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Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Darboux transformations in theory of integrable systems
Sergey V. Smirnov
Moscow State University

Research Seminar Nazarbaev University, Astana May 22, 2015

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Introduction Dynamical systems Hyperbolic PDEs in 2D Other examples

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

What is integrability?
Different definitions of integrability are used for different types of equations. There are two most general approaches: to consider properties of the equation itself, or properties of its solutions. Properties of equations: explicit integrability sufficient number of first integrals in involution (ODEs) existence of a Lax representation sufficient number of integrals along characteristics (hyperbolic PDEs in 2D) existence of higher symmetries 3D-consistency (PEs on quad-graphs in 2D) ...
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

First integrals
Function F (x1 , x2 , . . . , x system 1 x 2 x ... n x
n

) is called a first integral of the dynamical

= f1 (x 1 , x 2 , . . . , x n ) = f2 (x 1 , x 2 , . . . , x n ) = fn (x 1 , x 2 , . . . , x n )
d dt

if it is constant along the trajectories:

(F ) = 0.

Existence of a first integral allows to reduce the order of the system by 1.

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Important remark

According to standard rectification theorem in a neighborhood of any non-singular point of a smooth vector field there is a coordinate system such that this vector field becomes constant. But that's not the integrability we are going to discuss!

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Hamiltonian systems
A skew-symmetric tensor field ij defines a Poisson bracket
n

{f , g } =
i ,j =1



ij

f g x i x j

on a manifold M if the Jacobi identity {f , {g , h}} + {g , {h, f }} + {h, {f , g }} = 0 holds for arbitrary smooth functions f , g , h C (M ). A dynamical system x 1 = {H , x 1 }, x 2 = {H , x 2 }, . . . , x n = {H , x n } is called hamiltonian.
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Liouville's Theorem
Consider a hamiltonian system on a symplectic manifold M , 2n = dim M , and let I1 , I2 , . . . , In be it's first integrals such that each pair of them is in involution: {Ii , Ij } = 0. If a common level set Mc = {I1 = c1 , I2 = c2 , . . . , In = cn } is compact, connected and all integrals I1 , I2 , . . . , In are functionally independent on it, then Mc is diffeomorphic to n-dimensional torus and hamiltonian system can be integrated in quadratures.

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Example: finite Toda chain
The Toda chain Ё q1 = -e q1 -q2 qj = e qj -1 -qj - e Ё qn = e qn-1 -qn Ё

qj -qj

+1

,

j = 2, 3, . . . , n - 1

is hamiltionian with respect to the canonical Poisson bracket and the Hamiltomian H= where pj = qj . 1 2
n n -1

pj2 +
j =1 j =1

e

qj -qj

+1

,

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Toda chain, Lax representation
The Toda chain admits a Lax representation, i.e. it is equivalent to matrix equation: Lt = [A, L], where L=

a1 b1 . . . 0

b1 a2 .. . ...

... ... .. . bn-

0 0
1





bn- an

1

, A = 1 bj = e 2

0 -b1 . . . 0

b1 0 .. . ...

... ... .. . -bn-

0 0
1

,

bn- 0

1

1 aj = - pj , 2

j = 1, 2, . . . , n,

1 2

(qj -qj

+1

)

,

j = 1, 2, . . . , n-1.

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Toda chain: Liouville integrability
Theorem
Lax representation provides the sufficient number of first integrals in involution: Ik = tr Lk , k = 1, 2, . . . , n. The fact that I1 is a first integrals is obvious: d d (I1 ) = (tr L) = tr Lt = tr[A, L] = 0. dt dt If k > 1, then this follows from the fact that all eigenfunctions of the Lax matrix L are first integrals.

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Darboux transformations
Linear differential operators L= and dn + an dx n
-1

d n -1 d + · · · + a1 + a0 dx n-1 dx

dn d n -1 d ^ L = n + ^n-1 n-1 + · · · + ^1 a a + ^0 a dx dx dx are related by a Darboux transformation if there exists a differential operator D such that ^ L D = D L. The main property of Darboux transformations is that D maps ^ Ker L to Ker L.
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Decomposition theorem
Theorem
Any Darboux transformation of the order r can be represented as a composition of first-order Darboux transformations, i.e. if ^ L D = D L, then there exist a sequence of operators Lj , such that L0 = L, ^ Lr = L and Lj where Dj =
d dx +1

Dj = Dj Lj ,

j = 0, 1, . . . , r - 1,
r -1

+ j . In this case, D = Dr D

. . . D1 .

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Veselov-Shabat dressing chain
Consider a ShrЁ inger operator L0 = - od factorized as follows:
d2 dx 2

+ u0 (x ), it can be

+ L0 = D0 D0 - 0 , 0 = const, + where D0 = d /dx + f0 (x ) and D0 = -d /dx + f0 (x ). Define a new operator and refactorize it in the same way: + L1 = D0 D0 = -

d2 + + u1 = D1 D1 , dx 2
+1

etc. On each step operators Lj and Lj Darboux transformation: (Lj
+1

- j are related by a

- j )D

+ j

= Dj+ Lj .

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Veselov-Shabat dressing chain, continued
Periodic closure Lr = L0 of this sequence is equivalent to the following system of ODEs: (fr + f1 ) = fr2 - f12 + 1 (f1 + f2 ) = f12 - f22 + 2 . ... (fr -1 + fr ) = fr2 1 - fr2 + r - This system is called the Veselov-Shabat dressing chain. It is related to the famous KdV equation.
r

Denote =
j =1

j . Cases = 0 and = 0 are essentially

different.

