On the spectral parameter problem

On the Sp ectral P arameter Problem Mic hal Marv an Abstract W e consider the problem whether a nonpara metric zero-cur v ature representation can be embedded into a one-par ameter family within the same Lie algebra . After in tro ducing a co mputable coho mological obstruction, a metho d using the recursion oper ator t o inco rpora te the parameter is discuss ed. 1 In tro duction Despite of four d eca d es’ dev elopment of s oliton theo r y , a general algorithm to recognize in tegrabilit y stil l remains elusiv e. By an integ r able system in t wo ind epend en t v ariables x, y w e sh all mean one that can b e rep rodu ced as the compatibilit y condition A y − B x + [ A, B ] = 0 (1) of the auxiliary linear sys te m Ψ x = A Ψ , Ψ y = B Ψ . (2) ([47, 38]). Matrices A, B are su pp osed to b elong to a m atrix Lie algebra g . The g -v alued 1-form A dx + B dy is ca lled a ze r o curvatur e r e pr esentation . T o b e indicativ e of in tegrabilit y , matrices A, B m ust dep end on wh at is called the sp e c tr al p ar ameter . Sometimes nonlinear systems come equipp ed with a nonparametric zero- curv ature represent ation. In geo m et r y of immersed surfaces, the Gauss– Mainardi–Co dazzi equations are alw ays of the form (1) w hile the lin ea r problem (2) is s u pplied b y the Gauss–W eingarten equations [46]. Inte grab le systems of this kind relev an t to physics w ere fi rst noticed b y Lu n d and Regge [26]. F or n u merous classical and new examples see [4, 38]. F or a dif- feren t source (the n on p arametric Lax p ai r s [22]) see [24]. Th us we face the “sp ectral parameter p roblem” or w hether a giv en n on- parametric zero curv ature r epresen tation can b e emb edded in a n on-trivia l 1 one-parameter family . An arbitrary num b er of r emovable parameters can b e alw a ys introd uced through the gauge transformation A ′ = S x S − 1 + S AS − 1 , B ′ = S y S − 1 + S B S − 1 , where S is an in v ertible matrix b elonging to the Lie group G a s s ociated with the Lie algebra g . Th e gauge transformation is a left action of the group of G -v alued functions on the set of all solutio n s of s ystem (1). Ho w ev er, parameters introduced that w ay pla y no role in soliton theory . In th e geometric context , the ma jorit y of effectiv e app roac hes link the sp ectral parameter with a d eformatio n parameter (Bob enk o [4]). A nonre- mo v able p aramet er can b e also often iden tified with the group parameter of a suitable p oint symmetry , see, e.g., [25, 41]. Th e early nineties [23 , 5] witnessed a dev elopment of al gorithmic pro cedures based on existing al- gorithms to compute symmetries. T o d et ect in tegrable cases, Cie ´ sli ´ nski [5 ] compares dimensions of the p oin t sym m et ry algebras of the in p ut system and of its co v ering [20] indu ce d by the zero cur v ature repr ese n tation. If these are inequal, a candidate for in tegrabilit y is disco v ered. Obvio u sly , un- detected remain the cases when the sp ectral parameter cannot b e iden tified with a group parameter. Th e first counterexample to b e in vestig ated thor- oughly w as th at of inhomogeneous nonlinear Schr¨ odinger sys te m . This led Cie ´ sli ´ nski to consideration of extended sym metries [6 , 7] that op erate within a class of equations. Th is m et ho d is suitable for classification p roblems, but results can d epend on the initial class of equations considered (the extended symmetry m u st op erate within this class). In this paper w e p resen t a ge n eral non-existence criterio n fo r the sp ectral parameter problem within a giv en Lie algebra, a giv en jet order, and a giv en ord er of p o we r expansion. Our main result can b e view ed as a “restricting” pro cedure whic h says wh en a nonro- mo v able parameter cannot b e inserted. W e also consider a new “extending” pro cedure that seeks the w ay to insert the parameter, th us trying to fill the space left by th e restricting p r ocedure. The sp ectral p aramete r problem is solv ed (in certain jet ord er) when the results of r estric ting and extend in g pro cedures meet. 2 Characteristic eleme nts In this section we recall few necessary facts f rom th e geometric theory of partial differen tial equatio n s [21] and zero curv ature represen tations [2 7]. Consider an arbitrary system of partial differen tial equations E l ( x, y , u i , u i x , u i t , . . . ) = 0 (3) 2 of an arbitrary order. By u k I w e shall denote the deriv ativ e of u k sp ecified b y a symmetric m ultiindex I in x , y . In th e jet space with coord inates t, x, u k I w e consider total derivatives D