Integrability conditions for difference equations

Transcription

1 Integrability conditions for difference equations Pavlos Xenitidis School of Mathematics University of Leeds joint work with A. V. Mikhailov Discrete Integrable Systems a follow-up meeting 12 July2013 Xenitidis P. Leeds Integrability conditions 12 July / 22

2 Plan of talk Definitions Dynamical variables Symmetries& conservation laws Formal series& Residues Formal Recursion Operators General formulation of integrability conditions First and Second order integrability conditions Integrable quadrilateral equations u 01 u 10 u 00 + u = 0 v 11 v 01 + v v 10 v 00 + v 00 = 0 w 00 w 11 c 1 w 10 w 01 w 10 w 01 + w11 + w 00 c = 0 This talk is based on A.V. Mikhailov, J. P. Wang, P. Xenitidis2011 Recursion operators, conservation laws, and integrability conditions for difference equations Theor. Math. Phys. 167, A.V. Mikhailov, P. Xenitidis2013 Second order integrability conditions for difference equations. An integrable equation arxiv Xenitidis P. Leeds Integrability conditions 12 July / 22

3 Notation, assumptions & a definition Notation Independent variables: n, m The dependenceof function u on n, m : u pq := un+p, m + q, p, q Z 2 Shift operatorinthe n-and m-direction:sandt,respectively. Assumptions We deal with autonomous quadrilateral difference equations Qu 00, u 10, u 01, u 11 = 0 where Q isapolynomialwhich dependsexplicitlyonall ofits argumentsand is irreducible Definition By integrability of a difference equation, we understand the existence of an infinite hierarchy of its symmetries. Xenitidis P. Leeds Integrability conditions 12 July / 22

4 Dynamical variables Inthe theory ofdifferenceequations, u pq aretreatedas variables and we denotethe set of these variables as U = { u pq p, q Z 2 }. Any rational expression in variables U, which can be restricted on solutions of equation Qu 00, u 10, u 01, u 11 = 0,canbe reducedto an expressioninvolving onlyvariables u l0 and u 0l. Dynamical variables Wereferto variables u l0, u 0l as dynamical variables and denote theircorrespondingsets as U s ={u l0 l Z}, U t ={u 0l l Z}, U 0 = U s U t. Moreover, the fields of rational functions of the dynamical variables will be denoted as respectively. F s =CU s, F t =CU t, F 0 =CU 0, Xenitidis P. Leeds Integrability conditions 12 July / 22

5 Symmetries and conservation laws Symmetry A symmetry K ofequation Qu 00, u 10, u 01, u 11 = 0 isanelementoff 0 satisfyingthe equation Q upq S p T q K = 0, mod < Q = 0>. p,q Remark For quadrilateral equations, symmetries are always of the form K = K s + K t, K s F s, K t F t. A symmetry K maybe consideredas aflow t u 00 = K commuting withthe equation. Conservation law A local conservationlaw forequation Qu 00, u 10, u 01, u 11 = 0isapair offunctions ρ,σ Span C F 0, logf 0 suchthat T 1ρ S 1σ = 0, mod < Q = 0>. Xenitidis P. Leeds Integrability conditions 12 July / 22

6 Recursion operators Our definition of integrability is based on the existence of an infinite hierarchy of symmetries. One way to generate an infinite hierarchy of symmetries for a difference equation is to find an operator which maps symmetries to symmetries. Definition Arecursionoperatorofadifferenceequation Q = 0isapseudo-differenceoperatorRsuchthat R : DomR K Q K Q, wherek Q isthe linear spaceof symmetriesof this differenceequation. Inother words,ifthe action ofronasymmetry K F 0 isdefined,i.e.rk F 0,then RKisasymmetryofthe samedifferenceequation. Example of pseudo-difference operator: the recursion operator for H1 R = 1 S+1S S S 1 u 10 u 10 u 10 u 10 Xenitidis P. Leeds Integrability conditions 12 July / 22

