# Integrable non-abelian Laurent ODEs T. Wolf, Brock University (Canada) May 20, 2012

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Integrable non-abelian Laurent ODEs T. Wolf, Brock University (Canada) May 20, 2012
```Integrable non-abelian Laurent ODEs
O.V. Efimovskaya, Moscow State University (Russia)
May 20, 2012
Outline
Non-commutative ODEs
First Integrals and Lax Pairs
Symmetries, pre-Hamiltonian Operators and Recursion Operators
Classifications
Computer Runs
References
Examples of integrable matrix homogeneous ODEs
Integrable systems for unknown matrices u(t) and v (t) of arbitrary
size are
ut = uv ,
vt = −vu
(1)
and
ut = v 2 ,
vt = u 2 .
(2)
Examples of integrable matrix homogeneous ODEs
Integrable systems for unknown matrices u(t) and v (t) of arbitrary
size are
ut = uv ,
vt = −vu
(1)
and
ut = v 2 ,
vt = u 2 .
(2)
Another integrable system is
ut = u 2 v − v u 2 ,
vt = 0.
If u is a skew-symmetric and v is a diagonal matrix, then this
system coincides with the n-dimensional Euler top. The
integrability of this model was established by S.V. Manakov in
1976. The simplest first integrals are given by
I2,2 = Tr (2v 2 u 2 + vuvu),
I2,3 = Tr (v 3 u 2 + v 2 uvu),
where Tr stands for the trace of the matrix.
(3)
Block Matrix Equations
The cyclic reduction

0 u1 0 0
 0 0 u2 0

·
· ·
u =
 ·
 0 0 0 0
un 0 0 0
·
0
·
0
·
·
· un−1
·
0






 , v =




0 0
J1 0
0 J2
·
·
0 0
0
0
0
·
0

·
0
Jn
·
0
0 

·
0
0 

·
·
· 
· Jn−1 0
where uk and Jk are matrices converts (3) to the matrix Volterra
equation
d
uk = uk uk+1 Jk+1 − Jk−1 uk−1 uk ,
dt
k ∈ Zk .
If we assume n = 3, J1 = J2 = J3 = Id and u3 = −u1 − u2 then
the latter system is equivalent to (2) (Mikhailov, Sokolov).
Non-abelian Laurent Polynomials
In connection with the theory of non-commutative elliptic curves,
M. Kontsevich considered the following discrete map
u → uvu −1 ,
v → u −1 + v −1 u −1 ,
(4)
where u, v are non-commutative variables. His numerical computer
experiments have shown that this map may be integrable and
possess the Laurent property. The latter means that the right hand
sides for all iterations of (4) are polynomials in u, v , u −1 , v −1 .
Non-abelian Laurent Polynomials
In connection with the theory of non-commutative elliptic curves,
M. Kontsevich considered the following discrete map
u → uvu −1 ,
v → u −1 + v −1 u −1 ,
(4)
where u, v are non-commutative variables. His numerical computer
experiments have shown that this map may be integrable and
possess the Laurent property. The latter means that the right hand
sides for all iterations of (4) are polynomials in u, v , u −1 , v −1 .
Kontsevich observed also that (4) is a discrete symmetry of the
following non-abelian ODE system:
ut = uv − uv −1 − v −1 ,
vt = −vu + vu −1 + u −1
and conjectured that (5) is integrable itself.
(5)
Non-abelian Laurent Polynomials
In connection with the theory of non-commutative elliptic curves,
M. Kontsevich considered the following discrete map
u → uvu −1 ,
v → u −1 + v −1 u −1 ,
(4)
where u, v are non-commutative variables. His numerical computer
experiments have shown that this map may be integrable and
possess the Laurent property. The latter means that the right hand
sides for all iterations of (4) are polynomials in u, v , u −1 , v −1 .
Kontsevich observed also that (4) is a discrete symmetry of the
following non-abelian ODE system:
ut = uv − uv −1 − v −1 ,
vt = −vu + vu −1 + u −1
(5)
and conjectured that (5) is integrable itself.
Non-commutative polynomials in u, v , u −1 , v −1 like the right hand
sides of (5) are called non-abelian Laurent polynomials.
