Semidiscrete Toda lattices

Outline Introduction Lax pairs in the continuous case Symmetry approach in the continuous case Lax pairs in the semidiscrete case Symmetry approach in the semidiscrete case

Semidiscrete Toda lattices Sergey V. Smirnov Moscow State University

Matrix Models, Tau functions and Geometry University of Glasgow March 3, 2012

Sergey V. Smirnov

Semidiscrete Toda lattices


Introduction Lax pairs in the continuous case Symmetry approach in the continuous case Lax pairs in the semidiscrete case Symmetry approach in the semidiscrete case

Sergey V. Smirnov



The Toda chain I

qxx (j) = exp(q(j + 1) − q(j)) − exp(q(j) − q(j − 1)).

Sergey V. Smirnov



The Toda chain I

qxx (j) = exp(q(j + 1) − q(j)) − exp(q(j) − q(j − 1)).

I

Change of variables: (ln h(j))xx = h(j + 1) − 2h(j) + h(j − 1), where h(j) = exp(q(j + 1) − q(j)).

I

Another change of variables: uxx (j) = exp(u(j + 1) − 2u(j) + u(j − 1)), where h(j) = uxx (j).

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How to obtain a finite dynamical system?

I

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

I

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

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How to obtain a finite dynamical system?

I

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

I

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

I

Any other ideas?

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Integrable boundary conditions I

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

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0 0 0 .. .



   .   −2




I


0 0 0 .. .



   .   −2

Notice that −K is the Cartan matrix of an A-series Lie algebra.

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0 0 0 .. .



   .   −2

I

Notice that −K is the Cartan matrix of an A-series Lie algebra.

I

There are other simple Lie algebras!

I

Bogoyavlensky (1976): uxx = exp(K u) is integrable if −K is the Cartan matrix of a simple Lie algebra. 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 u(j)xy = exp(u(j + 1) − 2u(j) + u(j − 1)), where h(j) = uxy (j),

I

or uxy = exp(K u) (exponential system).

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Integrability in two-dimensional case Various approaches by I

Leznov (1980)

I

Mikhailov, Olshanetsky, Perelomov (1979-1981)

I

Shabat, Yamilov (1981)

I

and probably by many others. . .

give the same result: Boundary conditions corresponding to simple Lie algebras are integrable whatever definition of integrability is being used.

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Laplace invariants 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.

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Laplace invariants 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.

Lemma Two hyperbolic differential operators L1 and L2 are gauge-equivalent (i.e. L2 = ω −1 L1 ω for some function ω = ω(x, y )) iff their Laplace invariants are equal.

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DLT and Toda lattice Consider a sequence of hyperbolic operators Lj = ∂x ∂y + a(j)∂x + b(j)∂y + c(j) such that any two neighboring operators are linked 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 .

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Lax presentations for infinite lattice Darboux-Laplace transformation Dj = ∂x + b(j) : Lj → Lj+1 leads to the following Lax presentation for the Toda lattice: ψx (j) = ψ(j + 1) + qx (j)ψ(j) , ψy (j) = −h(j − 1)ψ(j − 1) where h(j) = exp(q(j + 1) − q(j)).

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Lax presentations for infinite lattice Darboux-Laplace transformation Dj = ∂x + b(j) : Lj → Lj+1 leads to the following Lax presentation for the Toda lattice: ψx (j) = ψ(j + 1) + qx (j)ψ(j) , ψy (j) = −h(j − 1)ψ(j − 1) where h(j) = exp(q(j + 1) − q(j)). Symmetry x ↔ y provides another Lax presentation: ϕx (j) = −h(j − 1)ϕ(j − 1) . ϕy (j) = ϕ(j + 1) + qy (j)ϕ(j) Sergey V. Smirnov



Lax pair consistent with a boundary condition Consider boundary condition of the type q(−1) = F (q(0), q(1), . . . , q(k)) In order to obtain a Lax presentation for Toda lattice satisfying this boundary condition, one has to express ψ(−1) in terms of ψ(0), ψ(1), . . . But this works only for the trivial boundary condition q(−1) = ∞.