7 Formal series Definition A formallaurentseriesoforder N isdefinedas the formalsemi-infinite sum A L = a N S N + +a 0 + a 1 S 1 +, a N 0, a k F 0. Pseudo-difference operators and formal series Any s-pseudo-difference operator A can be uniquely represented by a Laurent formal seriesa L. Fractional powers of formal series Inthedifferencecase,itisnotalwayspossibletofindthe NthrootofaNthorderformal series. Foragiven Nth orderformalseriesa L we cannot alwaysfind afirstorderformalseries B L = b 1 S+b 0 + b 1 S 1 +, b i F 0, such that A L =B N L. Xenitidis P. Leeds Integrability conditions 12 July / 22

8 Residues and difference analog of Adler s theorem Definition LetA L beaformalseriesof order N, i.e. A L = a N S N + +a 0 + a 1 S 1 +. The residueresa L and logarithmicresidue res loga L ofa L aredefined as resa L = a 0, res loga L = loga N. Difference analog of Adler s theorem Let A L = a N S N + a N 1 S N 1 +, B L = b M S M + b M 1 S M 1 +. Thenthe residueoftheir commutator isgivenby res[a L,B L ] = S 1σA L,B L, where N n M n σa L,B L = S k a n S n k b n S k b n S n k a n. n=1 k=1 n=1 k=1 Xenitidis P. Leeds Integrability conditions 12 July / 22

9 Integrability conditions for quadrilateral equations Let Qu 0,0, u 1,0, u 0,1, u 1,1 = 0be aquadrilateraldifferenceequation. 1 If thereexisttwo s pseudo-differenceoperatorsrandpoforder N such that 0 0 D Q R =P D Q, D Q S i T j uij, i=1 j=1 then R is a recursion operator of the difference equation. 2 The above relationis validifand only if TR R = [Φ R,Φ 1 ], where Φ = Q u1,1 S+Q u0,1 1 Q u1,0 S+Q u0,0. 3 Moreover, if R satisfies the above conditions, then it also satisfies the equations TR k R k = [Φ R k,φ 1 ], k = 2, 3,. Xenitidis P. Leeds Integrability conditions 12 July / 22

10 Formal series, residues and conservation laws TR k R k = [Φ R k,φ 1 ], k = 1, 2, 3,. These relationshold ifwe replacer,φby theirformalseriesr L,Φ L. TR k L Rk L = [Φ L R k L,Φ 1 L ], k = 1, 2, 3,. Taking the residues of these relations, we end up with infinitely many canonical conservation laws N 1 ρ 0 = res logr L, σ 0 = S k 1 logβ 0, k=0 ρ k = resr k L, σ k =σ Φ L R k L,Φ 1 L, k> 0 Φ L =α 0 +α 1 S 1 +α 2 S 2 + α 0 =S 1 Qu10 Q u11 α 1 =S 1 Qu00 Q u11 Qu 01 Q u11 S 1 Qu10 Q u11 α k+1 = 1 k S 1 Qu01 Q u11 α k Φ 1 =β L 0 +β 1 S 1 +β 2 S 2 + β 0 =S 1 Qu11 Q u10 β 1 =S 1 Qu01 Q u10 Qu 00 Q u10 S 1 Qu11 Q u10 β k+1 = 1 k S 1 Qu00 Q u10 β k, k 1 Xenitidis P. Leeds Integrability conditions 12 July / 22

11 Derivation of integrability conditions TR k L Rk L = [Φ L R k L,Φ 1 L ], k = 1, 2, 3,. R L = r N S N + r N 1 S N 1 + +r 0 + r 1 S 1 + Φ L =α 0 +α 1 S 1 +α 2 S 2 + α 0 =S 1 Qu10 Q u11 α 1 =S 1 Qu00 Q u11 Qu 01 Q u11 S 1 Qu10 Q u11 α k+1 = 1 k S 1 Qu01 Q u11 α k Φ 1 =β L 0 +β 1 S 1 +β 2 S 2 + β 0 =S 1 Qu11 Q u10 β 1 =S 1 Qu01 Q u10 Qu 00 Q u10 S 1 Qu11 Q u10 β k+1 = 1 k S 1 Qu00 Q u10 β k, k 1 This is the general formulation of integrability conditions for any quadrilateral equation Qu 00, u 10, u 01, u 11 = 0and aformalrecursionoperatorr L of order N. By specifying the order N one can immediately derive the corresponding N-th order integrability conditions. We cannot always find afirstorderformalseriesq L suchthatq N L =R L. Hence, we cannot use only first order integrability conditions. Xenitidis P. Leeds Integrability conditions 12 July / 22