Componentless Calculus
Consider two-component ODEs of the form
ut = P1 (u, v , u −1 , v −1 ),
vt = P2 (u, v , u −1 , v −1 )
(6)
on the free associative algebra M generated by two generators u
and v . Here Pi are some (non-commutative) polynomials from M.
Componentless Calculus
Consider two-component ODEs of the form
ut = P1 (u, v , u −1 , v −1 ),
vt = P2 (u, v , u −1 , v −1 )
(6)
on the free associative algebra M generated by two generators u
and v . Here Pi are some (non-commutative) polynomials from M.
I
(u −1 )t and (v −1 )t are given by
(u −1 )t = −u −1 ut u −1 ,
(v −1 )t = −v −1 vt v −1 .
Componentless Calculus
Consider two-component ODEs of the form
ut = P1 (u, v , u −1 , v −1 ),
vt = P2 (u, v , u −1 , v −1 )
(6)
on the free associative algebra M generated by two generators u
and v . Here Pi are some (non-commutative) polynomials from M.
I
(u −1 )t and (v −1 )t are given by
(u −1 )t = −u −1 ut u −1 ,
I
(v −1 )t = −v −1 vt v −1 .
f1 , f2 ∈ M are called equivalent (f1 ∼ f2 ) iff f1 can be obtained from f2 by cyclic permutations of factors in its monomials. The equivalence class of any f is denoted by Tr f .
Componentless Calculus
Consider two-component ODEs of the form
ut = P1 (u, v , u −1 , v −1 ),
vt = P2 (u, v , u −1 , v −1 )
(6)
on the free associative algebra M generated by two generators u
and v . Here Pi are some (non-commutative) polynomials from M.
I
(u −1 )t and (v −1 )t are given by
(u −1 )t = −u −1 ut u −1 ,
I
I
(v −1 )t = −v −1 vt v −1 .
f1 , f2 ∈ M are called equivalent (f1 ∼ f2 ) iff f1 can be obtained from f2 by cyclic permutations of factors in its monomials. The equivalence class of any f is denoted by Tr f .
(The transpose of a polynomial of matrices is obtained by
reversing the order of all factors in each monomial and
replacing the factors by their transpose.)
Outline
Non-commutative ODEs
First Integrals and Lax Pairs
Symmetries, pre-Hamiltonian Operators and Recursion Operators
Classifications
Computer Runs
References
First Integrals
An element I of M is called an M-integral of (6), if
dI
dt
= 0.
First Integrals
An element I of M is called an M-integral of (6), if
Example: for KS: I = uvu −1 v −1 , dI /dt = 0.
dI
dt
= 0.
First Integrals
An element I of M is called an M-integral of (6), if
Example: for KS: I = uvu −1 v −1 , dI /dt = 0.
dI
dt
= 0.
An element h of M is called a trace integral of (6), if dh
dt ∼ 0.
Trace integrals h1 and h2 are called equivalent if h1 − h2 ∼ 0.
Example: h = u + v + u −1 + v −1 + u −1 v −1 , Tr dh/dt = 0.
First Integrals
An element I of M is called an M-integral of (6), if
Example: for KS: I = uvu −1 v −1 , dI /dt = 0.
dI
dt
= 0.
An element h of M is called a trace integral of (6), if dh
dt ∼ 0.
Trace integrals h1 and h2 are called equivalent if h1 − h2 ∼ 0.
Example: h = u + v + u −1 + v −1 + u −1 v −1 , Tr dh/dt = 0.
Example: In the associative algebra Q generated by u, v with
u v = q v u, where q is a fixed constant (the “q-case”)
J = u + qv + qu −1 + v −1 + u −1 v −1
is a full integral for KS. (J is a q-deformation of h.)
Lax Pairs I
Two Laurant polynomials L, A are called a Lax pair if they satisfy
Lt = [A, L]
and if (7) is equivalent to the system (here KS).
(7)
Lax Pairs I
Two Laurant polynomials L, A are called a Lax pair if they satisfy
Lt = [A, L]
and if (7) is equivalent to the system (here KS).
In that case also Lk , A for any natural k satisfy (7).
(7)
Lax Pairs I
Two Laurant polynomials L, A are called a Lax pair if they satisfy
Lt = [A, L]
(7)
and if (7) is equivalent to the system (here KS).
In that case also Lk , A for any natural k satisfy (7).
