About Integrable Non-Abelian Laurent ODEs

Transcription

1 About Integrable Non-Abelian Laurent ODEs T. Wolf, Brock University September 12, 2013

2 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

3 Examples of integrable matrix homogeneous ODEs Integrable systems for unknown matrices u(t) and v(t) of arbitrary size are u t = uv, v t = vu (1) and u t = v 2, v t = u 2. (2)

4 Examples of integrable matrix homogeneous ODEs Integrable systems for unknown matrices u(t) and v(t) of arbitrary size are u t = uv, v t = vu (1) and Another integrable system is u t = v 2, v t = u 2. (2) u t = u 2 v v u 2, v t = 0. (3) 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 The simplest first integrals are given by I 2,2 = Tr (2v 2 u 2 + vuvu), I 2,3 = Tr (v 3 u 2 + v 2 uvu), where Tr stands for the trace of the matrix.

5 Block Matrix Equations The cyclic reduction 0 u u u = u n 1, v = u n J n J J J n 1 0 where u k and J k are matrices converts (3) to the matrix Volterra equation d dt u k = u k u k+1 J k+1 J k 1 u k 1 u k, k Z k. If we assume n = 3, J 1 = J 2 = J 3 = Id and u 3 = u 1 u 2 then the latter system is equivalent to (2) (Mikhailov, Sokolov).

6 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.

7 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: u t = uv uv 1 v 1, v t = vu + vu 1 + u 1 (5) and conjectured that (5) is integrable itself.

8 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: u t = uv uv 1 v 1, v t = 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.

9 Componentless Calculus Consider two-component ODEs of the form u t = P 1 (u, v, u 1, v 1 ), v t = P 2 (u, v, u 1, v 1 ) (6) on the free associative algebra M generated by two generators u and v. Here P i are some (non-commutative) polynomials from M.

10 Componentless Calculus Consider two-component ODEs of the form u t = P 1 (u, v, u 1, v 1 ), v t = P 2 (u, v, u 1, v 1 ) (6) on the free associative algebra M generated by two generators u and v. Here P i are some (non-commutative) polynomials from M. (u 1 ) t and (v 1 ) t are given by (u 1 ) t = u 1 u t u 1, (v 1 ) t = v 1 v t v 1.

11 Componentless Calculus Consider two-component ODEs of the form u t = P 1 (u, v, u 1, v 1 ), v t = P 2 (u, v, u 1, v 1 ) (6) on the free associative algebra M generated by two generators u and v. Here P i are some (non-commutative) polynomials from M. (u 1 ) t and (v 1 ) t are given by (u 1 ) t = u 1 u t u 1, (v 1 ) t = v 1 v t v 1. f 1, f 2 M are called equivalent (f 1 f 2 ) iff f 1 can be obtained from f 2 by cyclic permutations of factors in its monomials. The equivalence class of any f is denoted by Tr f.

12 Componentless Calculus Consider two-component ODEs of the form u t = P 1 (u, v, u 1, v 1 ), v t = P 2 (u, v, u 1, v 1 ) (6) on the free associative algebra M generated by two generators u and v. Here P i are some (non-commutative) polynomials from M. (u 1 ) t and (v 1 ) t are given by (u 1 ) t = u 1 u t u 1, (v 1 ) t = v 1 v t v 1. f 1, f 2 M are called equivalent (f 1 f 2 ) iff f 1 can be obtained from f 2 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.)

13 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

14 First Integrals An element I of M is called an M-integral of (6), if di dt = 0.

15 First Integrals An element I of M is called an M-integral of (6), if di dt = 0. Example: for the Kontsevich system (KS): I = uvu 1 v 1, di /dt = 0.

16 First Integrals An element I of M is called an M-integral of (6), if di dt = 0. Example: for the Kontsevich system (KS): I = uvu 1 v 1, di /dt = 0. An element h of M is called a trace integral of (6), if dh dt 0. Trace integrals h 1 and h 2 are called equivalent if h 1 h 2 0. Example: h = u + v + u 1 + v 1 + u 1 v 1, Tr dh/dt = 0.

17 First Integrals An element I of M is called an M-integral of (6), if di dt = 0. Example: for the Kontsevich system (KS): I = uvu 1 v 1, di /dt = 0. An element h of M is called a trace integral of (6), if dh dt 0. Trace integrals h 1 and h 2 are called equivalent if h 1 h 2 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.)

18 Lax Pairs I Two Laurant polynomials L, A are called a Lax pair if they satisfy L t = [A, L] (7) and if (7) is equivalent to the system (here KS).