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Lax pair consistent with a boundary condition Consider boundary condition of the type q(−1) = F (q(0), q(1), . . . , q(k)) In order to obtain a Lax presentation for Toda lattice satisfying this boundary condition, one has to express ψ(−1) in terms of ψ(0), ψ(1), . . . But this works only for the trivial boundary condition q(−1) = ∞. Habibullin’s idea (2005): why not to express ψ(−1) and ϕ(−1) in terms of both sets of variables: ψ(0), ψ(1), . . .

and ϕ(0), ϕ(1), . . .?

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Lax pairs for A − D series lattices Theorem The A − D series Toda lattices admit Lax presentation of the form B L A K Ψ Ψ Ψ Ψ , = , = Φ Φ y Φ Φ x M C N D


Lax pairs for A − D series lattices Theorem The A − D series Toda lattices admit Lax presentation of the form B L A K Ψ Ψ Ψ Ψ , = , = Φ Φ y Φ Φ x M C N D i.e. for any Toda lattice there exist integers k < k1 and m < m1 such that Toda equations together with corresponding boundary conditions are equivalent to the compatibility conditions for the above matrix equations, where Ψ = (ψ(k + 1), ψ(k + 2), . . . , ψ(k1 − 1))t , Φ = (ϕ(m + 1), ϕ(m + 2), . . . , ϕ(m1 − 1))t ; and matrices A, B, . . . , N are as follows: Sergey V. Smirnov





qx (k + 1) 1 0  0 qx (k + 2) 1   . . . .. A= .   0 0 Ak+1 Ak+2 Ak+3 

Bk+1 Bk+2 −h(k + 1) 0   . . .. .. B=   0 0 0 0

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

 0 0    ,  1 

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

Ak1 −1

 Bk1 −1 0    ,  0 0  −h(k1 − 2) 0 0 0





Cm+1 Cm+2 −h(m + 1) 0   . . . .. C = .   0 0 0 0 

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

 Cm1 −1 0    ,  0 0  −h(m1 − 2) 0

qy (m + 1) 1 0  0 q (m + 2) 1 y   .. .. D= . .   0 0 Dm+1 Dm+2 Dm+3

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0 0

... ... .. .

0 0

... ...

1 Dm1 −1


    ,  




0 0 .. .

   K =   0 Km+1 

Mk+1  0   M =  ...   0 0



 Lm+1  0   L =  ...   0 0

0 0

... ... .. .

0 0

0 Km+2

... ...

0 Km1 −1

Mk+2 0

... ... .. .

0 0

... ...

  0 Mk1 −1  0 0     .   , N =  ..    0 0  Nk+1 0

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   ,  

Lm+2 0

... ... .. .

0 0

... ...

0 0

... ... .. .

0 Nk+2

... ...


 Lm1 −1 0    ,  0  0  0 0    .  0  Nk1 −1


Remarks

I

Since boundary conditions on the left and on the right are independent, this approach provides Lax presentations for a wider class of systems than just A − D series Toda lattices.

Sergey V. Smirnov



Remarks

I


I

Moreover, Lax presentations of the above form lead to systems that are not necessarily exponential.

Sergey V. Smirnov



Remarks

I


I

Moreover, Lax presentations of the above form lead to systems that are not necessarily exponential.

I

If one seeks for Lax presentation for some boundary value problem for the Toda lattice, this approach gives no hint on how to choose indeterminate matrix elements.

Sergey V. Smirnov



Boundary conditions and involutions 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).

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Motivation for Habibullin’s idea

Each of the wave functions ψ(j) and ϕ(j) satisfy the hyperbolic ϕ differential equation Lψ j ψ(j) = 0 (Lj ϕ(j) = 0 resp.) In B- and C -series cases there are relations between the Laplace invariants of these equations.

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I

B-series: hϕ (0) = h(−1) = h(2) = hψ (2), k ϕ (0) = h(0) = h(1) = k ψ (2), hψ (1) = h(1) = h(0) = hϕ (1),

Sergey V. Smirnov

k ψ (1) = h(1) = h(0) = k ϕ (1).