Taking the trace of Lkt = [A, Lk ] gives
(Tr Lk )t = Tr (Lkt ) = Tr [A, Lk ] = Tr (ALk ) − Tr (Lk A) = 0
Each L gives us an infinite series of trace integrals, one for each k
even if (7) is not equivalent to the system.
Lax Pairs II
To find Lax pairs an ansatz for Laurent polynomials L, A up to
Lax Pairs II
To find Lax pairs an ansatz for Laurent polynomials L, A up to
The condition
Lt = [A, L]
leads to bi-linear polynomial (i.e. non-linear) algebraic systems for
the indefinite coefficients in L and A.
Infinite Sequences of Trace Integrals I
We found 12 pairs of polynomials L, A satisfying Lt = [A, L] and
thus providing 12 infinite sequences of trace first integrals. But
L, A were not equivalent to the Kontsevich system.
Infinite Sequences of Trace Integrals I
We found 12 pairs of polynomials L, A satisfying Lt = [A, L] and
thus providing 12 infinite sequences of trace first integrals. But
L, A were not equivalent to the Kontsevich system.
Apart from L = h (= u + v + u −1 + v −1 + u −1 v −1 ), A = −v − u −1
the Li of the other pairs are given by Li = h + ai :
a1
a2
a3
a4
a5
a6
=
=
=
=
=
=
[u −1 , vu]
[v , u −1 v −1 ]
[v , uv −1 ]
[u −1 , v −1 u]
a1 + a4 ,
a2 + a3 ,
a7
a8
a9
a10
a11
=
=
=
=
=
a2 + a4 + vu −1 v −1 u,
a2 + vu −1 v −1 u + [v , u −1 v 2 ],
a4 + vu −1 v −1 u + [u −1 , u −1 v −1 u],
u −1 v −1 uv + [v −1 , uv ],
uvu −1 v −1 + [u, vu −1 ]
Infinite Sequences of Trace Integrals II
Due to the discrete symmetry (inversion) of the Kontsevich system
u ↔ v , t ↔ −t this inversion applied to Li , Ai gives 12 new pairs
L̄i , Āi .
Infinite Sequences of Trace Integrals II
Due to the discrete symmetry (inversion) of the Kontsevich system
u ↔ v , t ↔ −t this inversion applied to Li , Ai gives 12 new pairs
L̄i , Āi .
For each pair L, A and any invertable Laurent polynomial f a new
pair L̂, Â is defined by conjugation
L̂ = fLf −1
Â = fAf −1 + ft f −1
satisfying L̂t = [Â, L̂].
But, two L, L̂ related by conjugation give the same trace integrals.
Infinite Sequences of Trace Integrals II
Due to the discrete symmetry (inversion) of the Kontsevich system
u ↔ v , t ↔ −t this inversion applied to Li , Ai gives 12 new pairs
L̄i , Āi .
For each pair L, A and any invertable Laurent polynomial f a new
pair L̂, Â is defined by conjugation
L̂ = fLf −1
Â = fAf −1 + ft f −1
satisfying L̂t = [Â, L̂].
But, two L, L̂ related by conjugation give the same trace integrals.
Identification through conjugation: 24 Li , Ai → 8 groups
up to inversion
→ 4 groups
represented by h, h + a1 , h + a7 , h + a11 .
Infinite Sequences of Trace Integrals III
Another infinite sequence of first integrals:
If P, Q ∈ M satisfy
Dt (P) = [I , Q],
where dI /dt = 0 then PI k are trace integrals ∀k.
(8)
Infinite Sequences of Trace Integrals III
Another infinite sequence of first integrals:
If P, Q ∈ M satisfy
Dt (P) = [I , Q],
where dI /dt = 0 then PI k are trace integrals ∀k.
trivial P: P = [I , R] for some R as Tr(PI k ) = 0.
(8)
Infinite Sequences of Trace Integrals III
Another infinite sequence of first integrals:
If P, Q ∈ M satisfy
Dt (P) = [I , Q],
(8)
where dI /dt = 0 then PI k are trace integrals ∀k.
trivial P: P = [I , R] for some R as Tr(PI k ) = 0.
non-trivial P:
P = uuvu −1 v −1 + uv vu −1 v −1 + uv u −1 u −1 v −1 + uvu −1 v −1 v −1
+u −1 v −1 uvu −1 v −1 .