19 Lax Pairs I Two Laurant polynomials L, A are called a Lax pair if they satisfy L t = [A, L] (7) and if (7) is equivalent to the system (here KS). In that case also L k, A for any natural k satisfy (7).

20 Lax Pairs I Two Laurant polynomials L, A are called a Lax pair if they satisfy L t = [A, L] (7) and if (7) is equivalent to the system (here KS). In that case also L k, A for any natural k satisfy (7). Taking the trace of L k t = [A, L k ] gives (Tr L k ) t = Tr (L k t) = Tr [A, L k ] = Tr (AL k ) Tr (L k 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.

21 Lax Pairs II To find Lax pairs an ansatz for Laurent polynomials L, A up to some degree has been made.

22 Lax Pairs II To find Lax pairs an ansatz for Laurent polynomials L, A up to some degree has been made. The condition L t = [A, L] leads to bi-linear polynomial (i.e. non-linear) algebraic systems for the indefinite coefficients in L and A.

23 Infinite Sequences of Trace Integrals I We found 12 pairs of polynomials L, A satisfying L t = [A, L] and thus providing 12 infinite sequences of trace first integrals. But L, A were not equivalent to the Kontsevich system.

24 Infinite Sequences of Trace Integrals I We found 12 pairs of polynomials L, A satisfying L t = [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 L i of the other pairs are given by L i = h + a i : a 1 = [u 1, vu] a 7 = a 2 + a 4 + vu 1 v 1 u, a 2 = [v, u 1 v 1 ] a 8 = a 2 + vu 1 v 1 u + [v, u 1 v 2 ], a 3 = [v, uv 1 ] a 9 = a 4 + vu 1 v 1 u + [u 1, u 1 v 1 u], a 4 = [u 1, v 1 u] a 10 = u 1 v 1 uv + [v 1, uv], a 5 = a 1 + a 4, a 11 = uvu 1 v 1 + [u, vu 1 ] a 6 = a 2 + a 3,

25 Infinite Sequences of Trace Integrals II Due to the discrete symmetry (inversion) of the Kontsevich system u v, t t this inversion applied to L i, A i gives 12 new pairs L i, Ā i.

26 Infinite Sequences of Trace Integrals II Due to the discrete symmetry (inversion) of the Kontsevich system u v, t t this inversion applied to L i, A i gives 12 new pairs L i, Ā i. For each pair L, A and any invertable Laurent polynomial f a new pair ˆL, Â is defined by conjugation ˆL = flf 1 Â = faf 1 + f t f 1 satisfying ˆL t = [Â, ˆL]. But, two L, ˆL related by conjugation give the same trace integrals.

27 Infinite Sequences of Trace Integrals II Due to the discrete symmetry (inversion) of the Kontsevich system u v, t t this inversion applied to L i, A i gives 12 new pairs L i, Ā i. For each pair L, A and any invertable Laurent polynomial f a new pair ˆL, Â is defined by conjugation ˆL = flf 1 Â = faf 1 + f t f 1 satisfying ˆL t = [Â, ˆL]. But, two L, ˆL related by conjugation give the same trace integrals. Identification through conjugation: 24 L i, A i 8 groups up to inversion 4 groups represented by h, h + a 1, h + a 7, h + a 11.

28 Infinite Sequences of Trace Integrals III Another infinite sequence of first integrals: If P, Q M satisfy D t (P) = [I, Q], (8) where di /dt = 0 then not only P is a trace integral but also PI k are trace integrals k.

29 Infinite Sequences of Trace Integrals III Another infinite sequence of first integrals: If P, Q M satisfy D t (P) = [I, Q], (8) where di /dt = 0 then not only P is a trace integral but also PI k are trace integrals k. trivial P: P = [I, R] for some R as Tr(PI k ) = 0.

30 Infinite Sequences of Trace Integrals III Another infinite sequence of first integrals: If P, Q M satisfy D t (P) = [I, Q], (8) where di /dt = 0 then not only P is a trace integral but also 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 + uvvu 1 v 1 + uvu 1 u 1 v 1 + uvu 1 v 1 v 1 +u 1 v 1 uvu 1 v 1.

31 Infinite Sequences of Trace Integrals III Another infinite sequence of first integrals: If P, Q M satisfy D t (P) = [I, Q], (8) where di /dt = 0 then not only P is a trace integral but also 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 + uvvu 1 v 1 + uvu 1 u 1 v 1 + uvu 1 v 1 v 1 +u 1 v 1 uvu 1 v 1. which is a merger of the full first integral and the trace first integral I = uvu 1 v 1 h = u + v + u 1 + v 1 + u 1 v 1.