I

B-series: hϕ (0) = h(−1) = h(2) = hψ (2), k ϕ (0) = h(0) = h(1) = k ψ (2), hψ (1) = h(1) = h(0) = hϕ (1),

I

k ψ (1) = h(1) = h(0) = k ϕ (1).

ψ ψ ϕ This means that pairs of operators Lϕ 0 and L2 , L1 and L1 are gauge-equivalent. Therefore, there exist the multipliers R = R(x, y ) and S = S(x, y ), such that

ϕ(0) = Rψ(2),

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ψ(1) = Sϕ(1).



I

C -series: hϕ (0) = h(−1) = h(1) = hψ (1),

k ϕ (0) = h(0) = k ψ (1),

k ψ (0) = h(−1) = h(1) = k ϕ (1), hψ (0) = h(0) = hϕ (1).

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I

C -series: hϕ (0) = h(−1) = h(1) = hψ (1),

k ϕ (0) = h(0) = k ψ (1),

k ψ (0) = h(−1) = h(1) = k ϕ (1), hψ (0) = h(0) = hϕ (1).

I

Similarly, these relations imply the existence of R = R(x, y ), and S = S(x, y ) such that ϕ(0) = Rψ(1),

I

ψ(0) = Sϕ(1).

Both examples are particular cases of Habibullin’s theorem.

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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).

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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),

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(1)

∂y bn (j) = ∂x h(j).



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),

(1)

∂y bn (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);

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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...

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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.

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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);

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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)) ;

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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



New set of variables In the new set of variables u = e −q(−1) ,

v = e q(0)

these symmetries are expressed in a more simple way: ut = −uxx − 2ru,

vt = vxx + 2rv ,

ut = uxxx + 3rux − 3su + 3rx u,

vt = vxxx + 3rvx + 3sv ,

where r = b (1) (0), s = b (2) (0) + r (ln v )x .

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Classification theorem What boundary conditions are compatible with the above symmetries?

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Classification theorem What boundary conditions are compatible with the above symmetries? I

Only trivial boundary condition u = 0 is compatible with the second order symmetry.

Sergey V. Smirnov





I

Habibullin, G¨ urel (1997): if boundary condition u = F (v , vx , vy , vxy ) is compatible with the third order symmetry, then the function F belongs to one of the following classes: i) F (v , vx , vy , vxy ) = 0 (A series); ii) F (v , vx , vy , vxy ) = a 6= 0, a = const (B series); iii) F (v , vx , vy , vxy ) = av , a = const 6= 0 (C series); vxy vvx vy iv) F (v , vx , vy , vxy ) = a−v 2 + (a−v 2 )2 (D series).

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I

Habibullin, G¨ urel (1997): if boundary condition u = F (v , vx , vy , vxy ) is compatible with the third order symmetry, then the function F belongs to one of the following classes: i) F (v , vx , vy , vxy ) = 0 (A series); ii) F (v , vx , vy , vxy ) = a 6= 0, a = const (B series); iii) F (v , vx , vy , vxy ) = av , a = const 6= 0 (C series); vxy vvx vy iv) F (v , vx , vy , vxy ) = a−v 2 + (a−v 2 )2 (D series).

I

More complicate boundary conditions lead to the systems that are not exponential. Sergey V. Smirnov



Discrete analogs There are several possible ways to discretize Toda systems. I

One-dimensional discrete Toda chains were considered by Suris (1990). Lax presentations were obtained for the chains corresponding to non-exceptional Lie algebras.

Sergey V. Smirnov





I

Purely discrete two-dimensional Toda lattices were considered by Ward (1995) and Habibullin (2005). Lax presentations for A- and C -series Toda lattices were obtained.

Sergey V. Smirnov





I

Purely discrete two-dimensional Toda lattices were considered by Ward (1995) and Habibullin (2005). Lax presentations for A- and C -series Toda lattices were obtained.

I

Semidiscrete case mysteriously had not been considered before!

Sergey V. Smirnov



The purpose of the talk

The rest of this talk will concern the semidiscrete case, i.e. the Toda lattice with one of the independent variables being continuous and the other being discrete. The purpose of this talk is to highlight the similarities and the differences between the continuous case and the semidiscrete case.

