From discrete differential geometry to the classification of discrete integrable systems

Transcription

1 From discrete differential geometry to the classification of discrete integrable systems Vsevolod Adler,, Yuri Suris Technische Universität Berlin Quantum Integrable Discrete Systems, Newton Institute, Cambridge, March 2009 DFG Research Unit Polyhedral Surfaces

2 Discrete Differential Geometry I Aim: Development of discrete equivalents of the geometric notions and methods of differential geometry. The latter appears then as a limit of refinements of the discretization. I Question: Which discretization is the best one? I (Theory): preserves fundamental properties of the smooth theory I (Applications): represent smooth shape by a discrete shape with just few elements; best approximation

3 Surfaces and transformations Classical theory of (special classes of) surfaces (constant curvature, isothermic, etc.) General and special Quad-surfaces special transformations (Bianchi, Bäcklund, Darboux) discrete! symmetric

4 Basic idea Do not distinguish discrete surfaces and their transformations. Discrete master theory. Example - planar quadrilaterals as discrete conjugate systems. Multidimensional Q-nets [Doliwa-Santini 97].

5 Basic idea Do not distinguish discrete surfaces and their transformations. Discrete master theory. Example - planar quadrilaterals as discrete conjugate systems. Multidimensional Q-nets [Doliwa-Santini 97].

6 Discretization Principles I Transformation Group Principle. Smooth geometric objects and their discretizations belong to the same geometry, i.e. are invariant with respect to the same transformation group (discrete Klein s Erlangen Program) I Consistency Principle. Discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets (Integrability) Multidimensional Q-nets (projective geometry) can be restricted to an arbitrary quadric () Discretization of classical geometries) [Doliwa 99]

7 Differential geometry Smooth limit: I Differential geometry follows from incidence theorems of projective geometry

8 Integrability as Consistency I Equation I Consistency c d a b Q(a; b; c; d) = 0

9 Integrability as Consistency I Equation I Consistency c d a b Q(a; b; c; d) = 0

10 Integrability as Consistency I Equation I Consistency c d a b Q(a; b; c; d) = 0

11 Integrability as Consistency I Equation I Consistency c d a b Q(a; b; c; d) = 0

12 Why integrability? Can be derived from consistency: I Lax representation I Bäcklund-Darboux transformations Bobenko-Suris [ 02], Nijhoff [ 02]

13 Book in DDG A.I. Bobenko, Y.B. Suris, Discrete differential geometry. Integrable structure. AMS, Graduate Studies in Mathematics, v.98, 2008, xxiv+404 pp.

14 New progress Adler, Bobenko, Suris I Integrable discrete nets in Grassmannians. arxiv: (Q-nets: points! vector subspaces) I Discrete nonlinear hyperbolic equations. Classification of integrable cases. Func. Analysis and Appl., v.43, 2009 (Classification 2D) I Classification of discrete integrable octahedron equations. in progress (Classification 3D)

15 Grassmann generalization of Q-nets The Grassmann manifold G d+1 r+1 is the variety of (r + 1)-dimensional linear subspaces of some (d + 1)-dimensional linear space. Definition. A mapping Z N! G d+1 r+1 ; N 2; d > 3r + 2; is called the N-dimensional Grassmann Q-net of rank r, if any elementary cell maps to four r-dimensional subspaces in P d which lie in a (3r + 2)-dimensional one. The images of any three vertices of a square cell are generic subspaces and their span contains the image of the last vertex.

16 3D system, 4D-consistency Theorem. Let seven r-dimensional subspaces X ; X i ; X ij, 1 i 6= j 3 be given in P d, d 4r + 3, such that dim(x + X i + X j + X ij ) = 3r + 2 for each pair of indices, but with no other degeneracies. Then the conditions dim(x i + X ij + X ik + X 123 ) = 3r + 2 define an unique r-dimensional subspace X 123. Theorem. The 3-dimensional Grassmann Q-nets are 4D-consistent. ) ND-consistent Proof: dim(a + B) = dim A + dim B dim(a \ B):