x = ∂ ∂ x + X i,I u xI ∂ ∂ u i I , D t = ∂ ∂ t + X i,I u tI ∂ ∂ u i I . Equations (3) and their differen tial consequences D I E l = 0 determine a submanifold (called diffiety ), whic h we sh all denote by E . Here D I denotes the comp osition of the op erators of total deriv ativ e D x , D y (in an y ord er, since they co m mute). Finally , operators D x , D y admit restriction to the diffiet y . A zero curv ature representat ion is d efined to b e a 1-form α = A dx + B dy suc h th at D y A − D x B + [ A, B ] = 0 (4) on E . T w o zero curv atur e r epresen tations A dx + B dy and A ′ dx + B ′ dy are said to b e gauge e quiv al e nt if one can b e obtained fr om the other by the gauge transformation A ′ = D x S S − 1 + S AS − 1 , B ′ = D y S S − 1 + S B S − 1 . The gauge equiv alence explains why system (4) is so muc h underdetermin ed (has t wice as many un kno wns as equations), which is the main obstacle to successful searc h for its solutions. T o compu te zero curv ature repr esenta- tions, we can c h oose b et we en the classical W ahlquist–Estabrook metho d of prolongation algebras [45, 9] (see [14 ] for implemen tation), its extensions suc h as [10], the metho d of co v erings by Krasil’shc h ik and Vinogrado v [20], the remark able deve lopment b y Igonin [17], the Sako vic h metho d of cyclic bases [39, 40], and also the metho d of c haracteristic elemen ts [27, 28]. Being a m odification of the well -kno wn c haracteristic function metho d to compute conserv ation la ws, the latter ap p roac h tries to fi x the gauge and cop es with classification pr oblems reasonably well. Giv en a zero curv ature representa tion A dx + B dy of system (3), the c haracteristic elements can b e computed using th e Sak ovi c h form ula [39]. Namely , w e necessarily h a v e D y A − D x B + [ A, B ] = P J,l C J l D J E l with suitable matrices C J l , and then the l th charac teristic element C ( l ) is giv en b y C ( l ) = X J ( − b D D ) J C J l     E . (5) 3 Here b D D x = D x − [ A, · ], b D D y = D y − [ B , · ], and b D D I is their ob vious comp osi- tion (in an y order since b D D x , b D D y necessarily comm ute once A, B form a zero curv ature represen tation). Finally , | E denotes the restriction to E . Example 1. Th e mKdV equation u t + u xxx − 6 u 2 u x = 0 has an sl (2)-v alued zero curv ature representati on A dx + B dt with A =  u λ 1 − u  , B =  − u xx + 2 u 3 − 4 λu 2 λu x + 2 λu 2 − 4 λ 2 − 2 u x + 2 u 2 − 4 λ u xx − 2 u 3 + 4 λu  . Actually , we h a v e D t ( A ) − D x ( B ) + [ A, B ] = ( u t + u xxx − 6 u 2 u x ) · C (1) , wh ere C (1) =  1 0 0 − 1  . By f orm ula (5), C (1) is the c h arac teristic elemen t. Th e only nonzero sum- mand in (5) corresp onds to the empty m u ltiindex J . This is t y p ica l for zero curv ature r ep resen tations computable b y the W ahlquist–Estabrook p roce- dure. Under gauge tr an s formatio n , characte r istic element s transform by conju- gation, w hic h allo ws us to brin g one of them in to the Jord an norm al form J . This usually lea v es some gauge fr ee d om, whic h can b e exploited to reduce one of the matrices A i b y gauge transformation with resp ect to the relativ ely small stabilizer subgroup S J of J . Th is v ery muc h resem bles th e well-kno wn “wild” problem of normal forms of pairs of matrices with resp ect to simul- taneous conjugation. In realit y our problem is simpler, since zero curv ature represent ations of use in soliton theory never tak e v alues in a resolv able sub - algebra, which allo w s u s to cut b r anc hes that fall into suc h a subalgebra. In the case of L ie algebra sl (2) and sl (3) all the normal form s were f ound in [28] and [43], resp ectiv ely . F or con ve n ience we tabulate them in the t wo columns headed k = 0 of T able 1 and 2. Th e remaining columns , headed k > 0, will b e explained in Section 3. The general case of Lie algebra sl ( n ) s til l remains largely unexplored except when the Jord an normal form J consists of a single Jordan blo c k, whic h h as b een completely inv estigated by Seb est y´ en [44]. All p ossible zero curv ature repr esen tations (4) along with their charac- teristic elemen ts (5) now can b e foun d from the determining system D y A − D x B + [ A, B ] | E = 0 , X I ,l ( − b D D ) I  ∂ E l ∂ u k I C ( l )      E = 0 , (6) 4 T able 1: Normal forms in the case of algebra sl (2) C A k = 0 k > 0 k = 0 k > 0  c 0 0 − c   c 0 0 − c   a 11 a 12 1 − a 11   a 11 a 12 0 − a 11   0 1 0 0   0 0 c 0   0 a 12 a 21 0   0 a 12 a 21 0  deriv ed in [27]. With A, B , C ( l ) restricted to their resp ectiv e norm al forms, the determining system (6) enjo