12 Explicit form of the N th order integrability conditions T 1 log r N N 1 = S 1 S k 1 logβ 0, k=0 Tr N 1 α 0 r N 1 S N 1 β 0 = α 0 r N S N β 1 +α 1 S 1 r N S N 1 β 0, Tr N 2 α 0 r N 2 S N 2 β 0 = α 0 r N S N β 2 + α 1 S 1 r N +α 0 r N 1 S N 1 β 1 + α 1 S 1 r N 1 +α 2 S 2 r N S N 2 β 0,.. Tr 1 α 0 r 1 Sβ 0 = α 0 r N S N β N 1 +α 1 S 1 r N S N 1 β N 2 +, T 1r 0 = S 1σ Φ L R L,Φ 1 L. Xenitidis P. Leeds Integrability conditions 12 July / 22

13 First order integrability conditions WhenR L isafirstorderformalseries then where TR k L Rk L = [Φ L R k L,Φ 1 L ], k = 1, 2, 3,. R L = r 1 S+r 0 + r 1 S 1 + r 2 S 2 + r 3 S 3 +, T 1log r 1 = S 1logβ 0, T 1r 0 = S 1S 1 r 1 α 0 Sβ 1, T 1 S 1 r 1 r 1 + r r 1Sr 1 = S 1S 1 σ 2, σ 2 = r 1 α0 r 0 +Sr 0 +α 1 S 1 r 1 Sβ 1 +S+1 r 1 S 1 r 1 α 0 Sβ 2. Xenitidis P. Leeds Integrability conditions 12 July / 22

14 Second order integrability conditions TR k L Rk L = [Φ L R k L,Φ 1 L ], k = 1, 2, 3,. WhenR L isasecondorderformalseries then R L = r 2 S 2 + r 1 S + r 0 + r 1 S 1 + r 2 S 2 +, T 1log r 2 = S 2 1 logβ 0, T r 1 α 0 r 1 Sβ 0 = α 0 r 2 S 2 β 1 +α 1 S 1 r 2 Sβ 0, T 1 r 0 = S 1S 1 σ 1, where σ 1 = α 0 r 1 +α 1 S 1 r 2 Sβ 1 + S α 0 r 2 S 2 β 2. Xenitidis P. Leeds Integrability conditions 12 July / 22

15 Non-local conservation law Theorem Supposethat equation Qu 00, u 10, u 01, u 11 = 0doesnotsatisfythe firstorder integrability conditions and admits a second order formal recursion operator Then R L = r 2 S 2 + r 1 S+r Thereexistsno firstorderformalseriesq = q 1 S+q 0 + q 1 S 1 +, q i F 0,such thatq 2 =R L. 2 The pair of functions ρ 1 = r 1 w, σ 1 = S+1S 1 [ w Qu01 Q u10 S 1 Qu10 Q u11 Q u 00 Q u10 defines anonlocalconservation lawforequation Qu 00, u 10, u 01, u 11 = 0,where w is an additional variable a potential such that Sw = r 2 w, T w = Q u 11 S 1 Qu10 w. Q u10 Q u11 ], Xenitidis P. Leeds Integrability conditions 12 July / 22

16 Integrable equations u 00 + u 11 u 10 u = 0 w 00 w 11 c 1 w 10 w 01 w 10 w 01 + w11 + w 00 c = 0 v 11 v 01 + v v 10 v 00 + v 00 = 0 Properties 1 They are not linearizable by point transformations. 2 They do not admit first integrals. 3 They are integrable in the sense that they satisfy the second order integrability conditions and theyadmitlaxpairs givenintermsof 3 3 matrices. In particular, for the first equation, itisthe resultofaclassification problem, its algebraic entropy vanishes checked by Claude Viallet, itcan be seenas adegenerationofthe second equation, itisrelatedto the thirdequationby amiuratransformation. Xenitidis P. Leeds Integrability conditions 12 July / 22

17 Lax pair, point symmetries & lower order conservation laws u 00 + u 11 u 10 u = 0 Lax pair Ψ 10 = u 00 u 00 u 10 λ u 00 Ψ 00, Ψ 01 = u 00 u 01 λ u λ Ψ 00 Point symmetries u 00 ǫ k =χ kn m u 00, k = 1, 2, χ 2 +χ+1 = 0 Lower order conservation laws ω =ω n m ω u 00 u 10 1 u 10, ς ω =ω n m u 00 u 01 ω u 01, ω 3 = 1. Xenitidis P. Leeds Integrability conditions 12 July / 22