17 Analytic description The Grassmann manifold can be defined as G d+1 r+1 = (V d+1 ) r+1 =GL r+1 where GL r+1 acts as the base changes in any (r + 1)-dimensional subspace of V d+1. Such subspaces are identified with (r + 1) (d + 1) matrices which are equivalent modulo left multiplication by matrices from GL r+1. Affine normalization by choosing the representatives x = 0 x 1;1 : : : x 1;d r 1 1 : : : 0 C A : x r+1;1 : : : x r+1;d r 0 : : : 1

18 Noncommutative discrete Darboux system Linear problem with matrix coefficients: Bogdanov, Konopelchenko [ 95], Doliwa [ 08] x ij = x + a ij (x i x) + a ji (x j x) Convenient description: discrete Lamé coefficients h i and discrete rotation coefficients b ij : a ij = h i j (hi ) 1 ; x i x = h i y i ; b ij = (h i j ) 1 (h j i h j ) bring the system to the form y i j = y i b ij y j : (1) Theorem. The compatibility conditions of equations (1) are equivalent to the birational mapping for the discrete rotation coefficients b ij k = (bij + b ik b kj )(I b jk b kj ) 1 ; b ij 2 Mat(r + 1; r + 1) which is multi-dimensionally consistent.

19 Classification 2D I hyperbolic nonlinear equation Q(a; b; c; d) = 0 I Q multi-affine (can be resolved with respect to any variable) I Problem. Classify integrable (i.e. consistent) equations.

20 Classification 2D In our previous work [ABS 03] the classification was obtained under additional assumptions: I equations on the opposite faces coincide I all equations possess the square symmetry I tetrahedron property (x 123 does not depend on x) Now we deduce these properties.

21 Classification. Method s overview n 6 - Q m;n(x m;n; x m+1;n; x m;n+1; x m+1;n+1) = 0 m

22 Classification. Method s overview n 6 - Q m;n(x m;n; x m+1;n; x m;n+1; x m+1;n+1) = 0 m integrability = - 3D-consistency

23 Classification. Method s overview n 6 - Q m;n(x m;n; x m+1;n; x m;n+1; x m+1;n+1) = 0 m integrability = - 3D-consistency analysis of singular solutions? list of integrable equations

24 Classification. Method s overview n 6 - Q m;n(x m;n; x m+1;n; x m;n+1; x m+1;n+1) = 0 m assumptions: multi-affine Q integrability = - 3D-consistency P PPPP P P Pq? analysis of singular solutions list of integrable equations

25 Classification. Method s overview n 6 - Q m;n(x m;n; x m+1;n; x m;n+1; x m+1;n+1) = 0 m assumptions: integrability = - 3D-consistency P PPPP multi-affine Q + some nondegeneracy condition P P Pq? analysis of singular solutions list of integrable equations

26 3D-consistency D-consistency: the values of x 123 computed in 3 possible ways coincide identically on the initial values x; x 1 ; x 2 ; x

27 3D-consistency D-consistency: the values of x 123 computed in 3 possible ways coincide identically on the initial values x; x 1 ; x 2 ; x

28 3D-consistency D-consistency: the values of x 123 computed in 3 possible ways coincide identically on the initial values x; x 1 ; x 2 ; x

29 Singular solutions We consider only multi-affine equations (= of the first degree on each unknown): Q(x; y ; z; t) = a 1 xyzt + + a 16 = 0: (2) The important role play biquadratic curves: h(x; y ) = Q z Q t QQ zt = h 1 x 2 y h 9 = 0: We associate such curve to each edge of the square cell t 6 Q - z x? y

30 Biquadratics The key observation is given by the following theorem. Theorem. Let the equations be 3Dconsistent and all involved biquadratics be not degenerate. Then, for each edge of the cube, the equations corresponding to adjacent faces give rise to one and the same biquadratic curve. The nondegeneracy assumption means that a biquadratic polynomial h(x; y ) must be free of the factors of the form x const and y const =) two types of equations.

31 Singular solutions The idea of the proof. Choose the singular initial data on the face (1; 2). This leads to an undetermined value of x 123. However, due to consistency, x 123 can be found without using this face. Therefore, the initial data on the faces (1; 3) and (2; 3) must be singular as well. Therefore, the singular curves on these faces have the same projections on the common edges.