ys the follo wing prop erties: – is a system of differenti al equations in total d eriv ativ es; – has the same n u m b er of unknowns as equations, akin to systems de- termining symmetries and conserv ation la ws; – is quasilinear in A, B , and linear in C ( l ) . Solution of s y s te m (6) is essen tially algorithmic giv en an up p er b ound on the j et order of the u nkno wns. Because of nonlinearit y in A and B , solution can b e troublesome ev en with the help of compu te r algebra. Remark 1. W ahlquist–Estabro ok t yp e zero curv ature repr esen tations hav e the ord er equal to the ord er of the equation minus one. The order of their normal form can b e t wice the order of the equation minus one at worst. Let us remark that the pro cedure do es not need to fix the gauge com- pletely . The general s olution of system (6) ma y still dep end on a remo v able “false” parameter. The parameter originates from linearit y of system (6) in the c h arac teristic element s as d emo nstarted in the follo win g example. Example 2. Consider the case of the Lie algebra sl (2) and the Jordan normal form J =  0 0 1 0  of the c haracteristic element. The 1-paramete r gauge group S =  s 0 0 s − 1  , s 6 = 0 , (7) preserve s the norm al form of A and m u ltiplie s J by s − 2 . By linearit y , if A, B , C ( l ) is a solution of system (6), then A, B , s 2 C ( l ) is a solution as wel l. 5 T able 2: Normal forms in the case of algebra sl (3) C A k = 0 k > 0 k = 0 k > 0   c 1 0 0 0 c 2 0 0 0 c 3     c 1 0 0 0 c 2 0 0 0 c 3     a 11 a 12 a 13 1 a 22 a 23 a 31 1 a 33     a 11 a 12 a 13 0 a 22 a 23 a 31 0 a 33     c 0 0 0 c 0 0 0 − 2 c     c 0 0 0 c 0 0 0 − 2 c     0 1 0 a 21 a 22 1 a 31 a 32 a 33     0 0 0 a 21 a 22 0 a 31 a 32 a 33     c 0 0 1 c 0 0 0 − 2 c     c c 1 0 0 c 0 0 0 − 2 c     a 11 a 12 0 a 21 a 22 1 a 31 a 32 a 33     0 a 12 0 a 21 a 22 1 a 31 0 a 33     a 11 a 12 0 a 21 a 22 0 a 31 a 32 a 33     0 a 12 a 13 a 21 a 22 0 a 31 a 32 a 33     0 0 0 1 0 0 0 0 0     c c 12 c 13 0 c 0 0 c 32 − 2 c     0 a 12 0 a 21 a 22 1 a 31 0 a 33     0 0 1 a 21 0 0 a 31 0 0     0 a 12 0 a 21 a 22 0 a 31 0 a 33     0 a 12 0 a 21 0 0 a 31 a 32 0     0 0 0 1 0 0 0 1 0     0 c 1 c 2 0 0 c 1 0 0 0     a 11 a 12 1 a 21 a 22 0 a 31 a 32 a 33     0 a 12 0 a 21 a 22 a 23 a 31 0 a 33     0 0 a 13 a 21 a 22 a 23 a 31 a 32 a 33     0 a 12 a 13 a 21 a 22 a 23 a 31 0 a 33   All matrices are sup posed to b e traceless (e.g., c 3 = − c 1 − c 2 ). 6 If C (1) = J and A is normalized according to T able 1, then A, B , s 2 C ( l ) is an un n ormaliz ed solution, but S AS − 1 , S B S − 1 , S s 2 C ( l ) S − 1 is a normal- ized solution. Thus (7) in tro duces one false p aramet er in to th e solution of system (6) und er normalization. So far, the only complete classification (with u n b ounded jet order) ob- tained is [30] (second ord er ev olution equations p ossessing a sl (2)-v alued zero curv ature r epresen tations; none admitting an essen tial parameter, hence all nonint egrable as naturally exp ected). 3 Restriction p ro cedure. F ormal remo v abilit y In the p r evio u s wo r k [29], a nontrivial horizont al gauge cohomology group H 1 w as associated with zero curv ature representa tions dep ending on a n on- remo v able p aramet er. Here we elaborate the idea fu rther in terms of p o we r series in the sp ectral parameter. Giv en a nonparametric zero curv ature representa tion A 0 , B 0 , consider all p ossible expansions A ( λ ) = ∞ X i =0 A i λ i , B ( λ ) = ∞ X i =0 B i λ i . (8) around zero. Obviously , A 0 = A (0), B 0 = B (0). Up on inserting expansions (8) in to the zero curv ature condition (4) w e obtain D y A k − D x B k + X i + j = k [ A i , B j ] = 0 (9) for all k ≥ 0. F or k = 0 this equation sa ys ju st that A 0 , B 0 is a zero cur v ature represent ation. F or k = 1, equation (9) reads D y A 1 − D x B 1 + [ A 1 , B 0 ] + [ A 0 , B 1 ] = 0 (10) (according to [11], this equation also p la ys an imp ortan t r ol e in the geometry of sur face s immersed in Lie algebras). Denoting b D D x = D x − [ A 0 , · ], b D D y = D y − [ B 0 , · ], equation (10) can b e rewritten as b D D y A 1 = b D D x B 1 (11) and in terpreted cohomologicall y [29]. 