Lax Pairs III
A proper Lax pair composed of 2×2 matrices L, A was finally
found:
−1
v +u
v λ + (v −1 u −1 + u −1 + 1)
,
L =
v −1 u −1 + u −1 + v + λ1
v −1 + u λ1
−1
v − v + u λv
A =
v −1
u
where λ is a free spectral parameter.
All Trace First Integrals
m\d
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0
?0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
?
◦
•◦
◦
•◦
?
◦
•◦
◦
?
◦
•◦
?
◦
?
?
?
◦−1
•+1
?
◦
•
?
◦
•◦−2
?
◦
•◦
•+2
•
?
◦
•◦
◦−3
•
?
◦
•◦
◦
•
•+3
?
◦
•◦
◦
•◦−4
•
•
?
◦
•◦
◦
•◦
Table: # of new trace first integrals of degree d generated from TrLm
that are not generated from TrLk , k < m.
Outline
Non-commutative ODEs
First Integrals and Lax Pairs
Symmetries, pre-Hamiltonian Operators and Recursion Operators
Classifications
Computer Runs
References
Symmetries I
The system
uτ = Q1 (u, v , u −1 , v −1 ),
vτ = Q2 (u, v , u −1 , v −1 )
(9)
is called an (infinitesimal) symmetry for
ut = P1 (u, v , u −1 , v −1 ),
vt = P2 (u, v , u −1 , v −1 )
if the flows (10) and (9) commute:
uτ t = utτ ,
vτ t = vtτ .
The vector (Q1 , Q2 )t is called generator of the symmetry.
(10)
Symmetries II
Infinitely many symmetries → indicator of integrability
Symmetries II
Infinitely many symmetries → indicator of integrability
Example:
uτ
=
−uvu − uv 2 + uv + (vu)−1 + v −2 + u 2 v −1 − uvu −1
+uv −2 + (vuv )−1 + u(vuv )−1 ,
vτ
=
vuv + vu 2 − vu − (uv )−1 − u −2 − v 2 u −1 + vuv −1
−vu −2 − (uvu)−1 − v (uvu)−1 .
Symmetries II
Infinitely many symmetries → indicator of integrability
Example:
uτ
=
−uvu − uv 2 + uv + (vu)−1 + v −2 + u 2 v −1 − uvu −1
+uv −2 + (vuv )−1 + u(vuv )−1 ,
vτ
=
vuv + vu 2 − vu − (uv )−1 − u −2 − v 2 u −1 + vuv −1
−vu −2 − (uvu)−1 − v (uvu)−1 .
Conjecture:
dim S4k
=
2k 2 ,
dim S4k+1
=
2k 2 + 2k,
dim S4k+2
=
2k 2 + 2k + 1,
dim S4k+3
=
2k 2 + 4k + 1.
Direct Symmetry Computations I
For Dt defined by
Dt u = uv − uv −1 − v −1 ,
Dt v = −vu + vu −1 + u −1
find polynomials Q1 , Q2 of u, v , u −1 , v −1 such that
Dτ u = Q1 ,
Dτ v = Q2
commutes with Dt :
[Dt , Dτ ]u = 0,
[Dt , Dτ ]v = 0.
(11)
Direct Symmetry Computations II
Shorter necessary conditions:
Direct Symmetry Computations II
Shorter necessary conditions:
As mentioned before I = uvu −1 v −1 is a first integral for KS and
thus I k especially I −1 = vuv −1 u −1 are first integrals.
Direct Symmetry Computations II
Shorter necessary conditions:
As mentioned before I = uvu −1 v −1 is a first integral for KS and
thus I k especially I −1 = vuv −1 u −1 are first integrals.
Furthermore, I k , k = 0, ±1, ±2, . . . are the only full first integrals
up to degree 14.
Direct Symmetry Computations II
Shorter necessary conditions:
As mentioned before I = uvu −1 v −1 is a first integral for KS and
thus I k especially I −1 = vuv −1 u −1 are first integrals.
Furthermore, I k , k = 0, ±1, ±2, . . . are the only full first integrals
up to degree 14.
A necessary condition for Dτ being a symmetry is
Dt Dτ I = Dτ Dt I = 0 → Dτ I =
k0
X
ak I k , k ∈ Z
k=−k0
and similarly
Dτ I
−1
=
k0
X
bk I k , k ∈ Z
k=−k0
for sufficiently high k0 . These conditions involve extra unknown
constants ak , bk but these are first order conditions involving fewer
terms than the full symmetry condition.
