Integrability & Symmetries of difference equations The Adler–Bobenko–Suris case Pavlos Xenitidis DepartmentofMathematics UniversityofPatras Greece October 27, 2008 XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 1/15 Introduction Continuous (cid:224) Discrete Integrable differential equations 3 Ba¨cklund transformation, Lax pair 3 Infinite hierarchies of generalized symmetries 3 Symmetry reductions to Painleve´ equations Adler-Bobenko-Suris (ABS) Difference Equations (cid:224) Multidimensional consistent (cid:217) Ba¨cklund transformation, Lax pair (cid:224) Infinite hierarchies of generalized symmetries (cid:224) Symmetry reductions to discrete & continuous Painleve´ equations XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 2/15 TheAdler-Bobenko-Surisequations General characteristics Q(u , u , u , u , α ,α ) = 0 n,m n+1,m n,m+1 n+1,m+1 1 2 (n,m+1) α1 (n+1,m+1) ¶Autonomous difference equations α2 α2 • α1, α2 : lattice parameters ‚ Affine linear w.r.t. the values of u (n,m) α1 (n+1,m) Anelementaryquadrilateralonthelattice XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 3/15 TheAdler-Bobenko-Surisequations Known equations Ø DISCRETE POTENTIAL KORTEWEG – DE VRIES EQUATION (H1) (u −u )(u −u ) − α + α = 0 n,m n+1,m+1 n+1,m n,m+1 1 2 HirotaR.(1977) NONLINEARPARTIALDIFFERENCEEQUATIONS.I.ADIFFERENCEANALOGUEOFTHEKORTEWEG-DEVRIESEQUATION J.Phys.Soc.Japan43 Ø DISCRETE SCHWARZIAN KORTEWEG – DE VRIES EQUATION (Q10) α1(un,m−un,m+1)(un+1,m−un+1,m+1) −α2(un,m−un+1,m)(un,m+1−un+1,m+1) = 0 QuispelG.,NijhoffF.,CapelH.,vanderLindenJ.(1984) LINEARINTEGRALEQUATIONSANDNONLINEARDIFFERENCE-DIFFERENCEEQUATIONS PhysicaA125 XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 4/15 TheAdler-Bobenko-Surisequations New cases Ø EQUATION H2 (un,m −un+1,m+1)(un+1,m −un,m+1)+(α2−α1)(un,m +un+1,m +un,m+1 +un+1,m+1)−α21+α22=0 Ø EQUATION Q2 α1(un,m −un,m+1)(un+1,m −un+1,m+1)−α2(un,m −un+1,m)(un,m+1 −un+1,m+1)+ α1α2(α1−α2)(un,m +un+1,m +un,m+1 +un+1,m+1)−α1α2(α1−α2)(α21−α1α2+α22)=0 XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 5/15 TheAdler-Bobenko-Surisequations Adler’s Equation: the master equation Ø Equation Q4 a u u u u 0 (0,0) (1,0) (0,1) (1,1) +a (u u u +u u u +u u u +u u u ) 1 (0,0) (1,0) (0,1) (1,0) (0,1) (1,1) (0,1) (1,1) (0,0) (1,1) (0,0) (1,0) +a (u u +u u )+a¯ (u u +u u ) 2 (0,0) (1,1) (1,0) (0,1) 2 (0,0) (1,0) (0,1) (1,1) +a˜ (u u +u u )+a (u +u +u +u )+a =0 2 (0,0) (0,1) (1,0) (1,1) 3 (0,0) (1,0) (0,1) (1,1) 4 AdlerVE(1998) BA¨CKLUNDTRANSFORMATIONFORTHEKRICHEVER–NOVIKOVEQUATION Int.Math.Res.Notices1 XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 6/15 TheAdler-Bobenko-Surisequations Integrability criterion Multidimensional consistency Theequationcanbeextendedtoathreedimensionallatticeinaconsistentway. (cid:231) Classificationofintegrablecases AdlerV.E.,BobenkoA.I.,SurisY.B.(2003) CLASSIFICATIONOFINTEGRABLEEQUATIONSONQUADGRAPHS.THECONSISTENCYAPPROACH Commun.Math.Phys.233 XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 7/15 TheAdler-Bobenko-Surisequations Multidimensional consistency (n,m+1,k+1) (n+1,m+1,k+1) (n,m,k+1) α3 (n+1,m+1,k) (n,m+1,k) α2 (n,m,k) α1 (n+1,m,k) XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 8/15 IntegrabilityaspectsoftheABSequations Auto-Ba¨cklund transformation and Lax pair ‹ CONSISTENCY (cid:221) AUTO–BA¨CKLUND TRANSFORMATION BobenkoA.,SurisYu.(2002) Atkinson(2008) INTEGRABLESYSTEMSONQUADGRAPHS BA¨CKLUNDTRANSFORMATIONSFORINTEGRABLELATTICEEQUATIONS Int.Math.Res.Notices11 J.Phys.A:Math.Theor.41 › CONSISTENCY (cid:221) LAX PAIR NijhoffF.(2002) BobenkoA.,SurisYu.(2002) LAXPAIRFORTHEADLER(LATTICEKRICHEVER-NOVIKOV)SYSTEM INTEGRABLESYSTEMSONQUADGRAPHS Phys.Lett.A297 Int.Math.Res.Notices11 XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 9/15 IntegrabilityaspectsoftheABSequations Auto-Ba¨cklund transformation and Lax pair revised fi AUTO–BA¨CKLUND TRANSFORMATION (cid:221) LAX PAIR (cid:224) Equation Q(u , u , u , u , α ,α ) = 0 n,m n+1,m n,m+1 n+1,m+1 1 2 B1 :=Q(un,m, un+1,m, u˜n,m, u˜n+1,m, α1,λ)=0 (cid:224) Auto-Ba¨cklund 8 < B2 :=Q(un,m, u˜n,m, un,m+1, u˜n,m+1, λ,α2)=0 : (cid:224) Lax pair B1 −B1 B2 −B2 1 ,4 ,34 1 ,4 ,24 Ψ1 = h1 0 B1 −B1 1Ψ, Ψ2 = h2 0 B2 −B2 1Ψ @ ,3 A @ ,2 A XenitidisP. (Un.ofPatras) Integrablelatticeequations October27,2008 10/15