Integrability & Symmetries of difference equations The Adler--Bobenko

16 Pages·2008·0.33 MB·English

by  Pavlos Xenitidis

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
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TheAdler-Bobenko-Surisequations
Integrability criterion
Multidimensional consistency
Theequationcanbeextendedtoathreedimensionallatticeinaconsistentway.
(cid:231) Classificationofintegrablecases
AdlerV.E.,BobenkoA.I.,SurisY.B.(2003)
CLASSIFICATIONOFINTEGRABLEEQUATIONSONQUADGRAPHS.THECONSISTENCYAPPROACH
Commun.Math.Phys.233
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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)
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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
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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
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