7 Recall th at the horizon tal gauge complex [29] consists of mo dules g ⊗ – Λ Λ k of h orizonta l forms with co efficien ts in the Lie algebra g . Here horizont al essen tially m ea n s that the forms con tain differen tials dx, dy only and – Λ Λ 0 is simply the add iti v e group of functions. The op erator ˆ d d acts on an arbitrary g -v alued function F ∈ g ⊗ – Λ Λ 0 as ˆ d d F = b D D x F dx + b D D y F dy ∈ g ⊗ – Λ Λ 1 , while for an arbitrary g -v alued horizont al 1-form U dx + V dy ∈ g ⊗ – Λ Λ 1 w e ha ve ˆ d d ( U dx + V dy ) = ( b D D x V − b D D y U ) dx ∧ dy ∈ g ⊗ – Λ Λ 2 . It is easy to see that ˆ d d ◦ ˆ d d = 0 once A 0 , B 0 is a zero curv ature rep r esen tation. The group H 1 = Ker ˆ d d : g ⊗ – Λ Λ 1 − → g ⊗ – Λ Λ 2 Im ˆ d d : g ⊗ – Λ Λ 0 − → g ⊗ – Λ Λ 1 is called the fi rst horizont al gauge cohomolog y group of the zero curv ature represent ation A 0 , B 0 . Elemen ts of Ker ˆ d d are called co cycles w hile those of Im ˆ d d are cob oundaries. W e show that H 1 = 0 imp lie s remo v abilit y of the parameter in a strict sense. T o circum v ent the issu e of con vergence, w e adopt the follo wing defi- nition. Definition 1. Th e parameter λ is said to b e formal ly r emovable if eve r y summand in the expansions (8) except A 0 , B 0 can b e lo ca lly annihilated b y a gauge transform ation with resp ect to some m atrix fu nctio n in λ . The follo wing prop osition is easy to pr o v e. Prop osition 1. L et A ( λ ) , B ( λ ) b e a zer o cu rvatur e r e pr esentation dep end- ing analytic al ly on a p ar ameter λ . L et the first horizontal g auge c ohomolo gy gr oup H 1 with r esp e ct to A 0 , B 0 b e zer o. Then λ is formal ly r emovable. Pr o of. Equation (11) sa ys that A 1 dx + B 1 dy is a co cycl e, hence also a cob oundary if H 1 = 0. T his means that A 1 dx + B 1 dy = ˆ d d S 1 for a su ita b le g -v alued matrix function S 1 , i.e., A 1 = D x S 1 − [ A 0 , S 1 ] , B 1 = D y S 1 − [ B 0 , S 1 ] . 8 Let S ( λ ) = E + S 1 λ . Th e gauge action with resp ect to S ( λ ) will ke ep the terms A 0 , B 0 , but remov e the terms A 1 , B 1 . This is readily c h ec k ed using the expansion ( E + S 1 λ ) − 1 = E − S 1 λ + S 2 1 λ 2 − S 3 1 λ 3 + · · · , w h ic h has a p ositiv e radius of con verge n ce in ev ery op en set where E + S 1 has b ounded eigen v alues, see Gan tmac her [13, Ch. V, § 4, Thm. 2]. This means that A 1 , B 1 can b e lo call y annihilated. O therwise said, if H 1 = 0, then without loss of generalit y we can assume that A 1 = B 1 = 0. No w we can pro ceed by ind u ctio n . If A i = B i = 0 f or all i < k , then equation (9) with k ≥ 2 reads D y A k − D x B k + [ A k , B 0 ] + [ A 0 , B k ] = 0 , meaning that A k dx + B k dy is a co cycle, hence a cob oundary , i.e., equal to ˆ d d S 1 for suitable S k . Reasoning as ab o v e, one easily sees that A k , B k can b e lo ca lly gauged out through the gauge matrix E + S k λ k . This concludes the pro of. Corollary 1. A zer o curvatur e r epr esentation A 0 , B 0 such that the first horizonta l gauge c ohomolo gy g r oup H 1 is zer o c annot b e a memb er of an an- alytic one-p ar ameter family that would dep end on a formal ly nonr emovable p ar ameter. A metho d to compute th e cohomolog y group H 1 of a giv en zero cu rv ature represent ation A 0 , B 0 follo ws from [29, Prop. 7], w h ic h w e r eprod uce b elo w in a sligh tly simplifi ed form: Prop osition 2 ([29]) . Given a zer o curvatur e r epr esentation A 0 dx + B 0 dy , the 1 -form A 1 dx + B 1 dy is a c o cycle if and only if matric es A [1] =  A 0 0 A 1 A 0  , B [1] =  B 0 0 B 1 B 0  , (12) c onstitute a zer o curvatur e r epr esentation: D y A [1] − D x B [1] + [ A [1] , B [1] ] = 0 . (13) Mor e over, two c o cycles ar e c ohom olo gic al (differ b y a c ob oundary) if and only if the c orr esp onding zer o curvatur e r epr esentations (12) ar e gauge e quivalent with r esp e ct to a gauge matrix of the b lo ck triangular form S [1] =  E 0 S E  (14) with the unit matrix E at the diagonal p ositions. 9 It follo ws th at cohomology classes of co cycles A 1 dx + B 1 dy can b e de- termined by solving equation (13) mo dulo gauge equ iv alence with resp ect to (14). If A 1 = 0, B 1 = 0 for all normalized solutions to (13), then H 1 = 0. Ho we ver, ther e can also b e a kind of “false” solutions to equation (13). Example 3. Con tin u ing Example 2, consider the gauge matrix S =  1 + c 1 λ + c 2 λ 2 + · · · 0 0 ( 1 + c 1 λ + c 2 λ 2 + · · · ) − 1  =  1 + c 1 λ + c 2 λ 2 + · · · 0 0 1 − c 1 λ + ( c 2 1 − c 2 ) λ 2 + · · ·  with constant co efficien ts c i . In Example 2 w e already demonstrated that S in tro duces a false parameter not detected by the charact eristic elemen t metho d. Wh at is then the action of S on the matrix A 0 =  a 11 a 12 a 21 − a 11  ? By expanding S A 0 S − 1 in the p o wers of λ w e get S A 0 S − 1 = A 0 +  0 − 2 c 1 a 12 2 c 1 a 21 0  λ +  0 ( 3 c 2 1 − 2 c 2 ) a 12 ( c 2 1 + 2 c 2 ) a 21 0  λ 2 + · · · . In particular, A 1 =  0 − 2 c 1 a 12 2 c 1 a 21 0  . If this A 1 exhausts solutions to (13), then H 1 = 0 again. If H 1 = 0, then Prop osition 1 applies. Otherwise, A 1 , B 1 are candidates for the first co efficien ts of the T a ylor expans ions (8). Ho w to c h eck whether they are go o d? It is ea s ily v erified that the system of equations (9) indexed by k = 0 , . . . , m is satisfied if and only if the blo c k triangular m at r ice s A [ m ] =       A 0 0 · · · 0 A 1 A 0 . . . . . . . . . . . . . . . 