Symmetry condition in terms of Lie brackets

Symmetry condition in terms of Lie brac k ets. P eter H. v an der Kamp Departmen t of Mathematics and Statistics, La T robe Univ ersit y , Victoria, 30 86, Australia Email: p eterh v anderk a mp@gmail.com Abstract A passiv e orthonomic system of PDEs defines a submanifold in th e cor- respond ing jet manifold, coordinated by so called p arametric deriv atives. W e restrict t he total differential op erators and the prolongation of an evo- lutionary vector field v t o this sub manifold. W e show that the v anishing of their commutators is eq u iv alent to v b eing a generalized symmetry of the system. 1 Concerning the status of this preprin t After writing this preprint, I learnt (from an a nonymous referee) that the Lie- brack et criterio n is v alid for all systems, no t only o rthonomic ones. Vinogradov- style der iv atio n ca n b e found for instance in [5, Cha pter 4 , § 3]. The ‘if ’ pa rt of the cr iterion follows from § 3 . 3, while lemma 3.6 in § 3 . 4 gives the ‘o nly if ’ par t. Therefore, the present result is not the most ge ne r al one. Still the r eader may appreciate its V an der K amp-style deriv ation. The intrinsic differential op erator s we will define, can b e useful in prac tica l situations . 2 The standard symmetry condition In the ma jorit y of cases where exact solutions of differen tial equations can b e found, the underlying pr op erty is a (contin uous) symmetry o f the equation [1 6, 10]. And, in the theory of integrable equations, the recognition and clas s ification metho ds based on the existence of symmetries have b een particular successful [8, 14, 18, 4, 7]. A sy mmetr y -group tra nsforms one solution of an equa tion to another solu- tion of the same equatio n. Although this idea go es ba ck to Sophus Lie , we refer to [10] for a go o d introduction to the sub ject, numerous examples, a pplications and references. And we quote: ’The gr eat p ower of L ie group theor y lies in the crucial observ ation that one can replace the complicated, nonlinear co nditions 1 for the in v ariance of the solution set of an equa tion under the gr oup transfor- mations by an equiv alent linear condition of infinitesimal inv aria nce under the corres p o nding infinit es imal generato rs of the group action’ [10]. In this pap er we provide a character ization of symmetries that is different from the standard one, genera lizing a s imilar characterizatio n in the sp ecia l se tting of or dinary differential equations [16, eq. (3 .35)], and evolution equations [10, P r op. 5.19] to the setting of passive orthonomic s ystems. The natura l framework in which symmetries of differential equations are studied is the so calle d jet-manifold M . Co o rdinates o n M cons ist of p indep en- dent v aria ble s x i , q depe ndent v ariables u α and the deriv atives o f the dep endent v ariables, which ar e denoted using multi-index no tation, e.g. u 2 1 , 0 , 3 = ∂ 4 w ∂ r∂ t 3 when x = ( r, s, t ) a nd u = ( v , w ). A typical point z ∈ M is z = ( x i , u α , u α K ). The r ing of smo o th functions on M will be denoted A . T o indicate f unctio na l depe ndence of f ∈ A we simply write f ( z ). Thus the system ∆( z ) = 0 , ∆ ∈ A n is a system of n P DEs. The actio n of a Lie g roup is defined on the spa ce of dep endent a nd indep en- dent v ariables, and then prolonged to an action o n the jet manifold. Lik ewise the infinitesimal generator of the sy mmetry gr oup is obtained by prolo ngation from an infinitesimal vector field on the base manifold. It turns out that any symmetry has an evolutionary representativ e [10, Prop. 5 .5]. In terms o f the total differential op era tors D i = ∂ ∂ x i + X α,K u α K i ∂ ∂ u α K , i = 1 , . . . , p, (1) the pr olongatio n pr Q of an evolutionary vector field ν Q = P α Q α ∂ /∂ u α is pr Q = X α,K D K Q α ∂ ∂ u α K . (2) A s imple computation shows that these der iv ations on A commute among each other, we hav e [ D i , D j ] = 0 , i, j = 1 , . . . , p a nd [ D i , pr Q ] = 0 , i = 1 , . . . , p, Q ∈ A . (3) In fact, up to a linea r co mbination of trans la tional fields ∂ /∂ x i , evolutionary vector fields ar e uniquely determined by prop erty (3), cf. [10, Lemma 5.1 2]. The condition of infinitesimal in v ariance, the standar d symmetry c ondition , is [10, Theor em 2.31 ] pr Q ∆ ≡ 0 mo d ∆ (4) in which case ν Q , or the tuple Q ∈ A n itself, is called a (generalized) symmetry of the system ∆ = 0. The tuple Q is a trivial symmetry if Q ≡ 0 mo d ∆, which defines a n equiv alence r elation on the space of symmetries. In section 2 3 we show this is w ell defined for p assive orthonomic systems . W e r estrict Q to be a function Q ∈ B o n the s ub-manifold o f the jet-ma nifold defined by o ur sy stem of PDEs. Although this is a more than standard pr o cedure, restricting the deriv ations to act on this sub-manifold is no t sta ndard at all, except p ossibly in the settings o f ODEs a nd e volution equations. This is, at least fro m a philosophical p oint of view , no t fully satisfying. In section 4 , for any pa ssive or tho nomic sys tem of par tial differential equa- tions, we define intrinsic total differential op erator s D i and an intrinsic prolon- gation pr Q , which are der iv atio ns o n the sub-space B . Subsequent ly we show that the v anishing of the Lie bra ck ets [ D i , pr Q ] = 0 , i = 1 , . . . , p, Q ∈ B is equiv alent to Q b eing a symmetry . 3 P assiv e orthonomic systems Restricting to the sub-manifold is par ticularly easy when dealing with ortho - nomic systems, in which case this amounts to using the eq uations as s ubs titution rules. Ho wev er, in g eneral the order of substituting and differentiating do es matter, one enc o unters integrabilit y conditions. F or ex a mple, for the system u x = X, u y = Y to b e fo r mally integrable w e need D y X = D x Y . In gener al, a finite num b er of integrability conditions suffices to ma ke the system for mally int eg rable, in which case the sy stem is called passive. The idea of a pa s sive o rthonomic s ystem is the main idea b ehind Riquier’s existence theorems [17] and the corres p o nding algor ithms for solving systems of PDEs due to Janet [2]. Riquier- Janet theory extended the w o r ks o f Cauch y and Kov alevs k a ya, it takes a pr ominent place in computer a lgebra applied to PDE theor y [12], and it has lea d to impo r tant developments in p olynomia l elimination theory [1]. The pas sive orthono mic s ystem w as the predecess o r o f what is now called a n inv olutive system of P DEs. F or our purp ose, the co ncept of inv olutivity do es not play a role. W e ado pt a similar philosophy as in [6], and stick to the setting of pa ssive orthonomic sys tems. In that pap er an efficient algorithm is g iven by which a ny o rthonomic s ystem can b e ma de pass ive by construction of a sufficient set integrabilit y c o nditions free of redundancies [6]. Let N = { 0 , 1 , 2 , . . . } a nd N q = { 1 , 2 , . . . , q } . W e denote the i -th comp onent of J ∈ N p by J i and addition in N p is denoted by concatenation. A set of basis vectors for N p is given by { 1 , 2 , . . . , p } , where i j = 1 w he n i = j and i j = 0 otherwise . Thu s , with J, K ∈ N p , we hav e ( J K ) i = J i + K i , and in particular ( K j ) i equals K i or, when j = i , K i + 1 . Also , when L = J K we write J = L/K . Since total differential op era tors comm ute we ca n define a m ulti-differ e nt ia l op era tor D K as D K = D K 1 1 D K 2 2 · · · D K p p , (5) and we hav e D K u j = u j K . 