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Outline Introduction Continuous case Semidiscrete case Purely discrete case

On Darboux integrability of discrete 2D-Toda lattices
Sergey V. Smirnov
Moscow State University

Geometry, Topology and Integrability SkolTech, Skolkovo October 21, 2014

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Introduction Continuous case Semidiscrete case Purely discrete case

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Two-dimensional Toda lattice
qxy (j ) = exp(q (j + 1) - q (j )) - exp(q (j ) - q (j - 1)), or (ln h(j ))xy = h(j + 1) - 2h(j ) + h(j - 1), where h(j ) = exp(q (j + 1) - q (j )), or uxy (j ) = exp(u (j + 1) - 2u (j ) + u (j - 1)) where h(j ) = uxy (j ). This system is two-dimensional analog of one-dimensional Toda chain.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

How to obtain a finite system?
Trivial boundary conditions: u (-1) = u (r ) = -. Periodic boundary conditions: u (j + r ) = u (j ).

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

How to obtain a finite system?
Trivial boundary conditions: u (-1) = u (r ) = -. Periodic boundary conditions: u (j + r ) = u (j ). These Toda lattices are particular cases of the so-called exponential system:
r

uxy (i ) = exp
j =1

aij u (j ) ,

i = 1, 2, . . . , r ,

where aij = const.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

How to obtain a finite system?
Trivial boundary conditions: u (-1) = u (r ) = -. Periodic boundary conditions: u (j + r ) = u (j ). These Toda lattices are particular cases of the so-called exponential system:
r

uxy (i ) = exp
j =1

aij u (j ) ,

i = 1, 2, . . . , r ,

where aij = const. A.Shabat, R.Yamilov (1981): What exponential systems are integrable?
Sergey V. Smirnov On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrable boundary conditions
Trivial boundary conditions correspond -2 1 0 1 -2 1 1 -2 M = (aij ) = 0 . . . 0 0 0 to the matrix ... 0 ... 0 ... 0 . . .. . . . . . . -2

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrable boundary conditions
Trivial boundary conditions correspond -2 1 0 1 -2 1 1 -2 M = (aij ) = 0 . . . 0 0 0 to the matrix ... 0 ... 0 ... 0 . . .. . . . . . . -2

Notice that -M is the Cartan matrix of an A-series Lie algebra. There are other simple Lie algebras! 1980: Many papers by various authors on finite 2D-Toda lattices.
Sergey V. Smirnov On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Various approaches to integrability

A.Mikhailov, M.Olshanetsky, Perelomov (1979-1981): Lax representation depending on spectral parameter A.Leznov (1980): explicit integration A.Shabat, R.Yamilov (1981): x-integrals, y-integrals and characteristic algebras

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Various approaches to integrability

A.Mikhailov, M.Olshanetsky, Perelomov (1979-1981): Lax representation depending on spectral parameter A.Leznov (1980): explicit integration A.Shabat, R.Yamilov (1981): x-integrals, y-integrals and characteristic algebras Boundary conditions corresponding to simple Lie algebras are integrable whatever definition of integrability is being used.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Laplace invariants and Toda lattice
Definition
Functions h = by - ab - c and k = ax - ab - c are called the Laplace invariants of the hyperbolic differential operator L = x y + ax + b y + c .

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Laplace invariants and Toda lattice
Definition
Functions h = by - ab - c and k = ax - ab - c are called the Laplace invariants of the hyperbolic differential operator L = x y + ax + b y + c . Hyperbolic operator can be factorized if one the Laplace invariants is zero: L = (x + b )(y + a) + k = (y + a)(x + b ) + h.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Consider a sequence of hyperbolic operators Lj = x y + a(j )x + b (j )y + c (j ) such that any two neighboring operators are related by a Darboux-Laplace transformation: Lj
+1

Dj = D

j +1

Lj ,

where Dj = x + b (j ). Then the Laplace invariants of these operators satisfy the two-dimensional Toda lattice: k (j + 1) = h(j ), h(j + 1) = 2h(j ) - k (j ) + (ln h(j ))xy .