18 Integrability conditions & Canonical conservation laws u 00 + u 11 u 10 u = 0 Usingthe second orderintegrabilityconditions,one can compute the coefficients r i of the formal recursion operator and derive Canonical conservation laws ρ 0,σ 0 = log F 00, logu 10 + u 01,ρ 2,σ 2 = u 00 where F 00 u 10 u 00 u Nonlocal conservation law ρ 1 = 1 w [ u u 10 F 10 F 00 F 00 1 F 10 F 00 + { + S 1 u10 u σ 1 = S+1S 1 00 u 10 u 2 + u u 10 w u 00 u 01 1 F 10 F 00 F 10, 1, F 00 F }], F 00 F 10 F 00 where wsw = u 10u 00 2, T w = u2 10 u 01u 10 +u 01 w. F 10 2 F 00 u 00 Xenitidis P. Leeds Integrability conditions 12 July / 22,

19 Symmetries : Miura transformation & a generalized Bogoyavlensky lattice The firstsymmetryisgivenby The Miura transformation t 1u 00 = u 00 S 1 u 10 u 00 u u 00 u 10 u v 00 = u 10 u 00 u brings these symmetries to the following polynomial form t 1v 00 = v v 00 v20 v 10 v 10 v 20, t 2v 00 = v v 00 B 2 + M 2, t 3v 00 = v v 00 B 3 + M 3 P 3. This hierarchy constitutes a generalization of the Bogoyavlensky lattices x i v 00 = v 00 B i, y i v 00 = v 2 00 Mi, B 1 = M 1 = v 20 v 10 v 10 v 20. These lattices do not commute, so the integrability of their combinations is an exceptional property. Similar observations for the Bogoyavlensky lattices were made also by Adler and Postnikov. Xenitidis P. Leeds Integrability conditions 12 July / 22

20 Miura transformation & an asymmetric equation The Miura transformation v 00 = u 10 u 00 u not only brings the symmetries to an elegant polynomial form which provide a generalization of the Bogoyavlensky lattices, but also maps equation to u 00 + u 11 u 10 u = 0 v 11 v 01 + v v 10 v 00 + v 00 = 0. This equation belongs in the class of equations studied recently by Garifulin& Yamilov but it cannot be found in that paper since they considered only first order integrability conditions. The firstsymmetryinthe other lattice directionisgivenby s 1v 00 = v 00 v T 1 v 01 + v v 00 + v v v G 00 G 0 1, where G 00 v 01 v 0 1 +v 01 + v v Xenitidis P. Leeds Integrability conditions 12 July / 22

21 A degeneration Equation u 00 + u 11 u 10 u = 0 may be seen as a degeneration of the discrete Tzitzeica equation proposed by Adler, namely equation Indeed,ifwe set w 00 w 11 c 1 w 10 w 01 w 10 w 01 + w11 + w 00 c = 0. in the second equation, we derive 1 ǫ u 00 u 11 w pq = ǫ u pq, c =ǫ 3 1 ǫ u 00 u 10 u 01 u 11 The limitǫ 0impliesthe firstequation. 1 u 00 u 10 u = 0. u 00 u 01 u 11 Xenitidis P. Leeds Integrability conditions 12 July / 22

22 Conclusions & Perspectives General formulation of integrability conditions for difference equations. TR k L Rk = [Φ L L R k L,Φ 1 ], k = 1, 2, 3,. L Second order integrability conditions : Integrable equations u 10 u 01 u 00 + u = 0 v 11 v 01 + v v 10 v 00 + v 00 = 0 w 00 w 11 c 1 w 10 w 01 w 10 w 01 + w11 + w 00 c = 0 A generalization of the Bogoyavlensky lattices It would be interesting to t 1v 00 = v v 00 v20 v 10 v 10 v 20 Classify quadrilateral equations admitting symmetries of order higher than -1,1 Classifydifferential-differenceequationsof the form t 1u 0 = Fu 2, u 1, u 0, u 1, u 2 Xenitidis P. Leeds Integrability conditions 12 July / 22