Sergey V. Smirnov



Laplace invariants Definition Functions kn (j) =

cn (j) an (j)

− (ln an (j))0x − bn (j) and

hn (j) = acnn (j) (j) − bn (j − 1) 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



Laplace invariants Definition Functions kn (j) =

cn (j) an (j)

− (ln an (j))0x − bn (j) and

hn (j) = acnn (j) (j) − bn (j − 1) 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 .

Lemma Two hyperbolic differential-difference operators L1 and L2 are gauge-equivalent (i.e. L2 = (T ω)−1 L1 ω for some function ω = ωn (x)) iff their Laplace invariants are equal. 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 linked by Darboux-Laplace transformation: Lj+1 Dj = Dj+1 Lj , where Dj = ∂x + bn (j).

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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

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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)).

Sergey V. Smirnov




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



Lax presentations for infinite lattice Darboux-Laplace transformation Dj = ∂x + b(j) : Lj → Lj+1 leads to the following Lax presentation for the Toda lattice: ψn,x (j) = qn,x (j)ψn (j) + ψn (j + 1) . ψn+1 (j) = ψ(j) + hn (j − 1)ψn (j − 1)

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Lax presentations for infinite lattice Darboux-Laplace transformation Dj = ∂x + b(j) : Lj → Lj+1 leads to the following Lax presentation for the Toda lattice: ψn,x (j) = qn,x (j)ψn (j) + ψn (j + 1) . ψn+1 (j) = ψ(j) + hn (j − 1)ψn (j − 1) There is another Lax presentation: ϕn,x (j) = exp(qn+1 (j) − qn+1 (j − 1))ϕn (j − 1) − ϕn (j) . ϕn−1 (j) = − exp(qn (j) − qn+1 (j))ϕn (j) + ϕn (j + 1)

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Involutions in the semidiscrete case 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).

Sergey V. Smirnov



Involutions in the semidiscrete case 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).

I

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).

Sergey V. Smirnov



I

In continuum limit the latter equation tends to the C -series boundary condition.

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I

In continuum limit the latter equation tends to the C -series boundary condition.

I

It looks as if there’s no semidiscrete analog to the B-series Toda lattice!

Sergey V. Smirnov



Remark

The C -series semidiscrete Toda lattice has an n-integral µ(x) = qn,x (−1) + qn,x (0) − hn (0). Although the condition µ = 0 is not equivalent to the condition hn (−2) = hn+1 (0), these conditions are “almost equivalent”. Therefore further we’ll consider µ = 0 as a boundary condition for the C -series lattice instead of the boundary condition hn (−2) = hn+1 (0).

Sergey V. Smirnov



Lax pair for the C -series lattice For the C -series semidiscrete Toda lattice Laplace invariants of hyperbolic equations corresponding to the above Lax pairs are related as follows: hϕ (0) = h(−1) = h(1) = hψ (1),

k ϕ (0) = h(0) = k ψ (1),

k ψ (0) = h(−1) = h(1) = k ϕ (1),

hψ (0) = h(0) = hϕ (1).

Sergey V. Smirnov




k ϕ (0) = h(0) = k ψ (1),

k ψ (0) = h(−1) = h(1) = k ϕ (1),

hψ (0) = h(0) = hϕ (1).

Hence, there exist the functions R = Rn (x) and S = Sn (x) such that ϕn (−1) = Rn ψn+1 (0), ψn (−1) = Sn ϕn (0).

Sergey V. Smirnov




k ϕ (0) = h(0) = k ψ (1),

k ψ (0) = h(−1) = h(1) = k ϕ (1),

hψ (0) = h(0) = hϕ (1).

Hence, there exist the functions R = Rn (x) and S = Sn (x) such that ϕn (−1) = Rn ψn+1 (0), ψn (−1) = Sn ϕn (0). This leads to the following Lax presentation for the C -series Toda lattice: Sergey V. Smirnov



Lemma The C -series semidiscrete Toda lattice is equivalent to the compatibility conditions for the following linear system: ∂x (Ψ) = AΨ,

T (Ψ) = BΨ + LΦ,

∂x (Φ) = MΨ + C Φ,

T −1 (Φ) = DΦ,

where

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

pn an+1 (1)e −x   0  A= ..  .   0 0

0 0 ... −1 0 ... an+1 (2) −1 . . . .. .. . . 0 0 ... 0 0 ...

bn (0) ex 0 0 ...  0 bn (1) −1 0 ...   0 0 b (2) −1 ... n  B= . . .. ...  ..   0 0 0 0 ... 0 0 0 0 ... 