3 Cho ose n p oints ( i α , J α ) ∈ N q × N p , with nonze r o J α , α = 1 , . . . , n . A deriv a- tive u j K is called princip al if there exist L ∈ N p such that ( j, K ) = ( i α , J α L ) for some α . The remaining ones are called p ar ametric . The set of all ( j, K ) such that u j K is parametr ic is deno ted S , and the subspace of A c o nsisting of smo oth functions of the para metric deriv atives is denoted B . W e a lso choos e a rank ing ≤ o n N q × N p , that is, a total or der relatio n which is p ositive: ∀ L, ( j, K ) ≤ ( j, K L ) , and, compa tible with differentiation: ( i, J ) ≤ ( j, K ) ⇔ ( i, J L ) ≤ ( j, K L ) , cf. [9, 13]. W e co nsider s y stems of n partial differential equations, with α = 1 , . . . , n , u i α J α = P α , P α ∈ B . (6) The system (6) will b e wr itten shor tly ∆ = 0, where ∆ α = u i α J α − P α . W e make the following assumptions : i ) the P α only dep end on u j K with ( j, K ) < ( i α , J α ), a nd ii ) ( i α , J α K ) = ( i β , J β L ) ⇒ D K P α = D L P β . Such s y stems are called p assive orthonomic syst ems . Their crucial pro p erty is that for any Q ∈ A , there is a unique b Q ∈ B such that b Q ≡ Q mo d ∆. This b Q can b e obtained from the following r eduction alg orithm. Algorithm 1 Input: Expr ession Q ∈ A . Output: Expr ession b Q ∈ B . ⋆ if n o princip al de rivative app e ars in Q then r etu r n Q ⋆ let u j K b e the ≤ -highest princip al derivative app e aring in Q , and let α, L b e su ch that j, K = i α , J α L ⋆ substitute u j K = D L P α in Q and c al l the r esu lt R ⋆ r eturn b R The a lgorithm terminates b ecause the highest principal deriv ative of R , if it exists, is ≤ -sma ller tha n u j K , due to ass umption i). And, a different choice o f α wouldn’t change the result b ecause of assumption ii). The following lemma states that differentiation is compatible with the ab ov e reduction A → B , cf. [6, Theorem 4.8 ]. Lemma 2 F or a n y Q ∈ A we have \ D K Q = \ D K b Q Pro of: Using a modified v ersion o f Algo rithm 1 w e can write Q = b Q + R (∆), where R is some differential function of ∆ such that R (0) = 0. Clearly \ D K R (∆) v anishes.  4 4 The in trinsic symmetry condition Definition 3 We define an intrinsic multi-diff e r ential op erator D K : B → B by D K P = \ D K P , P ∈ B F rom this definition and Lemma 2 we obtain the following prop erties. Prop ositi on 4 Intrinsic multi-differ ential op er ators ar e c omp atible with c on- c atenation, D K D L = D K L . Pro of: W e have D K D L P = \ D K [ D L P = \ D K D L P = \ D K L P = D K L P.  Corollary 5 We have the analo gue of e qu ation (5), D K = D K 1 1 D K 2 2 · · · D K p p . Corollary 6 Intrinsic total diffe re n tial o p er ators c ommute, [ D i , D j ] = 0 . W e would like to hav e a more in tr insic characterization of D i , that is, without reference to any principal deriv ative o r tota l differe nt ia l o p erator. F or L ∈ N p we denote S L = { ( α, K ) : ( α, K L ) ∈ S } , which is a subset of S . F rom eq ua tion (1) and Definition 3 it follows that D j = ∂ ∂ x j + X ( k,L ) ∈S j u k Lj ∂ ∂ u k L + X ( i α ,J α M /j ) ∈S \S j D M P α ∂ ∂ u i α J α M /j . (7) W e note that when ( k , L ) ∈ S \ S j there exis t α ∈ N n , M ∈ N p such that ( k , Lj ) = ( i α , J α M ), and, for any β ∈ N n , N ∈ N p such that ( k , Lj ) = ( i β , J β N ) we hav e i α = i β and D M P α = D N P β . Due to Coro llary 5 equa tion (7) pr ovides a re c ur sive schema for intrinsic total differentiation. Prop ositi on 7 The r e cursive sche ma (7) for D i is wel l define d. Pro of: W e show the s chema terminates using tra nsfinite induction. F or any Q ∈ A , to ev aluate D j Q , apart from some differen tiation and m ultiplica tio ns, we need to ev aluate a finite num b er of expr e ssions D I P α . W e assume that D L P β can b e ev alua ted for all ( i β , J β L ) < ( i α , J α I ). Supp ose P α depe nds on u j K . Tha t implies ( j, K ) < ( i α , J α ). Supp ose there are β ∈ N n and L ∈ N p such that ( j, K I ) = ( i β , J β L ). Then, to ev aluate D I P α one may need to ev aluate D L P β . By the induction hypothesis this can b e done.  