0 A m · · · A 1 A 0       , B [ m ] =       B 0 0 · · · 0 B 1 B 0 . . . . . . . . . . . . . . . 0 B m · · · B 1 B 0       , (15) constitute a zero cur v ature repr esen tation, i.e., D y A [ m ] − D x B [ m ] + [ A [ m ] , B [ m ] ] = 0 . (16) Hence, if A k , B k are already kno wn for all k < m , then (1 6) determines whether expansions (8) can b e extend ed one step further. 10 Prop osition 3. If, f or some m , no solution A [ m ] , B [ m ] of e quation (16) exists, then ther e is no p ossibility to extend the exp ansions (8) b e yond the first m terms. Prop ositions 1 and 3 reduce th e s pectral parameter problem to that of computation of zero curv ature r epresen tations m odu lo a gauge group. The substanti al b enefit is that the p r oblem b ecomes linear, b eing reducible to solution of a system (6) written on matrices A [1] , B [1] giv en b y (12), resp. on matrices A [ m ] , B [ m ] giv en b y (15). The sy s te m (6) is then indeed linear in all un kno wns including A k , B k , k ≥ 1 (b eing still n onlinea r in A 0 , B 0 , which are supp osed to b e kn o wn, ho wev er). T o solv e system (13) or (16) b y the c h arac teristic element metho d, we need to construct th e resp ectiv e normal forms. T h e c haracteristic elemen ts of the zero curv ature represent ation A [ k ] , B [ k ] are blo c k triangular matrices of the same form C [ k ] ( l ) =       C ( l )0 0 · · · 0 C ( l )1 C ( l )0 . . . . . . . . . . . . . . . 0 C ( l ) k · · · C ( l )1 C ( l )0       . (17) Ev ery diagonal blo c k C ( l )0 coincides with the corresp onding known c harac- teristic element of the inp u t zero curv ature representa tion A 0 , B 0 . The gauge transformations considered sh ould b e with resp ect to matrices of the form S [ k ] =       E 0 · · · 0 S 1 E . . . . . . . . . . . . . . . 0 S k · · · S 1 E       . In the case of the Lie algebras sl (2) and sl (3) the n orm al forms for C k and A k , k > 0, are tab elated in T able 1 and 2 ab o ve. Being a determined s y s te m of equations in total deriv ativ es, system (6) is routinely solv able if one s et s an upp er b ound on the jet order of the un- kno wn s (see the end of Section 2). It seems natural to assu me th at insertion of the paramater do es not require increasing the jet order of the zero cu r- v ature repr esen tation. How ev er, this is not n ec essarily the s ame thin g as setting the order of A k , B k equal to that of A 0 , B 0 , since norm al ization can increase the jet ord er (see R emark 1 and the example in th e next section). 11 4 An example. The inhomogeneous nonlinear Sc hr¨ o- dinger equation In th is section we consider Cie ´ sli ´ nski’s coun terexample [1, 6, 7, 8], the inho- mogeneous nonlinear S c hr¨ odinger equation q t = ( f q ) xx + 2 q r , p t = − ( f p ) xx − 2 pr , (18) r x = f ( pq ) x + 2 f x pq , where f ( t, x ) is an arb itrary function. Compared to Cie ´ sli ´ nski’s formula- tion [6 , 7, 8], w e ha ve p u t p = ¯ q q and m ade the formal c hange of co ordinates t ↔ i t , preservin g integrabilit y . The system is kno wn to b e integrable if f = f 1 ( t ) x + f 0 ( t ) is linear in x ; otherwise it is b eliev ed to b e n onin tegrable (see op. cit. and references therein). The initial nonp aramet r ic first order sl (2 ) -v alued zero cur v ature repre- sen tation is D t A 0 − D x B 0 + [ A 0 , B 0 ] = 0 with A 0 =  0 q − p 0  , B 0 =  r ( f q ) x ( f p ) x − r  , (19) v alid for all functions f ( t, x ). Let us demonstrate ho w the pro cedure outlined in the p revious section detects the exp ected int egrable cases in tw o steps. T o start w ith , we fin d the c haracteristic elemen t of A 0 dx + B 0 dy to b e the triple of sl (2)-matrices C (1)0 =  0 1 0 0  , C (2)0 =  0 0 − 1 0  , C (3)0 =  − 1 0 0 1  . Indeed, the expression D t A 0 − D x B 0 + [ A 0 , B 0 ] is equal to the sum of C ( j )0 m u ltiplied by j th equation (18) eac h , similarly as in Examp le 1 ab o ve. T o pr oceed fu rther, w e construct th e 4 × 4 matrices (12) and (17), k = 1, f rom the already kno wn blo c ks A 0 , B 0 , C (1)0 , C (2)0 , C (3)0 and th e blo c ks A 1 , B 1 , C (1)1 , C (2)1 , C (3)1 y et to b e foun d . These matrices are sub ject to system (6). A t th is p oin t there are essentia lly tw o natural w ays to p ro- ceed furth er dep ending on whic h p air of the unknown matrices w e c ho ose to normalize. Choic e 1. According to the second ro w of T able 1 ab o ve , matrices C (1)0 and A 0 are already in their resp ectiv e normal forms . How ev er, C (1)0 is a single Jordan blo c k, hence a “false” parameter should b e exp ected (see Ex- ample 2). 