Документ взят из кэша поисковой машины. Адрес оригинального документа : http://higeom.math.msu.su/people/ssmirnov/nordfjordeid.pdf Дата изменения: Thu Jun 6 21:47:56 2013 Дата индексирования: Sun Apr 10 00:44:43 2016 Кодировка:

Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Discretization of 2D-Toda lattice: symmetries and integrals
Sergey V. Smirnov
Moscow State University

Nonlinear Mathematical Physics: 20 years of JNMP Nordfjordeid, Norway June 6, 2013

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Introduction Continuous case Semidiscrete case Further perspectives

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

How to obtain a finite system?
Trivial boundary conditions: u (-1) = u (r ) = -. Periodic boundary conditions: u (j + r ) = u (j ). Any other ideas? Finite Toda lattice is a particular case 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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Integrable boundary conditions
Matrix form: u = exp(K u), -2 1 K = (aij ) = 0 . . .
xy

where 1 0 ... -2 1 . . . 1 -2 . . . .. . 0 0 ... 0 0 0 . . .

0

. -2

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Integrable boundary conditions
Matrix form: u = exp(K u), -2 1 K = (aij ) = 0 . . .
xy

where 1 0 ... -2 1 . . . 1 -2 . . . .. . 0 0 ... 0 0 0 . . .

0

. -2

Notice that -K is the Cartan matrix of an A-series Lie algebra. There are other simple Lie algebras! Mikhailov, Olshanetsky, Perelomov (1981): uxy = exp(K u) admits Lax representation if -K is the Cartan matrix of a simple Lie algebra.
Sergey V. Smirnov Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Various approaches to integrability

Mikhailov, Olshanetsky, Perelomov (1979-1981): Lax representation depending on spectral parameter Leznov (1980): explicit integration Shabat, Yamilov (1981): integrals and characteristic algebra

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Various approaches to integrability

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

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Nonlocal variables
In one-dimensional case symmetries of the Toda chain are expressed in terms of the dynamical variables, but in two-dimensional case symmetries are expressed in terms of non-local variables (Shabat, 1995). Rewrite the infinite Toda lattice as follows: by (j ) = h(j ) - h(j - 1), where b (j ) = qx (j ) and define the variables b x b (j ) = b
(1) (1)

(j ) by

(j ) - b

(1)

(j - 1),

y b

(1)

(j ) = x h(j ).

Consistency of these equations is provided by the Toda equation.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

One has to introduce further non-localities in order to define x b (1) (j ): x b
(1)

(j ) = b

(1)

(b (j + 1) - b (j )) + b

(2)

(j ) - b

(2)

(j - 1);

consistency leads to the following relation: y b etc...
(2)

(j ) = h(j )b

(1)

(j + 1) - h(j + 1)b

(1)

(j ),

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Lemma
Non-local variables b following relations: y b x b
(k ) (k ) (k )

(j ), where k = 2, 3, . . . , satisfy the

(j ) = h(j )b (k -1) (j + 1) - h(j + k - 1)b (k -1) (j ) (j ) = b (k ) (j ) (b (j + k ) - b (j )) + b (k +1) (j ) - b (k

+1)

(j - 1)

Consistency of these equations for any k = 2, 3, . . . is provided by the first of them for k + 1.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Integrals of the infinite lattice
The functions I2 =
j Z

b 2 (j ) + 2b

(1)

(j )

and I3 =
j Z

b 3 (j ) + 3b

(1)

(j )(b (j ) + b (j + 1)) + 3b

(2)

(j )

are formal y -integrals of the infinite lattice. More integrals can be obtained using non-localities of higher grading.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Symmetries of the infinite lattice
The infinite Toda lattice admits the following symmetries: second order symmetry qt (j ) = b 2 (j ) + b third order symmetry qt (j ) = b 3 (j ) + b +b +b
(1) (1) (2) (1)

(j ) + b

(1)

(j - 1);

(j ) + b

(2)

(j - 1) + b

(2)

(j - 2) +

(j ) (2b (j ) + b (j + 1)) + (j - 1) (2b (j ) + b (j - 1)) ;

higher-order symmetries are expressed in terms of higher non-localities.
Sergey V. Smirnov Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Symmetries of finite lattices

Gurel, Habibullin, 1997: Ё

Theorem
i) Only trivial boundary condition is compatible with the second order symmetry.


