Discretization of 2D-Toda lattice: symmetries and integrals

Outline Introduction Continuous case Semidiscrete case Further perspectives

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-Toda lattice: symmetries and integrals


Introduction Continuous case Semidiscrete case Further perspectives

Sergey V. Smirnov



Two-dimensional Toda lattice I

qxy (j) = exp(q(j + 1) − q(j)) − exp(q(j) − q(j − 1)),

I

or (ln h(j))xy = h(j + 1) − 2h(j) + h(j − 1), where h(j) = exp(q(j + 1) − q(j)),

I

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



How to obtain a finite system? I

Trivial boundary conditions: u(−1) = u(r ) = −∞.

I

Periodic boundary conditions: u(j + r ) = u(j).

I

Any other ideas?

Sergey V. Smirnov




Trivial boundary conditions: u(−1) = u(r ) = −∞.

I

Periodic boundary conditions: u(j + r ) = u(j).

I

Any other ideas?

Finite Toda lattice is a particular case of the so-called exponential system:   r X uxy (i) = exp  aij u(j) , i = 1, 2, . . . , r , j=1

where aij = const.

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Integrable boundary conditions I

Matrix form: uxy = exp(K u), where  −2 1 0 ...  1 −2 1 . . .   1 −2 . . . K = (aij ) =  0  .. ..  . . 0 0 0 ...

Sergey V. Smirnov

0 0 0 .. .



   .   −2



Integrable boundary conditions I

Matrix form: uxy = exp(K u), where  −2 1 0 ...  1 −2 1 . . .   1 −2 . . . K = (aij ) =  0  .. ..  . . 0 0 0 ...

0 0 0 .. .



   .   −2

I

Notice that −K is the Cartan matrix of an A-series Lie algebra. There are other simple Lie algebras!

I

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



Various approaches to integrability

I

Mikhailov, Olshanetsky, Perelomov (1979-1981): Lax representation depending on spectral parameter

I

Leznov (1980): explicit integration

I

Shabat, Yamilov (1981): integrals and characteristic algebra

Sergey V. Smirnov



Various approaches to integrability

I

Mikhailov, Olshanetsky, Perelomov (1979-1981): Lax representation depending on spectral parameter

I

Leznov (1980): explicit integration

I

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



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 + a∂x + b∂y + c.

Sergey V. Smirnov



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 = Dj+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



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: I

h(−1) = 0 for A-series (trivial);

I

h(−j) = h(j − 1) for B-series (involution);

I

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



Reductions of the A-series lattice

I

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);

I

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);

I

D-series lattice is also a reduction of the A series lattice (Habibullin, 2005).

Sergey V. Smirnov



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 (1) (j) by ∂x b(j) = b (1) (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



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 (2) (j) = h(j)b (1) (j + 1) − h(j + 1)b (1) (j), etc...

Sergey V. Smirnov



Lemma Non-local variables b (k) (j), where k = 2, 3, . . . , satisfy the following relations: ∂y b (k) (j) = h(j)b (k−1) (j + 1) − h(j + k − 1)b (k−1) (j) ∂x b (k) (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



Integrals of the infinite lattice The functions I2 =

X

b 2 (j) + 2b (1) (j)

j∈Z

and I3 =

X

b 3 (j) + 3b (1) (j)(b(j) + b(j + 1)) + 3b (2) (j)

j∈Z

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

Sergey V. Smirnov



Symmetries of the infinite lattice The infinite Toda lattice admits the following symmetries: I

second order symmetry qt (j) = b 2 (j) + b (1) (j) + b (1) (j − 1);

I

third order symmetry qt (j) = b 3 (j) + b (2) (j) + b (2) (j − 1) + b (2) (j − 2) + + b (1) (j) (2b(j) + b(j + 1)) + + b (1) (j − 1) (2b(j) + b(j − 1)) ;

I

higher-order symmetries are expressed in terms of higher non-localities. Sergey V. Smirnov



Symmetries of finite lattices

G¨ urel, Habibullin, 1997:

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



G¨ urel, Habibullin, 1997:

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



Non-local variables and cut-off constraints

I

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

Sergey V. Smirnov




I


I

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



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



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: b

(1)

(r ) =

b (2) (r ) =

r X j=0 r X

bx (j), ((r − i + 1)bxx (j) + b(j)bx (j)) .

j=0

Sergey V. Smirnov



Independence of integrals

Definition Integrals I1 , I2 , . . . , Ir of orders k1 , k2 , . . . , kr resp. are called independent, if ! ∂Ii det 6= 0, i, j = 1, 2, . . . , r . ∂(∂xki b(j))

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Lemma The y -integrals I1 , I2 , . . . , Ir for the since  1 1  1 2  !   1 3 ∂Ii det = det   1 4 ki ∂(∂x b(j))  . .  . .  . . 1 r

Sergey V. Smirnov

Toda lattice are independent 1 3 6 10

1 4 10 20

... ... ... ...

r (r −1) 2

...

...

1 r r (r −1) 2

...

      = 1.    

...