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Involutions and boundary conditions
In terms of the variables h(j ) all A - C -series Toda lattices are obtained by the trivial boundary condition h(r ) = 0 on the right edge and by the following boundary conditions on the left edge: h(-1) = 0 for A-series (trivial); h(-j ) = h(j - 1) for B -series (involution); h(-j ) = h(j ) for C -series (involution). Boundary condition for the D -series Toda lattice can not be expressed in terms of the variables h(j ).

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Reductions of the A-series lattice

B -series lattice is a reduction of the A series lattice of the length 2r defined by the boundary condition h(-j ) = h(j - 1); C -series lattice is a reduction of the A series lattice of the length 2r + 1 defined by the boundary condition h(-j ) = h(j ); D -series lattice is also a reduction of the A series lattice (Habibullin, 2005).

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrals along characteristics
Definition
Function I = I (ux , uxx , u
xxx xy

, . . . ) is called y -integral of the system = F (u, ux , uy )

u

if Dy (I ) = 0 on solutions of the system. Integrals in direction x are defined similarly.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrals along characteristics
Definition
Function I = I (ux , uxx , u
xxx xy

, . . . ) is called y -integral of the system = F (u, ux , uy )

u

if Dy (I ) = 0 on solutions of the system. Integrals in direction x are defined similarly.

Example
2 2 Functions I = uxx - 1 ux and J = uyy - 1 uy are y - and x -integrals 2 2 u. resp. of the Liouville equation uxy = e

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Darboux integrability
Definition
[V.Adler, V.Sokolov] Integrals I1 , I2 , . . . , Ik of orders d1 , d2 , . . . , dk resp. are called essentially independent, if rk Hyperbolic system u
xy

Ii
d (x i uj )

= k.

= F (u, ux , uy )

is called Darboux integrable if it admits complete families of essentially independent x - and y -integrals.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Generating function for y -integrals

Theorem
Coefficients of the differential operator B = (x - qx (r ))(x - qx (r - 1)) . . . (x - qx (0)) are y -integrals of the A-series Toda lattice defined by boundary conditions q (-1) = +, q (r ) = -. These integrals are essentially independent.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Pro of. q (-1) = +, q (r ) = - = h(-1) = k (0) = 0, h(r ) = 0 Therefore, L0 = (x + b (0))y , Lr = y (x + b (r )). In the Toda lattice b (j ) = -qx (j ). Hence, B = Dr Dr -1 . . . D0 . Now use DLTs: B y = Dr D
r -1

. . . D0 y = Dr . . . D1 L0 =
r -1

Dr . . . L1 D0 = · · · = Lr D

D

r -2

. . . D0 = y B .

[y , B ] = 0 = coeffs. of B do not depend on y .

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrability of finite lattices
A-series Toda lattice equations are symmetric about the change of variables x y . Therefore, the similar construction allows to obtain x -integrals J 1 , J2 , . . . J r as well. These integrals are independent.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrability of finite lattices
A-series Toda lattice equations are symmetric about the change of variables x y . Therefore, the similar construction allows to obtain x -integrals J 1 , J2 , . . . J r as well. These integrals are independent. Explicit formulas that define reductions of an A-series lattice to B or C -series lattice allows to prove the following

Theorem
B - and C -series Toda lattices are Darboux integrable.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Laplace invariants

Definition
c c Functions kn = an - (ln an )x - bn and hn = an - bn are called the n n Laplace invariants of hyperbolic differential-difference operator

L = x T + ax + bT + c , where T is the shift operator: T n =
n+1

.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

DLT and Toda lattice
Consider the sequence of hyperbolic operators Lj = x T + an (j )x + bn (j )T + cn (j ) such that any two neighboring operators are related by Darboux-Laplace transformation: Lj where Dj = x + bn (j ).
+1

Dj = D

j +1

Lj ,

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Then the Laplace invariants of these operators satisfy the following equations: kn (j + 1) = hn (j ) ln
hn (j ) hn+1 (j ) x

= hn

+1

(j + 1) - hn

+1

(j ) - hn (j ) + hn (j - 1)

.