12 According to T able 1 ab o ve , n ormal forms for C (1)1 and A 1 are C (1)1 =  0 0 c 0  , A 1 =  0 a 12 a 21 0  . Then there is n o ro om for further normalization and matrices B 1 , C (2)1 , C (3)1 ha ve to b e left arbitrary . Therefore, the 4 × 4 matrices A [1] =     0 q 0 0 − p 0 0 0 0 a 12 0 q a 21 0 − p 0     , B [1] =     r ( f q ) x 0 0 ( f p ) x − r 0 0 b 11 b 12 r ( f q ) x b 21 − b 11 ( f p ) x − r     , C [1] (1) =     0 1 0 0 0 0 0 0 0 0 0 1 c 0 0 0     , C [1] (2) =      0 0 0 0 − 1 0 0 0 c (2)11 c (2)12 0 0 c (2)21 − c (2)11 − 1 0      , C [1] (3) =      − 1 0 0 0 0 1 0 0 c (3)11 c (3)12 − 1 0 c (3)21 − c (3)11 0 1      in volv e 12 un kno wn functions, namely a 11 , a 12 , b 11 , b 12 , b 21 , c , c (2)11 , c (2)12 , c (2)21 , c (3)11 , c (3)12 , c (3)21 . The d etermin in g sys tem (6) consists of the same n u m b er 12 of indep endent linear equations in total deriv ativ es. Assum ing A 1 , B 1 of the second order at least (cf. Remark 1), we obtain a non tr ivia l solution A 1 = 0 − p x p 2 0 0 ! , B 1 =     ( f p ) x p ( f p ) xx p 2 + 2 f q − 2 f p − ( f p ) x p     (20) 13 (and its constant multiples) of system (6), v alid without any constr aint on f ( t, x ) . W e also get a solution of the form sh o wn in Example 3, which cor- resp onds to a false parameter. This implies that H 1 is alwa ys nonzero. This means that all inh omog e- neous n on lin ea r Sc h r¨ odinger equations admit a nonremov able parameter up to the firs t ord er . T o see w hether the p o wer expansion can b e pr ol onged fur ther, as a next step w e consider the 6 × 6 matrices A [2] , B [2] , C [2] (1) , C [2] (2) , C [2] (3) from the already kno wn blo c ks A i , B i , C (1) i , C (2) i , C (3) i , i ≤ 1 and the y et unkn o wn blo c ks A 2 , B 2 , C (1)2 , C (2)2 , C (3)2 . Normalizing A 2 , C (3)2 in the same wa y as A 1 , C (3)1 ab o v e, we consider the corresp ondin g sys te m (6). Ho wev er, this new system turns out to b e incompatible unless d 2 f dx 2 = 0 . (21) By Prop osition 3, condition (21) is n ec essary for the initial zero curv ature represent ation (19) to admit a nonremov able parameter, assumin g the jet order at most tw o. Choic e 2. Acco r ding to the firs t row of T able 1, C (3)0 is in a n orm al form, whilst A 0 is not. Norm alizing A 0 increases the j et order of A 0 , B 0 to t wo . This le ads us immediately to computations on the second order j et lev el, while a voiding the “false” p arameter solution. 5 Extension pr o cedures. Recursion op erators The pr ocedure allo ws us to compu te ev ery term of the T a ylor exp ansion of A ( λ ) , B ( λ ). In the case of p olynomial dep endence on the sp ectral p arame- ter we get a closed-form result after a fin ite num b er of steps. O th erwise w e are left with a trun ca ted T a ylor expans io n , whic h is, ho w ever, not en tirely useless. In principle, one can apply the c h aracteristic elemen t metho d to compute the full zero cu rv ature repr esen tation A ( λ ) , B ( λ ). Solving the n on- linear system (6) b y triangularization [16] usually inv olv es hea vy branching, but knowing the trun cat ed T a ylor expansion (8) p ermits cutting off many branc h es immed ia tely . The symmetry metho d [5] and its extension [6, 7, 8] by Cie ´ sli ´ nski tak e adv an tage of linearit y of the p roblem. Remark ably enou gh , certain asp ects of this metho d admit a cohomologica l interpretation, which will b e d iscussed elsewhere. 14 Apart f r om zero curv ature representati on s , t wo-dimensional integrable systems usually p ossess infin ite hierarc hies of symmetries, generated by re- cursion op erators. Giv en a zero curv ature representati on or a Lax pair, the recursion op erator R can b e derive d b y v arious m et ho ds, see [12, 40] an d references therein. An ot her relation w as observ ed to exist b et ween the zero curv ature repr esen tation and the inv erse ( R + λ I d ) − 1 [3, 31, 33], w hic h ev en extends to certain m ultidimensional systems [34]. This, ho we v er, requires in terpr eta tion of the recur sion op erator as a B¨ ac klun d autotransform at ion of the linearized equation (P apac h r istou [37], Guthrie [15]). Although this interpretatio n first ap p eared in multidimensional integ ra- bilit y theory [37], it turn ed u p to b e extremely f ruitful