Direct Symmetry Computations III
In utilizing the shortcut we calculated all symmetries, where Qi are
polynomials of degree up to 16 in u, v , u −1 , v −1 .
Direct Symmetry Computations III
In utilizing the shortcut we calculated all symmetries, where Qi are
polynomials of degree up to 16 in u, v , u −1 , v −1 .
The general ansatz for such a symmetry contains more then 172
million unknown coefficients satisfying over 1 billion linear
conditions. The set of all such symmetries forms a commutative
Lie algebra of dimension 32.
Pre-Hamiltonian Operators I
A pre-Hamiltonian operator P is any 2 × 2 matrix that satisfies
Pt = F∗ P + PF∗?
where F∗ is the Frechet derivative (linearization) of the right hand
side of the Kontsevich system and superscript ? stands for the
< (x, y ), (x, y ) > = Tr (x 2 ) + Tr (y 2 ).
Pre-Hamiltonian Operators I
A pre-Hamiltonian operator P is any 2 × 2 matrix that satisfies
Pt = F∗ P + PF∗?
where F∗ is the Frechet derivative (linearization) of the right hand
side of the Kontsevich system and superscript ? stands for the
< (x, y ), (x, y ) > = Tr (x 2 ) + Tr (y 2 ).
Pre-Hamiltonian operators map gradients of trace integrals like
σ = Tr (Lki ) to symmetries.
T
(u, v )T
(12)
Pre-Hamiltonian Operators II
Such operators need in general to multiply from the left and the
right, therefore the following definitions.
Pre-Hamiltonian Operators II
Such operators need in general to multiply from the left and the
right, therefore the following definitions.
For any a ∈ M we denote by La and Ra the operators of left and
right multiplications:
La (b) = a ◦ b,
Ra (b) = b ◦ a .
Lu , Ru , Lv , Rv generate the associative algebra O.
Pre-Hamiltonian Operators III
The operator
Ru Ru − Lu Lu ,
Lu Lv + Lu Rv − Lv Ru + Ru Rv
P=
Lu Rv − Lv Lu − Lv Ru − Rv Ru ,
Lv Lv − Rv Rv
is a pre-Hamiltonian operator for KS.
Pre-Hamiltonian Operators III
The operator
Ru Ru − Lu Lu ,
Lu Lv + Lu Rv − Lv Ru + Ru Rv
P=
Lu Rv − Lv Lu − Lv Ru − Rv Ru ,
Lv Lv − Rv Rv
is a pre-Hamiltonian operator for KS.
Lax pair L, A → trace integrals Tr Tr Ln + pre-Hamiltonian →
symmetries
Pre-Hamiltonian Operators III
The operator
Ru Ru − Lu Lu ,
Lu Lv + Lu Rv − Lv Ru + Ru Rv
P=
Lu Rv − Lv Lu − Lv Ru − Rv Ru ,
Lv Lv − Rv Rv
is a pre-Hamiltonian operator for KS.
Lax pair L, A → trace integrals Tr Tr Ln + pre-Hamiltonian →
symmetries
→ confirmation that the table of first integrals is complete (at
least up to degree 14).
Recursion Operators I
Recursion operators are operators that acting on a symmetry
generate a new (higher degree) symmetry.
Recursion Operators I
Recursion operators are operators that acting on a symmetry
generate a new (higher degree) symmetry.
A 2 × 2 matrix R, whose entries belong to O is called a recursion
operator if for any symmetry generator Q = (Q1 , Q2 )t (therefore
satisfying Dt Q = F∗ (Q)) also the vector RQ is a symmetry
generator and therefore satisfying
Dt (RQ) = F∗ (RQ),
i.e.
(Rt + RF∗ − F∗ R)Q = 0
for any symmetry Q.
Recursion Operators II
Example: For
ut = uv ,
vt = −vu
a recursion operator is given by
Ru + Rv ,
0
R=
.
0,
Ru + Rv
Recursion Operators III
Example: For the Kontsevich system a recursion operator is given
as follows.
Recursion Operators III
Example: For the Kontsevich system a recursion operator is given
as follows.
At first, a non-local vector B = (B1 , B2 )T is defined through
Pτ
= [I , B1 ]
Pτ
= [I −1 , B2 ]
with I , P from previous slides and P from involution
u ↔ v.