Introduce another set of variables by hn (j ) = exp(qn
+1

(j + 1) - qn (j ));

then the latter equation can be rewritten as follows: qn,x (j )-qn
+1,x

(j ) = exp(qn

+1

(j +1)-qn (j ))-exp(qn

+1

(j )-qn (j -1)).

These equations are two forms of the semidiscrete Toda lattice.
Sergey V. Smirnov On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Finite semidiscrete Toda lattices

How to obtain a finite system? Trivial boundary conditions: hn (-1) = hn (r ) = 0. Periodic boundary conditions. How to introduce semidiscrete analogs of Toda lattices, corresponding to simple Lie algebras?

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Semidiscrete exponential systems
In terms of the variable un (j ), where hn (j ) = un,x (j ) - un
+1,x

(j ),

semidiscrete Toda lattice can be rewritten as a particular case of semidiscrete analog of exponential system (Habibullin, Zheltukhin, Yangubaeva, 2011):
r

un,x (i ) - un

+1,x

(i ) = exp
j =1

+ (aij un

+1

(j ) + ai- un (j )) , j

+ - where i = 1, 2, . . . , r . Here M + = (aij ) and M - = (aij ) are upperand lower-triangular constant matrices with -1 on the diagonal.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Semidiscrete exponential systems
In terms of the variable un (j ), where hn (j ) = un,x (j ) - un
+1,x

(j ),

semidiscrete Toda lattice can be rewritten as a particular case of semidiscrete analog of exponential system (Habibullin, Zheltukhin, Yangubaeva, 2011):
r

un,x (i ) - un

+1,x

(i ) = exp
j =1

+ (aij un

+1

(j ) + ai- un (j )) , j

+ - where i = 1, 2, . . . , r . Here M + = (aij ) and M - = (aij ) are upperand lower-triangular constant matrices with -1 on the diagonal. For semidiscrete analogs of Toda lattices corresponding to simple Lie algebras M = M - + M + is the Cartan matrix.
Sergey V. Smirnov On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Involutions in the semidiscrete case
In the continuous case the Toda lattice admits two types of involutions: reflection about an integer point (e.g. h(-j ) = h(j ) that corresponds to C -series) and reflection about a half-integer point (e.g. h(-j ) = h(j + 1) that corresponds to B -series). In the semidiscrete case there are no reflections about half-integer points! Involution hn (-j ) hn+j -c (j - d ) defines a reduction of the Toda lattice only if c = -2d . The choice c = -1 leads to the boundary condition hn (-2) = hn+1 (0) corresponding to C -series. Rewrite it as follows: qn (-1) - qn
-1

(-2) = qn

+1

(1) - qn (0).

C -series lattice is a reduction of an A series lattice of the length 2r .
Sergey V. Smirnov On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Darboux integrability of a hyperbolic system
All definitions are similar to the continuous case: a function I = I (ux , uxx , . . . ) is called an n-integral, if its total difference derivative with respect to n vanishes on solutions. Integrals I1 , I2 , . . . Ik are called essentially independent, if rk Ii
d (n i uj )

= k,

where di is the order of Ii . System is called Darboux integrable if it admits complete families of essentially independent n- and x -integrals.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Generating functions for integrals
Theorem
[S., to appear in TMPh, 2015] Coefficients of the differential operator B = (x - qn,x (r ))(x - qn,x (r - 1)) . . . (x - qn,x (0)) form a complete family of n-integrals for the A-series semidiscrete Toda lattice defined by boundary conditions q (-1) = +, q (r ) = -. Coefficients of the difference operator A = (T + exp(qn (0) - qn
+1

