D φ ) 2 = 0 (6) this equation is our supersymmetric SK equation. T o see the connection with the original SK equation ( 1 ), w e let φ = θ u ( x, t ) + ξ ( x , t ) and write t he equation (6) out in comp onen ts u t + u xxxxx + 5 uu xxx + 5 u x u xx + 5 u 2 u x − 5 ξ xxx ξ x = 0 , (7a) ξ t + ξ xxxxx + 5 uξ xxx + 5 u x ξ xx + 5 u 2 ξ x = 0 . (7b) It is now ob vious that the system reduce s to the SK equation when ξ = 0. Therefore, our system (6) do es qualify as a sup ersymmetric SK equation. 4 Our system (6 ) is in tegrable in the sens e that it has a Lax repres en tation. In fact, the factorization (5) implies that the reduced Lax op erator has the following app ealing for m L = ( D 3 + φ )( D 3 + φ ) (8) or L = ∂ 3 x + ( D φ ) ∂ x − φ x D + ( D φ x ) 3 Conser v ed Quantities In general, an in t egr a ble system has infinite n umber of conserv ed quan tities. Since the sSK equation has a simple Lax op erator (8), it is natural to take adv an tage of t he fractiona l p ow er metho d of Gel’fand and Dic k ey [12] to find conserv ed quan tities. In the presen t situation, w e ha v e to w ork with the sup er residue of a p esudo differen tia l op erato r . The obvious c hoice in this case is to consider the op erators L n 3 and their sup er residues . Then, w e hav e the Prop osition 1 sres L n 3 ∈ Im D . wher e sres m e ans taki n g the sup er r esidue of a sup er pseudo differ ential op er ator. Pro of: As observ ed already in [13], there exists a unique o dd o p erator Λ = D + O(1), whose co efficien ts are all differential po lynomials of φ , suc h that ( D 3 + φ ) = Λ 3 , th us, the La x op erator (8) is written as L = ( D 3 + φ )( D 3 + φ ) = Λ 6 . F rom it w e ha v e sres L n 3 = sresΛ 2 n = 1 2 sres { Λ 2 n − 1 Λ + Λ Λ 2 n − 1 } = 1 2 sres[Λ 2 k − 1 , Λ] ∈ Im D . This completes the pro of. Remark: The tr ivialit y of L n 3 implies tha t the Lax op erator could not generate an y Hamil- tonian structures for the equation (6). T o find nontrivial conserv ed quan tities, w e now turn to L n 6 rather than L n 3 . It is easy to pro v e that ∂ ∂ t L 1 6 = [9( L 5 3 ) + , L 1 6 ] th us ∂ ∂ t L n 6 = [9 ( L 5 3 ) + , L n 6 ] . 5 Consequen tly , the sup er residue of L n 6 is conserv ed. By direct calculation, w e obtain the first tw o nontriv ial conserv ed quan tities Z sres L 7 6 d x d θ = − 1 9 Z [2( D φ xx ) + ( D φ ) 2 − 6 φ x φ ]d x d θ Z sres L 11 6 d x d θ = − 1 81 Z [6( D φ xxxx ) + 18 ( D φ xx )( D φ ) + 9( D φ x ) 2 + 4( D φ ) 3 − 18 φ xxx φ + 6 φ xx φ x − 36 φ x φ ( D φ )]d x d θ Remarks: 1. What is remark able is that the conserv ed quan tities found in this wa y , unlik e the sup ersymmetric KdV case [1 5][14], are lo cal. 2. All those conserv ed quan tities are fermioic. T o our knowle dge, this is the first sup er- symmetric in tegrable system whose only conserv ed quan tities are fermioic. 4 Recursion op e rator An integrable system of t en app ears as a part icular flo w of hierarc hy equations and an im- p ortant ingredien t in t his asp ect is the existence of recursion o p erators. In this section, we deduce the recursion operato r f or the sSK equation (6) following the metho d prop osed in [16]. W e first notice that the sSK hierarc h y can b e written as ∂ ∂ t n L = [( L n 3 ) + , L ] (9) where L is giv en by (8). It is easy to see that t he flow equations are nontrivial only if n is an in teger satisfying n 6 = 0 mo d 3 and n = 1 mo d 2 . Therefore the next flow whic h is ac hieve d by applying recursion op erator to (9) should be ∂ ∂ t n +6 L = [( L n +6 3 ) + , L ] . (10) But h ( L n +6 3 ) + , L i = h  L 2 ( L n 3 ) + + L 2 ( L n 3 ) −  + , L i =  L 2 ( L n 3 ) + , L  + h  L 2 ( L n 3 ) −  + , L i = L 2  ( L n 3 ) + , L  + [ R n , L ] = L 2 ∂ ∂ t n L + [ R n , L ] 6 where R n =  L 2 ( L n 3 ) −  + (11) is a differential op erator of O ( ∂ 5 x D ), that is, R n = ( α ∂ 5 x + β ∂ 4 x + γ ∂ 3 x + δ ∂ 2 x + ξ ∂ x + η ) D + a∂ 5 x + b∂ 4 x + c∂ 3 x + d∂ 2 x + e∂ x + f . Therefore, ∂ ∂ t n +6 L = L 2 ∂ ∂ t n L + [ R n , L ] . (12) Next we may determine the co efficien ts in R n . Using (12), w e obtain a = 1 3 ( D − 1 φ n ) , b = 2( D φ n ) , c = 44 9 ( D φ n,x ) + 5 3 ( D φ )( D − 1 φ n ) + 4 9 ( ∂ − 1 x φ x φ n ) , d = 55 9 ( D φ n,xx ) + 19 9 ( D φ )( D φ n ) + 5 9 φ x φ n + 10 9 ( D φ x )( D − 1 φ n ) e = 1 27 { 106( D φ n,xxx ) + 74 ( D φ )( D φ n,x ) , − 14 φ x φ n,x + 79( D φ x )( D φ n ) + 27 φ xx φ n + [23( D φ xx ) + 4( D φ ) 2 ]( D − 1 φ n ) + 16 ( D φ )( ∂ − 1 x φ x φ n ) + 2 D − 1 [( φ xxx + φ x ( D φ ))( D − 1 φ n ) − 3( D φ )( D − 1 φ x φ n ) − 2 φ x ( ∂ − 1 x φ x φ n ) + 2 D − 1 ( φ xxx φ n + 2 φ x ( D φ ) φ n )] } , f = 1 27 { 28( D φ n,xxxx ) + 32 ( D φ )( D φ n,xx ) − 2 0 φ x φ n,xx + 54( D φ x )( D φ n,x ) + 16 φ xx φ n,x + [30( D φ xx ) + 4( D φ ) 2 ]( D φ n ) + [8 φ xxx + 4 φ x ( D φ )] φ n + [10( D φ xxx ) + 10( D φ x )( D φ )]( D − 1 φ n ) − 8 φ x ( D − 1 φ x φ n ) + 12( D φ x )( ∂ − 1 x φ x φ n ) } , α = 0 , β = − 1 3 φ n , γ = 5 3 φ n,x , δ = − 1 9 { 29 φ n,xx + 5 φ n ( D φ ) + 5 φ x ( D − 1 φ n ) − 2( D − 1 φ x φ n ) } , ξ = − 1 9 { 26 φ n,xxx + 16 φ x ( D φ n ) + 3 φ n ( D φ x ) + 14 φ n,x ( D φ ) + 5 φ xx ( D − 1 φ n ) } , η = − 1 27 { 28 φ n,xxxx + 32( D φ ) φ n,xx + 28 φ x ( D φ n,x ) + 26 ( D φ x ) φ n,x + 28 φ xx ( D φ n ) + [2( D φ xx ) + 4( D φ ) 2 ] φ n + [10 φ xxx + 10 φ x ( D φ )]( D − 1 φ n ) − 2( D φ )( D − 1 φ x φ n ) + 12 φ x ( ∂ − 1 x φ x φ n ) − 2 D − 1 [ φ xxx φ n + 2 φ x ( D φ ) φ n ] } . where w e used the shorthand notation φ n = ∂ φ/∂ t n . 7 Finally , w e ha ve the recursion op erato r R = ∂ 6 x + 6( D φ ) ∂ 4 x + 9( D φ x ) ∂ 3 x + 6 φ xx ∂ 2 x D + { 5( D φ xx ) + 9( D φ ) 2 } ∂ 2 x + { 9 φ xxx + 12 φ x ( D φ ) } ∂ x D + { ( D φ xxx ) + 9( D φ x )( D Φ) } ∂ x + { 5 φ xxxx + 12 φ xx ( D φ ) + 6 φ x ( D φ x ) }D + { 4( D φ xx )( D φ ) + 4( D φ ) 3 − 3 φ xx φ x } + { φ xxxxx + 5 φ xxx ( D φ ) + 5 φ xx ( D φ x ) + 2 φ x ( D φ xx ) + 6 φ x ( D φ ) 2 }D − 1 − { 2( D φ xx ) + 2( D φ ) 2 }D − 1 φ x − 4 φ x ( D φ ) ∂ − 1 x φ x − 2( D φ ) D − 1 [ φ xxx + 2 φ x ( D φ )] − 2 φ x D − 1 { ( φ xxx + φ x ( D φ )) D − 1 − 3( D φ ) D − 1 φ x − 2 φ x ∂ − 1 x φ x + 2 D − 1 [ φ xxx + 2 φ x ( D φ )] } Remark: When calculating the coefficien ts of R n , one should solve a system of differen tial equations. Due to nonlo calit y (those underlined terms), there is certain am biguit y and to a v oid it, we used the t 7 -flo w φ t 7 = φ xxxxxxx + 7 φ xxxxx ( D φ ) + 1 4 φ xxxx ( D φ x ) + 14 φ xxx ( D φ xx ) +14 φ xxx ( D φ ) 2 + 7 φ xx ( D φ xxx ) + 28 φ xx ( D φ x )( D φ ) +14 φ x ( D φ xx )( D φ ) + 7 φ x ( D φ x ) 2 + 28 3 φ x ( D φ ) 3 . 5 Conclus ion Summarizing, we find a supersymmetric SK equation whic h has Lax represen tation. W e also obtain infinite conserv ed quan tities and a recursion op erator for this new prop osed system. These imply that the system is integrable. It is in teresting to establish other prop erties for it, suc h a s B¨ ac klund transformation, Hirota bilinear form, etc.. Ac kno wledgemen ts The calculations w ere done with the assistance of SUSY2 pack a ge of Popowicz [17]. W e w ould lik e to thank him fo r helpful discuss ion ab out his pack ag e. The commen ts of anon ymous referee has b een ve ry useful. 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