in t wo -dimensional case as well. Guthrie’s motiv ating question was how to app ly a recursion op erator to a symmetry in a safe wa y . The d ange r ous p oint is that the com- mon pseudo different ial form of a recursion op erator [36] inv olv es a purely formal inv erse D − 1 = D − 1 x . Th e u s ual answer is to interpret H = D − 1 x F as F = D x H , but this can easily lead to errors (‘b ogus’ symmetries [15]). Guthrie’s idea wa s to p ro vide also the v alue G = D t H , wh er e t is the other co ordinate. Th is suffi ce s to determine the pseudop oten tials up to an in tegra- tion constan t. The in tegration constan t then can b e safely omitted since it only adds R ( 0) to the resu lt. Ser gyey ev [42] discussed another in terpr eta tion problem absent u nder Gu thrie’s app roac h. As is w ell kn own, infinitesimal symmetries of the system (3) can b e p ut in the ve r tic al form U = X i,I D I U i ∂ ∂ u i I , (22) where I runs ov er all m ultiindices o ve r t, x , co efficien ts U i b eing functions defined on E . The vect or field (22) is a symmetry if and only if the functions U l satisfy the system 0 = X i,I ∂ E l ∂ u i I D I U i     E =: ℓ E l ( U ) | E . (23) F ormally , (3) and (23) can b e wr itte n together as a system of p artia l d iffer- en tial equations E l = 0 , X i,I ∂ E l ∂ u i I U i I = 0 , (24) 15 where the comp onen ts U i of a symmetry are add itio n al unkn o wns. Sys- tem (24) will b e called the line arize d system . T his allo w s us to in terpr et recursion op erators as B¨ ac klun d autotransformations for solutions U i of the linearized system (24) (Papac hristou [37]). Example 4. Considering the linearized Burgers equation u t = u xx + uu x , U t = U xx + uU x + u x U, it can b e easily c hec ke d that the w ell-known t -dep endent r ecursion op erator U ′ = tU x + 1 2 ( tu + x ) U + 1 2 ( 1 + tu x ) W , W x = U, W t = U x + uU, is actually a B¨ ac klun d autotransformation. General recur sion op erators of system (3) are to b e sought among B¨ ac k- lund autotransformations of the linearized system (24), w ith ps eu dopoten- tials su b ject to first-order linear systems of the form (2). The co effici en t ma- trices A, B are sup p osed to dep end on u i I and U i I no w. Hence the pair A, B constitutes a zero cur v ature representat ion o ver the linearized system (24). Numerous examples [2, 3, 31, 33, 34] (as well as unpub lish ed ones) su p- p ort the follo wing conjecture: Conjecture 1. F or every r e cursion op er ator R of a system (3) ther e exists a finite-dimensional Lie algebr a g and a ze r o curvatur e r epr esentation A, B of the system (3) such that the pseudp otentials of R c an b e assemble d in a single g -value d nonlo c al variable Φ subje ct to c onditions Φ x = [ A, Φ ] + ℓ A ( U ) , Φ t = [ B , Φ ] + ℓ B ( U ) . (25) Here U denotes a seed symmetry , [– , –] the comm u tat or in g , and ℓ the comp onen t w ise F r ´ ec het d er iv ativ e ℓ A ( U ) = X k ,I ∂ A ∂ u k I U k I . F or ‘con v entional’ recursion op erato r s th e algebra g is t ypically solv able and mostly ab elian, in which case the zero curv ature representa tion A, B reduces to a collectio n of conserv ation la ws (p ossibly n onlocal) [32]. E.g., 16 in Example 4 the alg ebra is 1-dimensional, u t = ( u x + 1 2 u 2 ) x b eing the corresp onding conserv ation la w. On the other hand, the in verse ( R + λ Id) − 1 of a con ve n tional recur sion op erator R + λ Id is u sually asso ciated with the λ -dep enden t zero curv ature represent ation A, B of system (3) (how ev er, it may happ en that the algebra g is solv able). This is another expression of the conv en tional wisdom that a recursion op erator R yields a zero cur v ature represen tation through th e eigen v alue problem R U = λU . What we hav e obtained is a w ay from a λ -ind epend en t zero cur v ature represent ation to a r ecursion op erator R , the inv erse ( R + λ Id) − 1 , and its asso cia ted λ -dep endent zero cu r v ature representa tion. T o mak e it in to a w orkin g algorithm we m ust fur ther restrict the form of the recursion op er- ator. Conjecture 2. F or every inte gr able system (3) ther e exists a zer o or der r e cursion op er ator with diagonal matrix, i.e., of the form R ( U ) = ( U i ′ ) = ( c i i U i + a i j Ψ j ) (26) with Ψ as in (25) , wher e c i i , a i j ar e functions on E . The nonlo cal ities Ψ j can tak e v alues in an extension of the algebra g of the in iti al zero-curv ature represen tation. L eaving this asp ect aside, we prop ose the follo wing p rocedur e. Pro cedure 1. Given an initial zero curv ature represen tation A = A 0 , B = B 0 of (3), 