(0))) . . . (T + exp(qn (r ) - qn

+1

(r )))

form a complete family of x -integrals for this system. Generating function for x -integrals is obtained by the use of DLTs of the other type, i.e. in the discrete variable n.
Sergey V. Smirnov On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrability of finite lattices

Theorem
[S., to appear in TMPh, 2015] C -series semidiscerete Toda lattices are Darboux integrable. This follows from the fact that sufficient number of integrals remain independent after the reduction from A-series to C -series.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Integrability of finite lattices

Theorem
[S., to appear in TMPh, 2015] C -series semidiscerete Toda lattices are Darboux integrable. This follows from the fact that sufficient number of integrals remain independent after the reduction from A-series to C -series. Darboux integrability of B -series lattices can not be proved similarly because in the semidiscrete case it's not known that B -series lattices can be obtained from A-series lattices by reduction.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Explicit formula for x -integrals
x -integrals for the A-series lattice of the length r are given by the following explicit formula: Jk =
0 i1 k

exp(cn (0)+· · ·+cn (i1 -1)+c
r n+2 (i2 n+1 (i2

n+1 (i1

+1)+· · · +

+c

- 1) + c

+ 1) + · · · + c +c
n +k

n+k -1 (ir

- 1)+
n +k

(ik + 1) + · · · + c

(r )),

where cn (j ) = qn (j ) - qn

+1

(j ).

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

DLT and Toda lattice
Sequence of difference operators Lj = T1 T2 + a(j )T1 + b (j )T2 + c (j ), related by Darboux­Laplace transformations Lj where Dj = T1 + bn hn,m+1 (j )hn hn,m (j )hn-1
- 1 ,m ,m+1 ,m - 1 +1

Dj = D

j +1

Lj ,

(j ), lead to purely discrete Toda lattice:
-1,m+1 -1,m+1

(j ) (1 + hn,m (j + 1))(1 + hn = (j ) (1 + hn,m (j ))(1 + hn

(j - 1)) . (j ))

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

In terms of the new set of variables hn,m (j ) = exp(qn+1,m-1 (j + 1) - qn,m (j )) this equation can be rewritten as follows: exp(qn
+1,m+1

(j ) + qn,m (j ) - qn+1,m (j ) - qn,m+1 (j )) = 1 + exp(qn+1,m (j + 1) - qn,m+1 (j )) . 1 + exp(qn+1,m (j ) - qn,m+1 (j - 1))

The substitution qn,m (j ) = un,m (j ) - un,m (j - 1) gives the following lattice: exp(un
+1,m+1

(j ) + un,m (j ) - un (j - 1) - un
,m+1

+1,m

(j ) - un
+1,m

,m+1

(j )) =
+1,m

= 1 + exp(un

,m+1

(j ) - un

(j ) + un

(j + 1)).

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Purely discrete exponential systems
Discrete finite Toda lattices are particular cases of the purely discrete exponential systems: exp(un
+1,m+1

(j ) + un,m (j ) - un
j -1

+1,m

(j ) - un
,m+1

,m+1

(j )) =
+1,m

= 1 + exp
k =1

ajk un

,m+1

(k ) +

ajj (un 2

(j ) + un

(j ))+

r

+
k =j +1

ajk un

+1,m

(k ) ,

where j = 1, 2, . . . , r (Garifullin, Habibullin, Yangubaeva, 2012).

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Darboux integrability in purely discrete case

Theorem
[S., to appear in TMPh, 2015] Purely discrete A-series Toda lattices are Darboux integrable. Darboux transformations provide generating functions for complete families of n- and m-integrals for purely discrete A-series Toda lattices. Sufficient number of integrals remain independent after reduction from A-series lattices to C -series lattices. Therefore discrete C -series lattices are Darboux integrable.

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices


Outline Introduction Continuous case Semidiscrete case Purely discrete case

Thank you!

Sergey V. Smirnov

On Darb oux integrability of discrete 2D-To da lattices