Recursion Operators IV
If G defines a symmetry through
dxα
= Gα (x1 , ..., xN , x1−1 , ..., xN−1 ),
dτ
Gα ∈ M
then a new symmetry of degree 2 higher than the degree of G is
given by
Rn B + Rl G
with B from the previous slide and
Lu − Ru Lu − Ru
Rn =
Lv − Rv Lv − Rv
Rl =
−Rv − Ru − Lu −1 − Ru −1 Rv −1 − Lu Lv −1 Lu −1 (1 + Ru −1 + Rv ) + Lu −1 Ru −1 Rv −1 Ru ,
Lu −1 Lv −1 Rv −1 Ru − Lu Lv −1 Rv −1 Ru −1 + Lu Lv −1 − Lv −1 Ru
Lv −1 Lu −1 Ru −1 Rv − Lv Lu −1 Ru −1 Rv −1 + Lv Lu −1 − Lu −1 Rv ,
−Ru − Rv − Lv −1 − Rv −1 Ru −1 − Lv Lu −1 Lv −1 (1 + Rv −1 + Ru ) + Lv −1 Rv −1 Ru −1 Rv
!
Outline
Non-commutative ODEs
First Integrals and Lax Pairs
Symmetries, pre-Hamiltonian Operators and Recursion Operators
Classifications
Computer Runs
References
Classification Problem
After a proper scaling of u, v and t the Kontsevich system can be
rewritten as
ut = uv − ε2 uv −1 − ε3 v −1 ,
vt = −vu + ε2 vu −1 + ε3 u −1 . (13)
This system can be regarded as a Laurent deformation of the
Mikhailov-Sokolov system
ut = uv ,
vt = −vu.
Classification Problem
After a proper scaling of u, v and t the Kontsevich system can be
rewritten as
ut = uv − ε2 uv −1 − ε3 v −1 ,
vt = −vu + ε2 vu −1 + ε3 u −1 . (13)
This system can be regarded as a Laurent deformation of the
Mikhailov-Sokolov system
ut = uv ,
vt = −vu.
We found several more Laurent deformations of this system having
infinitesimal symmetries.
Further Laurent Deformations
Theorem. All systems of the form:
ut = uv + P1 (u, u −1 , v , v −1 ),
vt = −vu + Q1 (u, u −1 , v , v −1 ),
where P1 , Q1 are the most general inhomogeneous degree 2
polynomials without quadratic term in u, v
Further Laurent Deformations
Theorem. All systems of the form:
ut = uv + P1 (u, u −1 , v , v −1 ),
vt = −vu + Q1 (u, u −1 , v , v −1 ),
where P1 , Q1 are the most general inhomogeneous degree 2
polynomials without quadratic term in u, v which have commuting
flows of the form
uτ = P2 (u, u −1 , v , v −1 ),
vτ = Q2 (u, u −1 , v , v −1 ),
where P2 , Q2 are the most general inhomogeneous degree 4
polynomials,
Further Laurent Deformations
Theorem. All systems of the form:
ut = uv + P1 (u, u −1 , v , v −1 ),
vt = −vu + Q1 (u, u −1 , v , v −1 ),
where P1 , Q1 are the most general inhomogeneous degree 2
polynomials without quadratic term in u, v which have commuting
flows of the form
uτ = P2 (u, u −1 , v , v −1 ),
vτ = Q2 (u, u −1 , v , v −1 ),
where P2 , Q2 are the most general inhomogeneous degree 4
polynomials, are shown in the following list:
A Listing of Laurent Deformations I
uv + a1 uv −1 + a2 v −1
ut
=
vt
= −vu + b1 u −1 v − a2 u −1 ,
ut
=
vt
= −vu + a2 u −1 v ,
ut
=
vt
= −vu + a1 u −1 v − a1 ,
ut
=
vt
= −vu + b1 vu −1 + a1 ,
uv + a1 v −1 u
uv − a1 v −1 u + a1
uv + a1 uv −1 + b1
A Listing of Laurent Deformations II
ut
=
vt
=
uv + a1 uv −1 + a2 u + a3
−vu − a2 u + a3 vu −1 + b1 u −1 v − a1 v −1 u + (a2 a3 + a2 b1 )u −1
+(a1 a3 + a1 b1 )u −1 v −1 + (a32 + a3 b1 )u −2 + b2 ,
ut
=
uv + a1 u + a2 v + a3
vt
=
−vu − a1 u − a2 v − a3 ,
ut
=
uv + 2a1 uv −1 − a1 v −1 u − a1
vt
=
−vu − 2a1 vu −1 + a1 u −1 v + a1 ,
ut
=
uv + a1 uv −1 − a2 v −1 + a3 u
vt
=
−vu + b1 vu −1 + a2 u −1 + b2 v ,
Outline
Non-commutative ODEs
First Integrals and Lax Pairs
Symmetries, pre-Hamiltonian Operators and Recursion Operators
Classifications
Computer Runs
References
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Identifying L, A pairs through conjugation → many medium
linear systems.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Identifying L, A pairs through conjugation → many medium
linear systems.