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Symmetries of finite lattices

Gurel, Habibullin, 1997: Ё

Theorem
i) Only trivial boundary condition is compatible with the second order symmetry. ii) Boundary conditions corresponding to A - D -series are compatible with the third order symmetry. This is complete classification of boundary conditions of a certain type compatible with this test symmetry.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Non-local variables and cut-off constraints

Under the A-series reduction non-local variables become local.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Non-local variables and cut-off constraints

Under the A-series reduction non-local variables become local. Since B - D -series lattices are reductions of the A-series lattice, non-local variables for each of them also become local.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

y -integrals
Since h(r ) = h(r + 1) = · · · = 0 for A-series lattice, functions b (k ) (r ) are y -integrals for any k > 0: y b
(k )

(r ) = h(j )b

(k -1)

(r + 1) - h(r + k - 1)b

(k -1)

(r ) = 0.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

y -integrals
Since h(r ) = h(r + 1) = · · · = 0 for A-series lattice, functions b (k ) (r ) are y -integrals for any k > 0: y b
(k )

(r ) = h(j )b

(k -1)

(r + 1) - h(r + k - 1)b

(k -1)

(r ) = 0.

Explicit formulas for y -integrals:
r

b

(1)

(r ) =
j =0 r

bx (j ), ((r - i + 1)bxx (j ) + b (j )bx (j )) .
j =0

b

(2)

(r ) =

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Independence of integrals

Definition
Integrals I1 , I2 , . . . , Ir of orders k1 , k2 , . . . , kr resp. are called independent, if det Ii
k (x
i

b (j ))

= 0,

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

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Lemma
The y -integrals I1 , I2 , . . . , Ir for since 1 1 1 Ii det = det 1 ki (x b (j )) . . . the Toda lattice are independent 1 2 3 4 . . . 1 3 6 10
r (r -1) 2

1 4 10 20 ...

. . . .

. . . .

. . . .

1 r
r (r -1) 2

= 1.

... ...

1r

...

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Darboux integrability

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Darboux integrability

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.

Theorem
A - D -series Toda lattices are Darboux integrable.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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 Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

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

- Here K + = (ai+ ) and K - = (aij ) are upper- and lower-triangular j constant matrices with -1 on the diagonal.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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

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

- Here K + = (ai+ ) and K - = (aij ) are upper- and lower-triangular j constant matrices with -1 on the diagonal. For semidiscrete analogs of Toda lattices corresponding to simple Lie algebras K = K - + K + is the Cartan matrix.
Sergey V. Smirnov Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

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 Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Non-local variables

In the semi-discrete case symmetries and integrals are also (1) expressed in terms of non-local variables. Non-localities bn (j ) are defined as follows: x bn (j ) = bn (j ) - bn (j - 1),
(1) (1)

n bn (j ) = x hn (j ).

(1)

Consistency of these equations is provided by the Toda lattice.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Further non-localities are defined by the following

Lemma

Non-local variables bn (j ), where k = 2, 3, . . . , satisfy the following relations: ( ( ( n bnk ) (j ) = hn (j )bnk -1) (j + 1) - hn+k -1 (j + k - 1)bnk -1) (j ) +1 +1 ( ( bnk ) (j ) = bnk ) (j ) (b (j + k ) - bn (j )) + x n +k (k ) (k ) + hn+k -1 (j + k - 1) bn (j - 1) - bn (j ) + (k +1) (k +1) + bn (j ) - bn (j - 1) Consistency of these equations for any k = 2, 3, . . . is provided by the first of them for k + 1.

(k )

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Integrals of the infinite lattice
The functions I2 =
j Z 2 bn (j ) + 2bn (j ) (1)

and I3 =
j Z 3 bn (j ) + 3bn (j )(bn (j ) + bn (j + 1) - hn (j + 1)) + 3bn (j ) (1) (2)

are formal n-integrals of the infinite lattice. More integrals can be obtained using nonlocalities of higher grading.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Symmetries of the infinite lattice
The infinite semidiscrete Toda lattice admits the following symmetries: second order symmetry
2 qn,t (j ) = bn (j ) + bn (j ) + bn (j - 1); (1) (1)

third order symmetry
3 qn,t (j ) = qn (j ) + bn (j ) + bn (j - 1) + bn (j - 2) + (2) (2) (2)

+ bn (j ) (2bn (j ) + bn (j + 1)) + = bn (j - 1) (2bn (j ) + bn (j - 1)) -
(1) (1) (1) (1)

(1)

- bn (j )hn (j + 1) - bn (j - 1)hn (j ) - bn (j - 2)hn (j - 1

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Symmetries of finite lattices

Theorem
i) Only trivial boundary condition is compatible with the second order symmetry.