32 Möbius transformations The classification is made modulo (PSL 2 (C)) 8, that is the variables in all vertices of the cube are subjected to independent Möbius transformations. It is important that the following commutative diagram is compatible with the action of this group.

33 Möbius transformations r 4 (x 4 ) 3 h 34 (x 3 ; x 4 ) 4! r 3 (x 3 ) 1 x??? x??? 12 x??? 2 h 14 (x 1 ; x 4 ) 23 Q(x 1 ; x 2 ; x 3 ; x 4 ) 14! h 23 (x 2 ; x 3 ) 4???y???y 34???y 3 r 1 (x 1 ) 2 h 12 (x 1 ; x 2 ) 1! r 2 (x 2 ) where ij (Q) = Q xi Q xj QQ xi ;x j ; i (h) = h 2 x i 2hh xi ;x i :

34 List of 2D integrable equations Theorem. Up to Möbius transformations, any 3D-consistent system with nondegenerate biquadratics is one of the following list ( = (i), = (j), sn() = sn(; k )): (x x j )(x i x ij ) (x x i )(x j x ij ) = ( ) (Q 1 ) (x x j )(x i x ij ) (x x i )(x j x ij ) +( )(x + x i + x j + x ij ) = ( )( ) (Q 2 )

35 List of 2D integrable equations Theorem. Up to Möbius transformations, any 3D-consistent system with nondegenerate biquadratics is one of the following list ( = (i), = (j), sn() = sn(; k )): 1 (xx i + x j x ij ) (xx ij + x i x j ) 1 (xx j + x i x ij ) = 0; (Q 3 ) sn() sn() sn( )(k 2 xx i x j x ij + 1) + sn()(xx i + x j x ij ) sn()(xx j + x i x ij ) sn( )(xx ij + x i x j ) = 0: (Q 4 )

36 List of 2D integrable equations Q1 [Quispel-Nijhoff-Capel-Van der Linden 84] Q2 [Adler-Bobenko-Suris 03] Q3 =0 [Quispel-Nijhoff-Capel-Van der Linden 84] Q3 6=0 [Adler-Bobenko-Suris 03] Q4 [Adler 98]

37 List of 2D integrable equations Equations with degenerated biquadratics: (x x ij )(x i x j ) = (H 1 ) (x x ij )(x i x j ) + ( )(x + x i + x j + x ij ) = 2 2 (H 2 ) (xx i + x j x ij ) (xx j + x i x ij ) = ( 2 2 ) (H 3 )

38 4D consistency of 3D cube equations Hirota-Miwa (or dbkp) equations. The following set of four equations x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0; x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0; x 1 x 34 x 2 x 14 + x 4 x 13 xx 134 = 0; x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 is 4D-consistent, that is the value x 1234 as the function on initial data x, x i, x ij does not depend on the order of computation. These equations imply a similar equation on odd/even sublattices in Z 4 : x 14 x 23 x 13 x 24 + Alexander x 12 x 34 Bobenko xx 1234 DDG = and 0: Classification of discrete integrable equations

39 4D consistency of 3D cube equations Hirota-Miwa (or dbkp) equations. The following set of four equations x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0; x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0; x 1 x 34 x 2 x 14 + x 4 x 13 xx 134 = 0; x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 is 4D-consistent, that is the value x 1234 as the function on initial data x, x i, x ij does not depend on the order of computation. These equations imply a similar equation on odd/even sublattices in Z 4 : x 14 x 23 x 13 x 24 + Alexander x 12 x 34 Bobenko xx 1234 DDG = and 0: Classification of discrete integrable equations

40 4D consistency of 3D cube equations Hirota-Miwa (or dbkp) equations. The following set of four equations x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0; x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0; x 1 x 34 x 2 x 14 + x 4 x 13 xx 134 = 0; x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 is 4D-consistent, that is the value x 1234 as the function on initial data x, x i, x ij does not depend on the order of computation. These equations imply a similar equation on odd/even sublattices in Z 4 : x 14 x 23 x 13 x 24 + Alexander x 12 x 34 Bobenko xx 1234 DDG = and 0: Classification of discrete integrable equations