32 Lax Pairs III A proper Lax pair composed of 2 2 matrices L, A was finally found: ( v L = 1 + u vλ + (v 1 u 1 + u 1 ) + 1) v 1 + u 1 λ v 1 u 1 + u 1 + v + 1, λ ( v A = 1 ) v + u λv v 1 u where λ is a free spectral parameter.

33 All Trace First Integrals m\d Table: # of new trace first integrals of degree d generated from TrL m that are not generated from TrL k, k < m.

34 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

35 Symmetries I The system u τ = Q 1 (u, v, u 1, v 1 ), v τ = Q 2 (u, v, u 1, v 1 ) (9) is called an (infinitesimal) symmetry for u t = P 1 (u, v, u 1, v 1 ), v t = P 2 (u, v, u 1, v 1 ) (10) if the flows (10) and (9) commute: u τt = u tτ, v τt = v tτ. The vector (Q 1, Q 2 ) t is called generator of the symmetry.

36 Symmetries II Infinitely many symmetries indicator of integrability

37 Symmetries II Infinitely many symmetries indicator of integrability Example: u τ = uvu uv 2 + uv + (vu) 1 + v 2 + u 2 v 1 uvu 1 +uv 2 + (vuv) 1 + u(vuv) 1, v τ = vuv + vu 2 vu (uv) 1 u 2 v 2 u 1 + vuv 1 vu 2 (uvu) 1 v(uvu) 1.

38 Symmetries II Infinitely many symmetries indicator of integrability Example: u τ = uvu uv 2 + uv + (vu) 1 + v 2 + u 2 v 1 uvu 1 +uv 2 + (vuv) 1 + u(vuv) 1, v τ = vuv + vu 2 vu (uv) 1 u 2 v 2 u 1 + vuv 1 vu 2 (uvu) 1 v(uvu) 1. Conjecture: The dimension of the vector space S i of symmetries of degree i is dim S 4k = 2k 2, dim S 4k+1 = 2k 2 + 2k, dim S 4k+2 = 2k 2 + 2k + 1, dim S 4k+3 = 2k 2 + 4k + 1.

39 Direct Symmetry Computations I For D t defined by D t u = uv uv 1 v 1, D t v = vu + vu 1 + u 1 find polynomials Q 1, Q 2 of u, v, u 1, v 1 such that D τ u = Q 1, D τ v = Q 2 commutes with D t : [D t, D τ ]u = 0, [D t, D τ ]v = 0. (11)

40 Direct Symmetry Computations II Shorter necessary conditions:

41 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.

42 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.

43 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 D τ being a symmetry is and similarly D t D τ I = D τ D t I = 0 D τ I = D τ I 1 = k 0 k 0 k= k 0 b k I k, k Z k= k 0 a k I k, k Z for sufficiently high k 0. These conditions involve extra unknown constants a k, b k but these are first order conditions involving fewer terms than the full symmetry condition.

44 Direct Symmetry Computations III In utilizing the shortcut we calculated all symmetries, where Q i are polynomials of degree up to 16 in u, v, u 1, v 1.

45 Direct Symmetry Computations III In utilizing the shortcut we calculated all symmetries, where Q i 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.

46 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

47 The Algebra O of local Operators We will need to multiply from the left and the right, therefore the following definitions.

48 The Algebra O of local Operators We will need to multiply from the left and the right, therefore the following definitions. For any a M we denote by L a and R a the operators of left and right multiplications: L a (b) = a b, R a (b) = b a. L u, R u, L v, R v generate the associative algebra O.

49 Frechet Derivative For any element a = a(x) M the Frechet derivative a is an 1 N-matrix a defined through: d dɛ a(x + ɛ δx) ɛ=0 = a (δx). (12)

50 Frechet Derivative For any element a = a(x) M the Frechet derivative a is an 1 N-matrix a defined through: d dɛ a(x + ɛ δx) ɛ=0 = a (δx). (12) For example, for h = u + v + u 1 + v 1 + u 1 v 1 we have h (δx) = δu u 1 (δu)u 1 u 1 (δu)u 1 v 1 + δv +.. h = ( 1 L u 1R u 1 L u 1R v 1R u 1,.. ).