Computing symmetries → large linear systems.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Identifying L, A pairs through conjugation → many medium
linear systems.
Computing symmetries → large linear systems.
Computing a pre-Hamiltonian structure → small linear
systems.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Identifying L, A pairs through conjugation → many medium
linear systems.
Computing symmetries → large linear systems.
Computing a pre-Hamiltonian structure → small linear
systems.
Computation of recursion operator → medium linear systems.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Identifying L, A pairs through conjugation → many medium
linear systems.
Computing symmetries → large linear systems.
Computing a pre-Hamiltonian structure → small linear
systems.
Computation of recursion operator → medium linear systems.
Application of recursion operator → medium linear systems.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Identifying L, A pairs through conjugation → many medium
linear systems.
Computing symmetries → large linear systems.
Computing a pre-Hamiltonian structure → small linear
systems.
Computation of recursion operator → medium linear systems.
Application of recursion operator → medium linear systems.
2. Classification problem.
Computational Problems
We have two classes of problems.
1. Integrability of a given equation.
Computing first integrals → large linear systems.
Searching for a Lax-pair → bi-linear inhomogeneous systems.
Identifying L, A pairs through conjugation → many medium
linear systems.
Computing symmetries → large linear systems.
Computing a pre-Hamiltonian structure → small linear
systems.
Computation of recursion operator → medium linear systems.
Application of recursion operator → medium linear systems.
2. Classification problem.
In this case we need to solve bi-linear algebraic systems to find
systems with symmetries, both at the same time.
Direct Symmetry Computations, the Conditions:
N
v
e1
t1
e2
t2
4
322
430
616
1,412
3,706
5
970
1,294
1,904
4,448
12,914
6
2,914
3,886
5,784
13,878
44,098
7
8,746
11,662
17,440
43,052
148,346
8
26,242
34,990
52,424
132,954
493,162
9
78,730
104,974
157,392
409,470
1,623,842
10
236,194
314,926
472,312
1,258,526
5,304,562
11
708,586
944,782 1,417,088
3,862,086 17,212,778
12 2,125,762 2,834,350 4,251,432 11,835,758 55,535,578
13 6,377,290 8,503,054 12,754,480 36,228,892 178,298,450
14 19,131,874 25,509,166 38,263,640 110,777,292 569,970,466
15 57,395,626 76,527,502 > 1.1×108 > 3.3 × 108 > 1.7 × 109
16 > 172×106 > 229×106 > 3.4×108
> 1×109 > 5.7×109
p
2
4
5
7
8
12
13
17
18
24
25
31
32
Table: For a symmetry ansatz of degree N are listed the # v of variables,
the # e1 of equations and # t1 of terms of system Dτ (I ) = 0 and the #
e2 of equations and # t2 of terms of system [Dt , Dτ ](u, v ) = 0 and the
# p of free parameters of the solution.
Quantitative Progress
N
3
4
5
6
7
8
9
10
11
12
13
stream solve
solve
subst.
.02
.01
.14
.01
1.58
.17
20.79
1.46
205.73
8.97
2,742.27
82.50
53,935.88 1,191.47
sorting by size,
stream solve
solve
subst.
.01
.00
.15
.02
1.29
.23
15.30
2.36
85.16
9.00
807.62
92.77
13,335.13 1,481.95
1-term equ. (A),
1-term equ. (B),
sorting by size,
sorting by size,
stream solve
stream solve
solve
subst.
solve
subst.