41 4D consistency of 3D cube equations Hirota-Miwa (or dbkp) equations. The following set of four equations x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0; x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0; x 1 x 34 x 2 x 14 + x 4 x 13 xx 134 = 0; x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 is 4D-consistent, that is the value x 1234 as the function on initial data x, x i, x ij does not depend on the order of computation. These equations imply a similar equation on odd/even sublattices in Z 4 : x 14 x 23 x 13 x 24 + Alexander x 12 x 34 Bobenko xx 1234 DDG = and 0: Classification of discrete integrable equations

42 Hirota-Miwa and double cross-ratio equations The double cross-ratio equation (x x ij )(x jk x ki ) (x ij x jk )(x ki x) = (x ijk x k )(x i x j ) (x k x i )(x j x ijk ) : is also 4D consistent. The value x 1234 does not depend on the order of computation. Double cross-ratio and Hirota-Miwa equations are related via certain difference substitution and there exist several other modifications of these equations. The classification of 4D consistent 3D cube equations seems to be a difficult problem, and we address to a bit more simple class of octahedron equations.

43 Hirota (dkp) equation dkp equation can be obtained from dbkp x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 through a limiting process. Notice that the scaling x(m; n; k )! a mn b nk c mk x(m; n; k ) brings dbkp to the form bx 1 x 23 cx 2 x 13 + ax 3 x 12 abc xx 123 = 0; so that the last term is dropped under the limit a = b = c! 0. But this changes the combinatorics of equation. How to define the consistency?

44 Hirota (dkp) equation dkp equation can be obtained from dbkp x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 through a limiting process. Notice that the scaling x(m; n; k )! a mn b nk c mk x(m; n; k ) brings dbkp to the form bx 1 x 23 cx 2 x 13 + ax 3 x 12 abc xx 123 = 0; so that the last term is dropped under the limit a = b = c! 0. But this changes the combinatorics of equation. How to define the consistency?

45 Independent equations A hint comes from the consistent quadruple of dbkp equations. One obtains, by the same limiting process for all equations: x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 * x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0 x 1 x 34 x 3 x 14 + x 4 x 13 xx 134 = 0 x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 x 14 x 23 x 13 x 24 + x 12 x 34 = 0 + However, this set of four equations is not independent: one equation follows from the other three. Moreover, the equation on odd/even sublattice also becomes a consequence of these three.

46 Independent equations A hint comes from the consistent quadruple of dbkp equations. One obtains, by the same limiting process for all equations: x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 * x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0 x 1 x 34 x 3 x 14 + x 4 x 13 xx 134 = 0 x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 x 14 x 23 x 13 x 24 + x 12 x 34 = 0 + However, this set of four equations is not independent: one equation follows from the other three. Moreover, the equation on odd/even sublattice also becomes a consequence of these three.

47 Independent equations A hint comes from the consistent quadruple of dbkp equations. One obtains, by the same limiting process for all equations: x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 * x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0 x 1 x 34 x 3 x 14 + x 4 x 13 xx 134 = 0 x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 x 14 x 23 x 13 x 24 + x 12 x 34 = 0 + However, this set of four equations is not independent: one equation follows from the other three. Moreover, the equation on odd/even sublattice also becomes a consequence of these three.

48 Independent equations A hint comes from the consistent quadruple of dbkp equations. One obtains, by the same limiting process for all equations: x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 * x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0 x 1 x 34 x 3 x 14 + x 4 x 13 xx 134 = 0 x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 x 14 x 23 x 13 x 24 + x 12 x 34 = 0 + However, this set of four equations is not independent: one equation follows from the other three. Moreover, the equation on odd/even sublattice also becomes a consequence of these three.

49 Independent equations A hint comes from the consistent quadruple of dbkp equations. One obtains, by the same limiting process for all equations: x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 * x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0 x 1 x 34 x 3 x 14 + x 4 x 13 xx 134 = 0 x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 x 14 x 23 x 13 x 24 + x 12 x 34 = 0 + However, this set of four equations is not independent: one equation follows from the other three. Moreover, the equation on odd/even sublattice also becomes a consequence of these three.