51 Frechet Derivative For any element a = a(x) M the Frechet derivative a is an 1 N-matrix a defined through: d dɛ a(x + ɛ δx) ɛ=0 = a (δx). (12) For example, for h = u + v + u 1 + v 1 + u 1 v 1 we have h (δx) = δu u 1 (δu)u 1 u 1 (δu)u 1 v 1 + δv +.. h = ( 1 L u 1R u 1 L u 1R v 1R u 1,.. ). In other words, if u satisfies u t = F then the t derivative of any a M is given by D t (a) = a (F). (13)

52 Gradients The gradient grad (a) is the vector uniquely defined by: d dɛ a(x + ɛ δx) ɛ=0 = a (δx) δx, grad (a(x)).

53 Gradients The gradient grad (a) is the vector uniquely defined by: d dɛ a(x + ɛ δx) ɛ=0 = a (δx) δx, grad (a(x)). For example, for h = u + v + u 1 + v 1 + u 1 v 1 we had and thus h (δx) = δu u 1 (δu)u 1 u 1 (δu)u 1 v 1 + δv +.. δx, grad h = (δu)(1 u 2 u 1 v 1 u 1 ) + (δv)(...).

54 Adjoints I An adjoint operation on Mat N (O) is uniquely defined by the quadratic form and the formula < (x, y), (x, y) > = Tr(x 2 ) + Tr(y 2 ) where Q Mat N (O), p i, q i M. p, Q(q) Q (p), q, (14)

55 Adjoints II For example, to get F from ( R F = v R v 1 L u + L u L v 1R v 1 + L v 1R v 1 L v L v L u 1R u 1 L u 1R u 1 R u + R u 1 one takes the transpose of F, in each product reverses all L factors and all R factors, swaps L R )

56 Adjoints II For example, to get F from ( ) R F = v R v 1 L u + L u L v 1R v 1 + L v 1R v 1 L v L v L u 1R u 1 L u 1R u 1 R u + R u 1 one takes the transpose of F, in each product reverses all L factors and all R factors, swaps L R and gets ( ) F L = v L v 1 R v L u 1R u 1R v L u 1R u 1 R u + L v 1R v 1R u + L v 1R v 1 L u + L u 1

57 Pre-Hamiltonian Operators I A pre-hamiltonian operator P is any 2 2 matrix that satisfies P t = F P + PF where F is the Frechet derivative (linearization) of the right hand side of the Kontsevich system and F stands for the adjoint of F.

58 Pre-Hamiltonian Operators I A pre-hamiltonian operator P is any 2 2 matrix that satisfies P t = F P + PF where F is the Frechet derivative (linearization) of the right hand side of the Kontsevich system and F stands for the adjoint of F. Pre-Hamiltonian operators map gradients of trace integrals like σ = Tr (L k i ) to symmetries. (u, v) T τ = P(grad u σ, grad v σ) T. (15)

59 Pre-Hamiltonian Operators II The operator ( R P = u R u L u L u, L u R v L v L u L v R u R v R u, is a pre-hamiltonian operator for KS. L u L v + L u R v L v R u + R u R v L v L v R v R v )

60 Pre-Hamiltonian Operators II The operator ( R P = u R u L u L u, L u R v L v L u L v R u R v R u, L u L v + L u R v L v R u + R u R v L v L v R v R v ) is a pre-hamiltonian operator for KS. Lax pair L, A trace integrals Tr Tr L n symmetries + pre-hamiltonian

61 Pre-Hamiltonian Operators II The operator ( R P = u R u L u L u, L u R v L v L u L v R u R v R u, L u L v + L u R v L v R u + R u R v L v L v R v R v ) is a pre-hamiltonian operator for KS. Lax pair L, A trace integrals Tr Tr L n symmetries + pre-hamiltonian confirmation that the table of first integrals is complete (at least up to degree 14).

62 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

63 Recursion Operators I Recursion operators are operators that acting on a symmetry generate a new (higher degree) symmetry.

64 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 = (Q 1, Q 2 ) t (therefore satisfying D t Q = F (Q)) also the vector RQ is a symmetry generator and therefore satisfying i.e. for any symmetry Q. D t (RQ) = F (RQ), (R t + RF F R)Q = 0

65 Recursion Operators II Example: For u t = uv, v t = vu a recursion operator is given by ( Ru + R R = v, 0 0, R u + R v ).

66 Recursion Operators III Example: For the Kontsevich system a recursion operator is given as follows.

67 Recursion Operators III Example: For the Kontsevich system a recursion operator is given as follows. At first, a non-local vector B = (B 1, B 2 ) T is defined through P τ = [I, B 1 ] P τ = [I 1, B 2 ] with I, P from previous slides and P from involution u v.