.00
.00
.00
.00
.02
.02
.01
.00
.20
.16
.01
.00
1.72
1.31
.07
.10
16.25
15.43
.16
.03
240.78
222.81
0.63
0.12
4,645.20 2,674.81
2.10
0.42
60,141.15 20,356.36
7.22
1.23
26.67
4.24
91.27
14.22
402.70
50.80
Table: Times in sec for solving the symmetry conditions
[Dt , Dτ ](u, v ) = 0 for each degree N by different methods
Comparison with other Programs
N
3
4
5
6
7
8
9
10
11
12
13
Maple 14
LinBox
LinSolve
solve
total
default
sparse
(Reduce)
Sym+FI
Sym Sym+FI
Sym Sym+FI
Sym Sym+FI
Sym Sym+FI
.03
.03
.06
.06
.00
.09
.10
.19
.22
.02
.02
.02
.01
.30
.31
.63
.70
.12
.13
.01
1.07
.96
7.81
2.04
30.6
1.1
.90
.07
6.01
6.88
11.02
13.81
14.9
12.9
.16
21.55
17.23
47.28
35.65
3080
283.5
210
.63
78.34
69.08 154.40 131.90
2318
1812
2.10
312.3
273.1
587.6
508.4
21610
21210
7.22
1237
1127
2262
2015
26.67
91.27
402.70
Table: Times in sec for solving the symmetry conditions
[Dt , Dτ ](u, v ) = 0 for each degree N by different programs
Iterating the Formulation and Solution of Systems I
formulation of ansatz for Dτ
computation of Dτ−1
5 × computing + splitting
Dτ (I ) = 0, [Dt , Dτ ](u) = 0,
Dτ (I ) = 0, [Dt , Dτ ](v ) = 0,
each time only extracting and
using 1-term conditions
computation and splitting
of Dτ (I ) = 0, extracting and
using 1-term conditions
computation and splitting
of [Dt , Dτ ](u) = 0
extracting and using
1-term conditions,
keeping others
computation and splitting
of [Dt , Dτ ](v ) = 0
complete solution of all
remaining equations
[Dt , Dτ ](u, v ) = 0 [Dt , Dτ ](u, v ) = 0 Dτ (I ) = 0 Iteration
at once
in two stages
first
first
61
61
61
61
122
122
122
122
−
−
−
587
−
−
391
1
1216
1216
235
2
−
536
22
1
1216
63
2
2
365
253
241
216
Table: Times in sec of the whole symmetry computation for N = 13
Iterating the Formulation and Solution of Systems II
(continuation of the table above:)
[Dt , Dτ ](u, v ) = 0 [Dt , Dτ ](u, v ) = 0 Dτ (I ) = 0 Iteration
at once
in two stages
first
first
substitution of solution
in Dτ (u, v )
total CPU time
total garbage collection time
overall time
57
3037
578
3615
33
2284
210
2494
31
1105
9
1114
Table: Times in sec of the whole symmetry computation for N = 13
26
1018
16
1034
Outline
Non-commutative ODEs
First Integrals and Lax Pairs
Symmetries, pre-Hamiltonian Operators and Recursion Operators
Classifications
Computer Runs
References
References
1. A.V. Mikhailov and V.V. Sokolov, Integrable ODEs on
Associative Algebras, Comm in Math Phys. 211 (2000), 231-251.
2. O.V. Efimovskaya, Integrable cubic ODEs on Associative
Algebras, Fundamentalnaya i Prikladnaya Matematika 8, no. 3
(2002), 705-720
3. T. Wolf, An online tutorial for the package Crack,
http://lie.math.brocku.ca/crack/demo (2004)
4. T. Wolf, Applications of Crack in the Classification of
Integrable Systems, CRM Proceedings and Lecture Notes, Centre
de Recherches Mathematiques/Montreal, 37, (2004), 283-300,
also preprint on arXiv: nlin.SI/0301032
5. T. Wolf and O.V. Efimovskaya, On integrability of the
Kontsevich non-abelian ODE system, Lett. in Math. Phys., vol
100, no 2 (2012), p 161-170, DOI:10.1007/s11005-011-0527-4
6. P.J. Olver and V.V. Sokolov, V. V., Integrable evolution
equations on associative algebras, Comm. in Math. Phys., 1998,
193, no.2, 245-268.
7. P.J. Olver, Applications of Lie Groups to Differential Equations,
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