50 Independent equations A hint comes from the consistent quadruple of dbkp equations. One obtains, by the same limiting process for all equations: x 1 x 23 x 2 x 13 + x 3 x 12 xx 123 = 0 * x 1 x 24 x 2 x 14 + x 4 x 12 xx 124 = 0 x 1 x 34 x 3 x 14 + x 4 x 13 xx 134 = 0 x 2 x 34 x 3 x 24 + x 4 x 23 xx 234 = 0 x 14 x 23 x 13 x 24 + x 12 x 34 = 0 + However, this set of four equations is not independent: one equation follows from the other three. Moreover, the equation on odd/even sublattice also becomes a consequence of these three.

51 Consistency of octahedron equations It is natural to introduce the notion of consistency in terms of three equations which remain independent. This is a logical step back to the 3D-consistency situation. Definition. Equations x 12 = f (x 1 ; x 2 ; x 4 ; x 14 ; x 24 ); x 13 = g(x 1 ; x 3 ; x 4 ; x 14 ; x 34 ); x 23 = h(x 2 ; x 3 ; x 4 ; x 24 ; x 34 ) are called 4D-consistent if the equalities x 123 = f (: : :) = g(: : :) = h(: : :) hold identically on the initial data x 1 ; x 2 ; x 3 ; x 4 ; x 14 ; x 24 ; x 34 ; x 44 ; x 144 ; x 244 ; x 344 : The role of 4-th coordinate is distinguished, but we will see soon that the symmetry can be restored.

52 Consistency of octahedron equations It is natural to introduce the notion of consistency in terms of three equations which remain independent. This is a logical step back to the 3D-consistency situation. Definition. Equations x 12 = f (x 1 ; x 2 ; x 4 ; x 14 ; x 24 ); x 13 = g(x 1 ; x 3 ; x 4 ; x 14 ; x 34 ); x 23 = h(x 2 ; x 3 ; x 4 ; x 24 ; x 34 ) are called 4D-consistent if the equalities x 123 = f (: : :) = g(: : :) = h(: : :) hold identically on the initial data x 1 ; x 2 ; x 3 ; x 4 ; x 14 ; x 24 ; x 34 ; x 44 ; x 144 ; x 244 ; x 344 : The role of 4-th coordinate is distinguished, but we will see soon that the symmetry can be restored.

53 Consistency of octahedron equations It is natural to introduce the notion of consistency in terms of three equations which remain independent. This is a logical step back to the 3D-consistency situation. Definition. Equations x 12 = f (x 1 ; x 2 ; x 4 ; x 14 ; x 24 ); x 13 = g(x 1 ; x 3 ; x 4 ; x 14 ; x 34 ); x 23 = h(x 2 ; x 3 ; x 4 ; x 24 ; x 34 ) are called 4D-consistent if the equalities x 123 = f (: : :) = g(: : :) = h(: : :) hold identically on the initial data x 1 ; x 2 ; x 3 ; x 4 ; x 14 ; x 24 ; x 34 ; x 44 ; x 144 ; x 244 ; x 344 : The role of 4-th coordinate is distinguished, but we will see soon that the symmetry can be restored.

54 From triple to quintuple Theorem 1. If the triple is consistent then some equations are fulfilled automatically. k (x 1 ; x 2 ; x 3 ; x 12 ; x 13 ; x 23 ) = 0; l(x 12 ; x 13 ; x 14 ; x 23 ; x 24 ; x 34 ) = 0 The 4-th direction is actually on equal footing with the other ones and moreover, the odd/even sublattices carry an equation of Hirota type as well. The picture becomes completely symmetric if we consider the embedding Z 4! Z 5 accordingly to the rule x i! x i5. )consistent quintuples

55 Desargues configuration H(a; b; c; d; e; f ) = Equations (a b)(c d)(e f ) (b c)(d e)(f a) : H(x ij ; x ik ; x kj ; x kl ; x jl ; x il ) = 1; are consistent. i; j; k ; m 2 f1; 2; 3; 4; 5g Geometrically, each equation expresses Menelaus theorem and the whole consistent quintuple is contained in Desargues configuration. King-Schief [ 03]