1. write do wn the linearized system (24) and the asso ciated co vering (25); 2. find all op erators (26) su c h that U ′ = R ( U ) satisfies the linearized system (24); 3. solv e equation U ′ = R ( U ) + λU for U in terms of U ′ to obtain the in verse recursion op erator ( R + λ Id) − 1 ; 4. transform ( R + λ Id) − 1 to its Guthrie form and iden tify the asso ciate d zero curv ature represen tation (whic h o ccurs in adjoint represent ation here). This pro cedure w as p o werful enough to lead to all recursion op erators published in the wo r ks [2, 3, 31, 33, 34]. Ho we v er, not alw ays a recursion op erator could b e asso cia ted with ev ery v alue of the sp ectral parameter (for an ins ta n ce see [31]). Consequ en tly , for certain λ -indep endent zero cu r v ature represent ations the ab o v e pro cedure fails ev en though the corresp onding λ - dep enden t family exists. This phenomenon must b e fu rther inv estigated. 17 Nev ertheless, in the follo wing section w e sho w that our pro cedure copes w ell with the zero curv ature represen tation of the inhomogeneous nonlinear Sc hr ¨ odinger equation. 6 The inhomogeneous n onli n ear Sc hr¨ odinger equa- tion c on tin ued Con tinuing the example of the inhomogeneous nonlinear Sc h r¨ odinger equa- tion, w e consider the nonparametric sl (2 ) -v alued zero curv atur e representa - tion (19) v alid for all fun cti on s f ( t, x ). Step 1. The linearized system (24) is q t = ( f q ) xx + 2 q r , p t = − ( f p ) xx − 2 pr , r x = f ( pq ) x + 2 f x pq , Q t = ( f Q ) xx + 2 ( r Q + q R ) , P t = − ( f P ) xx − 2 ( r P + pR ) , R x = f ( q P + pQ ) x + 2 f x ( q P + pQ ) . The co ve r ing (25) is d etermin ed by equations   ψ 11 ψ 12 ψ 21   x =   0 p q − 2 q 0 0 − 2 p 0 0     ψ 11 ψ 12 ψ 21   +   0 Q − P   ,   ψ 11 ψ 12 ψ 21   t =   0 ( f p ) x − ( f q ) x ( f q ) x − 2 r 0 − ( f p ) x 0 2 r     ψ 11 ψ 12 ψ 21   +   0 ( f Q ) x ( f P ) x   , (27) where ψ ij are comp onen ts of an sl (2)-matrix Ψ =  ψ 11 ψ 12 ψ 21 − ψ 11  . Step 2. The equations to b e solv ed are D t Q ′ − f D xx Q ′ − 2 f x D x Q ′ − ( f xx + 2 r ) Q ′ − 2 q R ′ = 0 , D t P ′ + f D xx P ′ + 2 f x D x P ′ + ( f xx + 2 r ) P ′ + 2 pR ′ = 0 , D x R ′ − f q D x P ′ − ( f q x + 2 f x q ) P ′ − f pD x Q ′ − ( f p x + 2 f x p ) Q ′ = 0 (28) 18 where, according to (26), we assume P ′ = c P P + a 11 P ψ 11 + a 12 P ψ 12 + a 21 P ψ 21 , Q ′ = c Q Q + a 11 Q ψ 11 + a 12 Q ψ 12 + a 21 Q ψ 21 , R ′ = c R R + a 11 R ψ 11 + a 12 R ψ 12 + a 21 R ψ 21 . F unctions c P , c Q , c R , a ij P , a ij Q , a ij R are the unkn o wns and can dep end on t , x, p, q , r and their deriv ative s p x , q x , r t , p xx , q xx , r tt , . . . . System (28) can b e solv ed r outinely . As a r esult w e obtain that R ( U ) is a constant multiple of U unless f xx = 0 again. On the other hand, if f = f 1 ( t ) x + f 0 ( t ), then we obtain a nontrivial s ol u tio n P ′ = g 1 P + ψ 21 , Q ′ = g 1 Q + ψ 12 , R ′ = g 1 R + f 1 ψ 11 + f pψ 12 + f q ψ 21 , (29) where g 1 = Z f 1 ( τ ) dτ . (30) The recursion op erator give n by f ormulas (29) an d (27) can b e applied to an arb itrary seed sy m metry . Ho w ever, generated symmetries will b e n onlocal as a rule, which is usually the case with inv erse recursion op erators. Step 3. T o in vert this op erator, w e reinte r pret form u las (29) and (27) so th at P ′ , Q ′ , R ′ are give n and P , Q, R are to b e foun d. F rom equation (29) w e get P = P ′ − ψ 21 g 1 , Q = Q ′ − ψ 12 g 1 , R = R ′ − ( f 1 ψ 11 + f pψ 12 + f q ψ 21 ) g 1 , (31) while ψ ij will satisfy a system of the same form (25 ). The term λ Id is absorb ed in the integrat ion constant of (30). Herefrom we can reconstruct the corresp onding zero curv ature represen tation as A ( λ ) =     − 1 2 g 1 q − p 1 2 g 1     , B ( λ ) =     r + f 2 g 2 1 ( f q ) x − f q g 1 ( f p ) x + f p g 1 − r − f 2 g 2 1     , (32) 19 where g 1 is the integral (30) and the in tegration constant for g 1 serv es as the sp ectral parameter λ . This result is gauge equ iv alen t to that obtained b y Cie ´ sli ´ nski [6, 7 ]. That the parameter is nonremo v able can b e seen from the expansion of A in p o wers of λ . Indeed, A 1 dx + B 1 dy is then gauge equiv alen t to the previously computed generator (20) of H 1 . In p artic u lar, the upp er and lo wer b ounds established in Sections 4 and 6 coincide, meaning that the answer is complete (within the Lie group sl (2) and the second jet order of the zero curv ature represen tation). Remark ably enough, the inv erse of the recursion op erator (31) still f ails to pro duce a local hierarc hy . 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