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Symmetries of finite lattices

Theorem
i) Only trivial boundary condition is compatible with the second order symmetry. ii) Boundary conditions corresponding to A- and C -series are compatible with the third order symmetry. This is complete classification of boundary conditions of a certain type compatible with this test symmetry.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Non-local variables and cut-off constraints

Under the A-series reduction non-local variables become local.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Non-local variables and cut-off constraints

Under the A-series reduction non-local variables become local. Since C -series lattice is a reduction of the A-series lattice, non-local variables for it also become local.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

n-integrals
Since h(r ) = h(r + 1) = · · · = 0 for A-series lattice, functions (k ) bn (r ) are n-integrals for any k > 0: n bn (r ) = hn (r )bn
(k ) (k -1) +1

(r + 1) - hn

+k - 1

(r + k - 1)bn

(k -1) +1

(r ) = 0

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

n-integrals
Since h(r ) = h(r + 1) = · · · = 0 for A-series lattice, functions (k ) bn (r ) are n-integrals for any k > 0: n bn (r ) = hn (r )bn
(k ) (k -1) +1

(r + 1) - hn

+k - 1

(r + k - 1)bn

(k -1) +1

(r ) = 0

Explicit formulas for n-integrals:
r (1) bn r

(r ) =
j =0

bn,x (j ),

(2) bn

(r ) =
j =0

((r - i + 1)bn

,xx

(j ) + b (j )bn,x (j )) .

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

n-integrals
Since h(r ) = h(r + 1) = · · · = 0 for A-series lattice, functions (k ) bn (r ) are n-integrals for any k > 0: n bn (r ) = hn (r )bn
(k ) (k -1) +1

(r + 1) - hn

+k - 1

(r + k - 1)bn

(k -1) +1

(r ) = 0

Explicit formulas for n-integrals:
r (1) bn r

(r ) =
j =0

bn,x (j ),

(2) bn

(r ) =
j =0

((r - i + 1)bn

,xx

(j ) + b (j )bn,x (j )) .

Theorem

Functions bn (r ), bn (r ), . . . form a complete family of independent n-integrals for A- and C -series in semidiscrete case.
Sergey V. Smirnov Discretization of 2D-To da lattice: symmetries and integrals

(1)

(2)


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

x -integrals in semidiscrete case
Denote cn (j ) = qn (j ) - qn+1 (j ); then the following functions are x -integrals for the A-series lattice:
r r

J1 =
j =0

cn (j ),

J2 =
j =0

e

c

n +j

(j )

.

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

x -integrals in semidiscrete case
Denote cn (j ) = qn (j ) - qn+1 (j ); then the following functions are x -integrals for the A-series lattice:
r r

J1 =
j =0

cn (j ),

J2 =
j =0

e

c

n +j

(j )

.

How to construct another set of non-local variables c
(1) n

(j ), c

(2) n

(j ), . . .

in order to obtain the complete family of independent x -integrals?

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Purely discrete case
Sequence of linear hyperbolic difference operators linked by Darboux-Laplace transformations lead to the equations for discrete Laplace invariants: hn,m+1 (j )hn hn,m (j )hn-1
- 1 ,m ,m+1

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

Substitution hn,m (j ) = exp(qn+1,m-1 (j + 1) - qn,m (j )) allows to rewrite this lattice in another form: 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))

These are two forms of purely discrete 2D-Toda lattice. This system is a particular case of the so-called purely discrete exponential system (Garifullin, Habibullin, Yangubaeva, 2012).
Sergey V. Smirnov Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Symmetries and integrals in purely discrete case
Functions
r r

I=
j =0

(qn,m (j ) - qn

,m+1

(j )), J =
j =0

(qn,m (j ) - qn

+1,m

(j ))

are n- and m-integrals resp. Proper definition of non-local variables will allow to obtain a complete family of n- and m-integrals for A-series lattice due to the symmetry n -m; to obtain symmetries for purely discrete lattices. C -series lattice is also a reduction of the A-series lattice. Therefore, Darboux integrability of the A-series lattice implies Darboux integrability of the C -series lattice.
Sergey V. Smirnov Discretization of 2D-To da lattice: symmetries and integrals


Outline Introduction Continuous case Semidiscrete case Further p ersp ectives

Thank you!

Sergey V. Smirnov

Discretization of 2D-To da lattice: symmetries and integrals