56 Desargues configuration H(a; b; c; d; e; f ) = Equations (a b)(c d)(e f ) (b c)(d e)(f a) : H(x ij ; x ik ; x kj ; x kl ; x jl ; x il ) = 1; are consistent. i; j; k ; m 2 f1; 2; 3; 4; 5g Geometrically, each equation expresses Menelaus theorem and the whole consistent quintuple is contained in Desargues configuration. King-Schief [ 03]

57 Desargues configuration H(a; b; c; d; e; f ) = Equations (a b)(c d)(e f ) (b c)(d e)(f a) : H(x ij ; x ik ; x kj ; x kl ; x jl ; x il ) = 1; are consistent. i; j; k ; m 2 f1; 2; 3; 4; 5g Geometrically, each equation expresses Menelaus theorem and the whole consistent quintuple is contained in Desargues configuration. King-Schief [ 03]

58 Desargues configuration H(a; b; c; d; e; f ) = Equations (a b)(c d)(e f ) (b c)(d e)(f a) : H(x ij ; x ik ; x kj ; x kl ; x jl ; x il ) = 1; are consistent. i; j; k ; m 2 f1; 2; 3; 4; 5g Geometrically, each equation expresses Menelaus theorem and the whole consistent quintuple is contained in Desargues configuration. King-Schief [ 03]

59 Desargues configuration H(a; b; c; d; e; f ) = Equations (a b)(c d)(e f ) (b c)(d e)(f a) : H(x ij ; x ik ; x kj ; x kl ; x jl ; x il ) = 1; are consistent. i; j; k ; m 2 f1; 2; 3; 4; 5g Geometrically, each equation expresses Menelaus theorem and the whole consistent quintuple is contained in Desargues configuration. King-Schief [ 03]

60 Desargues configuration H(a; b; c; d; e; f ) = Equations (a b)(c d)(e f ) (b c)(d e)(f a) : H(x ij ; x ik ; x kj ; x kl ; x jl ; x il ) = 1; are consistent. i; j; k ; m 2 f1; 2; 3; 4; 5g Geometrically, each equation expresses Menelaus theorem and the whole consistent quintuple is contained in Desargues configuration. King-Schief [ 03]

61 Classification theorem Any 4D-consistent irreducible nonlinear autonomous octahedron equation is equivalent, up to nonautonomous point transformations, to one of the following: x 12 x 3 + x 13 x 2 + x 23 x 1 = 0 ( 1 ) (x 13 x 12 )x 1 + (x 12 x 23 )x 2 + (x 23 x 13 )x 3 = 0 ( 2 ) x 13 x 12 x 1 + x 12 x 23 x 2 + x 23 x 13 x 3 = 0 ( 0 2 ) (x 12 x 13 )(x 23 x 3 )(x 2 x 1 ) (x 13 x 23 )(x 3 x 2 )(x 1 x 12 ) = 1 ( 3) x 13 x = x 12 x 3 x 2 x 1 ( 4 )

62 Classification theorem Possible consistent equations, up to point transformations and permutations of indices are (i; j; k 2 f1; 2; 3; 4g): x ij x kl x ik x jl + x jk x il = 0; 1 i < j < k < l 4 ( 1 ) (x ik x ij )x i + (x ij x jk )x j + (x jk x ik )x k = 0 ( 2 ) x ik x ij x i + x ij x jk x j + x jk x ik x k = 0 ( 0 2 ) H(x ij ; x ik ; x kj ; x km ; x jm ; x im ) = 1 ( 3 ) x i4 x j4 x 4 = x ij 1 x j 1 x i ; i; j = 1; 2; 3; x 13 x 12 x 1 + x 12 x 23 x 2 + x 23 x 13 x 3 = 0: ( 4 )

63 Comments I We assume that in some domain equations can be solved with respect to each variable and are irreducible. This means that it is not of the form ab = 0 where a and b depend on incomplete sets of variables. I We do not assume any specific form (for ex. rational) of the equations. I In contrast to 2D case, 3D equations do not contain essential parameters. I All equations from the list can be derived from the auxiliary linear problems like 2 = u( 1 ); 3 = v ( 1 ) and are related to each other via difference substitutions. I Equations of Hirota type like H(x 12 ; x 13 ; x 23 ; x 34 ; x 24 ; x 14 ) = 1 hold on even/odd sublattices (fifth equation of the quintuple)