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



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



Laplace invariants

Definition Functions kn = acnn − (ln an )0x − bn and hn = acnn − bn are called the Laplace invariants of hyperbolic differential-difference operator L = ∂x T + a∂x + bT + c, where T is the shift operator: T ψn = ψn+1 .

Sergey V. Smirnov



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+1 Dj = Dj+1 Lj , where Dj = ∂x + bn (j).

Sergey V. Smirnov



Then the Laplace invariants of these operators satisfy the following equations: ( kn (j + 1) = hn (j) 0 . n (j) = hn+1 (j + 1) − hn+1 (j) − hn (j) + hn (j − 1) ln hhn+1 (j) x

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



Finite semidiscrete Toda lattices


Trivial boundary conditions: hn (−1) = hn (r ) = 0.

I

Periodic boundary conditions.

I

How to introduce semidiscrete analogs of Toda lattices, corresponding to simple Lie algebras?

Sergey V. Smirnov



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 X un,x (i)−un+1,x (i) = exp  aij+ un+1 (j) + aij− un (j) , i = 1, 2, . . . , r . j=1

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

Sergey V. Smirnov



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 X un,x (i)−un+1,x (i) = exp  aij+ un+1 (j) + aij− un (j) , i = 1, 2, . . . , r . j=1

Here K + = (aij+ ) and K − = (aij− ) are upper- and lower-triangular 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



Involutions in the semidiscrete case I

I

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).

I

C -series lattice is a reduction of an A series lattice of the length 2r . Sergey V. Smirnov



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: (1)

(1)

∂x bn (j) = bn (j) − bn (j − 1),

(1)

∂n bn (j) = ∂x hn (j).

Consistency of these equations is provided by the Toda lattice.

Sergey V. Smirnov



Further non-localities are defined by the following

Lemma

(k)

Non-local variables bn (j), where k = 2, 3, . . . , satisfy the following relations:  (k) (k−1) (k−1)  ∂n bn (j) = hn (j)bn+1 (j + 1) − hn+k−1 (j + k − 1)bn+1 (j)     ∂ bn(k) (j) = bn(k) (j) (b x n+k (j + k) − bn (j)) + (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.

Sergey V. Smirnov



Integrals of the infinite lattice The functions I2 =

X

(1) bn2 (j) + 2bn (j)

j∈Z

and I3 =

X

(1) (2) bn3 (j) + 3bn (j)(bn (j) + bn (j + 1) − hn (j + 1)) + 3bn (j)

j∈Z

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

Sergey V. Smirnov



Symmetries of the infinite lattice The infinite semidiscrete Toda lattice admits the following symmetries: I

second order symmetry (1)

(1)

qn,t (j) = bn2 (j) + bn (j) + bn (j − 1); I

third order symmetry (2)

(2)

(2)

qn,t (j) = qn3 (j) + bn (j) + bn (j − 1) + bn (j − 2) + (1)

+ bn (j) (2bn (j) + bn (j + 1)) + (1)

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

(1)

(1)

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

Sergey V. Smirnov




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



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




I


Sergey V. Smirnov




I


I

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

Sergey V. Smirnov



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

(k−1)

(k−1)

∂n bn (r ) = hn (r )bn+1 (r + 1) − hn+k−1 (r + k − 1)bn+1 (r ) = 0

Sergey V. Smirnov




(k−1)

(k−1)

∂n bn (r ) = hn (r )bn+1 (r + 1) − hn+k−1 (r + k − 1)bn+1 (r ) = 0 Explicit formulas for n-integrals: (1) bn (r )

=

r X

bn,x (j),

(2) bn (r )

=

j=0

r X

((r − i + 1)bn,xx (j) + b(j)bn,x (j)) .

j=0

Sergey V. Smirnov




(k−1)

(k−1)

∂n bn (r ) = hn (r )bn+1 (r + 1) − hn+k−1 (r + k − 1)bn+1 (r ) = 0 Explicit formulas for n-integrals: (1) bn (r )

=

r X

bn,x (j),

(2) bn (r )

=

j=0

Theorem

(1)

r X

((r − i + 1)bn,xx (j) + b(j)bn,x (j)) .

j=0

(2)

Functions bn (r ), bn (r ), . . . form a complete family of independent n-integrals for A- and C -series in semidiscrete case. Sergey V. Smirnov



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: J1 =

r X

cn (j),

j=0

Sergey V. Smirnov

J2 =

r X

e cn+j (j) .

j=0



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: J1 =

r X

cn (j),

J2 =

j=0

r X

e cn+j (j) .

j=0

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

(2)

cn (j), cn (j), . . . in order to obtain the complete family of independent x-integrals?

Sergey V. Smirnov



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



Symmetries and integrals in purely discrete case Functions I =

r X

(qn,m (j) − qn,m+1 (j)), J =

j=0

r X

(qn,m (j) − qn+1,m (j))

j=0

are n- and m-integrals resp. Proper definition of non-local variables will allow I

to obtain a complete family of n- and m-integrals for A-series lattice due to the symmetry n ↔ −m;

I

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



Thank you!

Sergey V. Smirnov