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0 0 0

0 0 0



    , 0 0  −1 0 an+1 (r ) −1 0 0 0



    ,   −1  bn (r )





qn,x (0) 1 0 ...  0 qn,x (1) 1 . . .   .. .. .. C = . . .   0 0 ... 0 0 ... 

1 0 ... hn (0) 1 . . .   .. .. D =  ... . .   0 0 ... 0 0 ...

Sergey V. Smirnov

0 0



   ,  1  qn,x (r )

 0 0   , 0 0  1 0 hn (r − 1) 1 0 0



 fn 0 0 . . . 0 0 0 ...  L=. ..  .. . 0 0 0 ...

 0 0  ,  0

 gn 0 0 . . .  0 0 0 ...  M=. ..  .. . 0 0 0 ...

 0 0    0

and an (j) = exp(qn (j) − qn (j − 1)), bn (j) = − exp(qn (j) − qn+1 (j)), fn = (−1)n exp(−qn (−1)),

gn = (−1)n exp(qn+1 (0)),

pn = un,x (−1) + un+1,x (0).

Sergey V. Smirnov



Non-local variables

In the semi-discrete case symmetries are also expressed in terms of non-local variables.

Sergey V. Smirnov



Non-local variables

In the semi-discrete case symmetries are also expressed in terms of (1) 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)  + h (j + k − 1) b (j − 1) − b (j) + n n n+k−1     (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



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);

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



New set of variables In the new set of variables un = e −qn (−2) ,

vn = e qn+1 (−1) ,

wn = q −qn (0) ,

zn = e qn+1 (1)

symmetries are expressed in a more simple way: un,t = un,xxx + 3rn un,x − 3sn un + 3rn,x un , vn,t = vn,xxx + 3rn+1 vn,x + 3sn+1 vn wn,t = wn,xxx + 3ρn wn,x − 3σn wn + 3ρn,x wn , zn,t = zn,xxx + 3ρn+1 zn,x + 3σn+1 zn . where (1)

sn = bn (−2) + rn (bn (−1) − hn (−1)),

(1)

σn = bn (0) + ρn (bn (1) − hn (1)).

rn = bn (−2), ρn = bn (0),

(2)

(2)

Sergey V. Smirnov



Classification theorem What boundary conditions are compatible with the above symmetries?

Sergey V. Smirnov




Only trivial boundary condition u = 0 is compatible with second order symmetry.

Sergey V. Smirnov




Only trivial boundary condition u = 0 is compatible with second order symmetry.

I

If the variables vn−1 , vn , wn , wn+1 , zn , zn+1 , vn−1,x , wn,x , zn,x are independent, then only trivial boundary condition u = 0 is compatible with the third order symmetry (A series).

Sergey V. Smirnov



I

If vn−1,x = H(vn−1 , vn , wn , wn+1 , zn , zn+1 , wn,x , zn,x ) and boundary condition u = F (vn−1 , vn , wn , wn+1 , zn , zn+1 ) is compatible with the third order symmetry, then H= and F =

wn,x vn−1 + zn wn vn−1 wn wn+1 zn+1 vn

Sergey V. Smirnov

(C series).



I

If vn−1,x = H(vn−1 , vn , wn , wn+1 , zn , zn+1 , wn,x , zn,x ) and boundary condition u = F (vn−1 , vn , wn , wn+1 , zn , zn+1 ) is compatible with the third order symmetry, then H= and F =

wn,x vn−1 + zn wn vn−1 wn wn+1 zn+1 vn

(C series).

Note that the equation for the function H is equivalent to the “modified” C -series boundary condition qn,x (−1) + qn,x (0) − hn (0) = 0.

Sergey V. Smirnov



Thank you!

Sergey V. Smirnov