Definition 8 We define intrinsic pr olongation, denote d pr Q : B → B , of an evolutionary ve ctor field ν Q with Q ∈ B by pr Q P = \ pr Q P , P ∈ B . 5 F r om equa tion (2) and Definition 8 we g e t the intrinsic formula pr Q = X j,K ∈ S D K Q j ∂ ∂ u j K . W e now state and prov e o ur main theorem. Theorem 9 A tuple Q ∈ B is a symmetry of e quation (6) iff [ D j , pr Q ] = 0 for al l j . Pro of: ⇐ W e ca lc ulate pr Q ∆ α mo dulo ∆ \ pr Q ∆ α = \ D J α Q i α − \ pr Q P α = D J α Q i α − p r Q P α (8) Next we c alculate the commutator [ D j , pr Q ]. N eg lecting se cond or der deriv atives, we get D j pr Q = X ( k,L ) ∈S D Lj Q k ∂ ∂ u k L , and pr Q D j = X ( k,L ) ∈S j D Lj Q k ∂ ∂ u k L + X ( i α ,J α M /j ) ∈S \S j pr Q D M P α ∂ ∂ u i α J α M /j . Hence we get [ D j , pr Q ] = X ( i α ,J α M /j ) ∈ S \S j  D J α M Q i α − p r Q D M P α  ∂ ∂ u i α J α M /j . Suppo se that J α j 6 = 0. Then the ac tion of the a bove vector field on u i α J α /j yields the right hand side o f equation (8). Since we hav e chosen J α 6 = 0 ∈ N p this pr ov es our case . ⇒ Supp os e D J α Q i α = pr Q P α . Then [ D j , pr Q ] = X ( i α ,J α M /j ) ∈S \S j  D M , pr Q  P α ∂ ∂ u i α J α M /j . W e will pr ove that  D M , pr Q  P α = 0 for all α ∈ N n and M ∈ N p . The statement is certainly true for M = 0. A s sume tha t  D N , pr Q  P β = 0 6 for all ( i β , J β N ) < ( i α , J α M ). When M 6 = 0 there exists j such that M /j ∈ N p . W e write  D M , pr Q  P α = D j  D M /j , pr Q  P α +  D j , pr Q  D M /j P α The first term is zero b y the induction h yp othesis, so w e concentrate o n the se cond, which is X ( i β ,J β N ) ∈ S \ S j  D N , pr Q  P β ∂ ∂ u i β J β N/j D M /j P α . Suppo se P α depe nds on u k L . Then ( k , L ) < ( i α , J α ). The function D M /j P α may dep end on the deriv ative u k LM /j (namely , if u k LM /j ∈ S , otherwise it dep ends on smaller deriv atives). B ut when ( i β , J β N /j ) ≤ ( k , LM /j ) < ( i α , J α M /j ) by the induction h y po thesis  D N , pr Q  P β v an- ishes.  5 Discussion W e hav e s how that the infinitesimal conditio n [ D j , pr Q ] = 0 , for all j , is completely equiv alent to the more standar d infinitesima l symmetry condition pr Q ∆ ≡ 0 mo d ∆ in the setting o f pa ssive o rthonomic systems (6). These conditions a re sufficient for the corresp o nding group of transforma tions to b e a symmetry , in the sense o f transfor ming so lutions into so lutions [10, Theor em 2.31]. In genera l, o ne needs to make non-degener acy assumptions on the systems ∆ to ensure that they ar e also necessa ry , s ee [10, Sectio n 2 .6 a nd 5.1]. Howev er, in the setting o f passive o rthonomic systems lo cal so lv abilit y is ensured by the existence theor ems of Riquier a nd Janet [17, 2]. This means that infinitesimal metho ds c an b e used for classifying all the symmetries of passive orthonomic systems. W e do not prop ose that the bracket condition should repla ce the prolo n- gation condition for pr actical calcula tions. On the other hand, the int r insic differential op er ators do provide a con venient wa y of restric ting expressions to the sub-manifold defined by the equation. F o r example, consider the Bo ussinesq equation u tt = P , P = u xxxx − 2 u x u xx If we let the prolonga tion pr Q , with Q = u txx − u t u x , a ct on it we find pr Q ( u tt − P ) = u tttxx − 2 u t u 2 xx − 6 u tx u x u xx + 6 u txxx u xx − 2 u txx u 2 x − 2 u t u x u xxx + 3 u txxxx u x + 6 u txx u xxx + 4 u tx u xxxx + u t u xxxxx − u txxxxxx − 2 u tx u tt − u ttx u t − u ttt u x . T o see whether this v anishes mo dulo the equa tion we ha ve to substitute iter- atively u tt = P whenever we s ee t wo t -der iv atives. An alternative is to work with 7 int r insic differential op er ators. The parametric deriv atives are u, u x , u tx , u xx , . . . , u x n , u tx n , . . . . Clearly we hav e D x = D x . F o r the in trinsic total t -deriv ative we hav e D t = ∂ ∂ t + X n u tx n ∂ ∂ u x n + D n x P ∂ ∂ u tx n which can str a ightforw ar dly b e implement in any symbolic ma nipula tion pack- age. Without having to p erform any further simplifica tions it follows that D 2 t Q − pr Q P = 0 . W e do not w ant to chec k that [ D t , pr Q ] = 0 using a computer since it in volv e s infinite vector-fields, which w e can only de a l with if we let them act on finite expressions . The infinitesimal symmetry Q is th e first of an infinite hierarc hy . The next mem b er of this hierar ch y is g iven by R = u 2 t − 8 5 u xxxxx + 3 u 2 xx + 4 u xxx u x − 2 3 u 3 x , In fact, the symmetries of this hierar ch y commute mo dulo the Bo ussinesq equa - tion, we have [ pr Q , pr R ] = 0. Using our bracket formulation of infinitesimal symmetry together with the Jac o bi-identit y one ca n conclude that the bracket of tw o symmetries should ag ain b e a symmetr y . How ever, it is not a prio ri clear that this br ack et is the intrinsic pro longation of something. W e hav e observed that up to some s caling, the ’spacial part’ of R equals the right hand side o f the in tegrable p otential Kaup-K upe r shmidt equation [8, Equation 4.2 .5] u t = u xxxxx + 1 0 u x u xxx + 15 2 u 2 xx + 20 3 u 3 x , and the spatial parts of the o ther sy mmetries in the hierar ch y of the Boussine s q equation a re the symmetries of th e p otential Kaup-Kup er s hmidt equation. At orders w her e the Kaup-Kup ers hmidt equation has ga ps in its hier a rch y , the symmetries o f Bouss inesq do not have a spacia l par t. Since the F r´ echet deriv ative of the Boussinesq equatio n is s elf-adjoint, its symmetries are also co-symmetries of the equatio n. Therefor e they ar e in so me sense dua l to the Bo ussinesq equatio n. In the discr ete setting this kind of duality was firs t introduced in [1 1]. Conside r ing the dua l equation u txx = u t u x , amongst its symmetries we find the hierar ch y of the Saw ada -Kotera equation [15] u t = u xxxxx − 5 u xxx u x + 5 3 u 3 x . W e note that a s imilar observ ation, using Lax-pair s, was ma de in [3]. These examples show the impor tance of extending the techniques develop ed fo r es - tablishing the integrability o f evolution equations and classifying them, see [8, 1 4, 18, 4, 7] and references in there, to the rea lm of pas s ive or thonomic systems, o r b eyond. 8 Ac knowledgm ent I sincerely thank Peter O lver for p ointing me in the direction of pass ive or- thonomic systems. This research has been funded b y the Australian Research Council through the Cent r e of Ex cellence for Mathematics and Statistics of Complex Systems. Also thanks to the a nonymous refer ee for bringing to my attent io n a more general r esult, see se ction 1. References [1] V. P . Gerdt a nd Y u. A. Blinko v, Inv olutive Bas es of Polynomial Ideals, Mathematics and Co mputers in Simulation 45 (1998 ), 5 19-5 4 2. [2] M. Janet, Leco ns s ur les syst` emes d’´ equations aux d´ eriv ´ ees par tielles, Gauthier- Villa rs, Paris (19 29). [3] A.N.W. Hone and J.P . 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