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Dressing chain: Lax representation
Suppose = 0. Consider an eigenfunction 0 such that L0 0 = 0 for some R and denote j = Dj+ 1 j , - j = 1, 2, . . . , r . Hence Lj = j j , where j = - 1 - 2 - · · · - j , j = 1, 2, . . . , r .

The scalar equation -j + uj j = j j is equivalent to the matrix equation j = Uj j , where j = (j , j )T , Uj = and j = j 0 uj -
-1

j

1 0

=

0 fj + fj2 - + j

1 0

.

- j .
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Dressing chain: Lax representation, continued
The equation j +1 = Dj+ j is equivalent to the matrix equation j +1 = Wj j , where Wj = fj fj - uj + -1 fj = fj fj - uj + - j -1 fj .

j

On one hand, j
+1

= (Wj j ) = Wj j +Wj j = Wj j +Wj Uj j = Wj + Wj Uj j ,

and on the other hand, j
+1

= Uj

+1

j

+1

= Uj

+1

Wj j .

Hence, the dressing chain is equivalent to the sequence of matrix equations Wj = Uj +1 Wj - Wj Uj .
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Dressing chain: Lax representation, continued
Theorem
[Shabat, Yamilov, 1990] If = 0, then the periodic dressing chain admits a parametric dependent Lax representation, i.e. it is equivalent to the following matrix equation d W () = [U1 , W ()], dx where W () = Wn W
r -1

. . . W1 .

Due to the obvious fact that tr W () = 0, such Lax representation provides a generating function for first integrals: () = tr W () = (-1)r (I0 + I1 + I2 2 + · · · + Ir ). Note that these integrals are a direct consequence of the structure of the dressing chain related to Darboux transformations.
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Dressing chain: Liouville integrability
Theorem
[Veselov, Shabat, 1993] Dressing chain is a hamiltonian system with respect to the Poisson structure ij = (-1)i
-j (mod r )

if j = i ,

ii = 0

and with the following hamiltonian:
r

H=
j =1

13 f + j fj 3j

.

If = 0 and r is odd, then it is Liouville integrable. Integrability follows from the fact that the first integrals provided by the Lax representation appear to be in involution.
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

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 + 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

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

2D-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 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: (ln h(j ))xy = h(j + 1) - 2h(j ) + h(j - 1).

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Different forms of 2D-Toda lattice
Toda lattice can be represented in various forms in terms of different sets of variables. 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 ).

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

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. Shabat, Yamilov (1981): What exponential systems are integrable?

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

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

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

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

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Darboux integrability
Definition
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

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Generating function for y -integrals

Theorem
[S., 2015] 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

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Proof: use of DLTs
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

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

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
[S., 2015] B - and C -series Toda lattices are Darboux integrable.

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Discrete dressing chain
Consider second-order difference operators Lj = Aj A+ - j , related j by Darboux transformations (Lj
+1

- j )A+ = A+ Lj , j j

where Aj = aj (n) + bj (n)T , A+ = aj (n) + bj (n - 1)T -1 and T is a j shift operator on 1D-lattice: T (n) = (n + 1). Periodic closure of this Darboux chain is equivalent to the system of difference equations: aj2 (n) + bj2 (n) = aj2-1 (n) + bj2-1 (n - 1) + j . aj (n)bj (n - 1) = aj -1 (n - 1)bj -1 (n - 1) Darboux transformations imply a Lax representation for this system and therefore allow to obtain conserved quantities for the discrete dynamics [S., 2005].
Sergey V. Smirnov Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Darboux q-chain
In the discrete case consider a modified operator relation Lj = Aj A+ - j = qA+ 1 Aj j j-
-1

,

where 0 < q < 1, with the following closure condition: Lj
+r

=T

-r

Lj T r .

This system is called the Darboux q -chain. All operators Lj are bounded and have purely discrete spectrum that can be obtained using the Darboux scheme [Dynnikov, S., 2002, S.,2003]

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Discrete Toda lattices

In the semidiscrete case DLTs for hyperbolic operators L = x T + ax + bT + c , lead to the following differential-difference equations ln hn (j ) hn+1 (j ) = hn
x +1

(j + 1) - hn

+1

(j ) - hn (j ) + hn (j - 1), n Z.

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Discrete Toda lattices, continued
In the purely discrete case Lj = T1 T2 + a(j )T1 + b (j )T2 + c (j ), we obtain similarly the following system of PEs: hn,m+1 (j )hn hn,m (j )hn-1
- 1 ,m ,m+1

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

-1,m+1 -1,m+1

(j - 1)) . (j ))

In both case the DT function for integrals Darboux integrability reductions [S., 2012,

approach allows to obtain a generating along characteristics and therefore to prove on the A-series lattice and of some its 2015].

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Discretization using DTs

Consequent application of Darboux transformations to solutions of integrable models always produce a discrete system. In some cases it is trivial, but sometimes it is very nontrivial and inherits some properties of initial integrable system (Adler, Mikhailov, Sokolov,. . . )

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems


Outline Introduction Dynamical systems Hyp erb olic PDEs in 2D Other examples

Thank you!

Sergey V. Smirnov

Darb oux transformations in theory of integrable systems