68 Recursion Operators IV If G defines a symmetry through dx α dτ = G α(x 1,..., x N, x1 1 1,..., xn ), G α M then a new symmetry of degree 2 higher than the degree of G is given by R n B + R l G with B from the previous slide and ( ) Lu R R n = u L u R u L v R v L v R v R l = ( Rv R u L u 1 R u 1 R v 1 L ul v 1 L u 1 (1 + R u 1 + R v ) + L u 1 R u 1 R v 1 R u, L u 1 L v 1 R v 1 R u L ul v 1 R v 1 R u 1 + L ul v 1 L v 1 R u L v 1 L u 1 R u 1 R v L v L u 1 R u 1 R v 1 + L v L u 1 L u 1 R v, R u R v L v 1 R v 1 R u 1 L v L u 1 L v 1 (1 + R v 1 + R u) + L v 1 R v 1 R u 1 R v )

69 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

70 Classification Problem After a proper scaling of u, v and t the Kontsevich system can be rewritten as u t = uv ε 2 uv 1 ε 3 v 1, v t = vu + ε 2 vu 1 + ε 3 u 1. (16) This system can be regarded as a Laurent deformation of the Mikhailov-Sokolov system u t = uv, v t = vu.

71 Classification Problem After a proper scaling of u, v and t the Kontsevich system can be rewritten as u t = uv ε 2 uv 1 ε 3 v 1, v t = vu + ε 2 vu 1 + ε 3 u 1. (16) This system can be regarded as a Laurent deformation of the Mikhailov-Sokolov system u t = uv, v t = vu. We found several more Laurent deformations of this system having infinitesimal symmetries.

72 Further Laurent Deformations Theorem. All systems of the form: u t = uv + P 1 (u, u 1, v, v 1 ), v t = vu + Q 1 (u, u 1, v, v 1 ), where P 1, Q 1 are the most general inhomogeneous degree 2 polynomials without quadratic term in u, v

73 Further Laurent Deformations Theorem. All systems of the form: u t = uv + P 1 (u, u 1, v, v 1 ), v t = vu + Q 1 (u, u 1, v, v 1 ), where P 1, Q 1 are the most general inhomogeneous degree 2 polynomials without quadratic term in u, v which have commuting flows of the form u τ = P 2 (u, u 1, v, v 1 ), v τ = Q 2 (u, u 1, v, v 1 ), where P 2, Q 2 are the most general inhomogeneous degree 4 polynomials,

74 Further Laurent Deformations Theorem. All systems of the form: u t = uv + P 1 (u, u 1, v, v 1 ), v t = vu + Q 1 (u, u 1, v, v 1 ), where P 1, Q 1 are the most general inhomogeneous degree 2 polynomials without quadratic term in u, v which have commuting flows of the form u τ = P 2 (u, u 1, v, v 1 ), v τ = Q 2 (u, u 1, v, v 1 ), where P 2, Q 2 are the most general inhomogeneous degree 4 polynomials, are shown in the following list:

75 A Listing of Laurent Deformations I u t = uv + a 1 uv 1 + a 2 v 1 v t = vu + b 1 u 1 v a 2 u 1, u t = uv + a 1 v 1 u v t = vu + a 2 u 1 v, u t = uv a 1 v 1 u + a 1 v t = vu + a 1 u 1 v a 1, u t = uv + a 1 uv 1 + b 1 v t = vu + b 1 vu 1 + a 1,

76 A Listing of Laurent Deformations II u t = uv + a 1 uv 1 + a 2 u + a 3 v t = vu a 2 u + a 3 vu 1 + b 1 u 1 v a 1 v 1 u + (a 2 a 3 + a 2 b 1 )u 1 +(a 1 a 3 + a 1 b 1 )u 1 v 1 + (a a 3 b 1 )u 2 + b 2, u t = uv + a 1 u + a 2 v + a 3 v t = vu a 1 u a 2 v a 3, u t = uv + 2a 1 uv 1 a 1 v 1 u a 1 v t = vu 2a 1 vu 1 + a 1 u 1 v + a 1, u t = uv + a 1 uv 1 a 2 v 1 + a 3 u v t = vu + b 1 vu 1 + a 2 u 1 + b 2 v,

77 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

78 Computational Problems We have two classes of problems. 1. Integrability of a given equation.

79 Computational Problems We have two classes of problems. 1. Integrability of a given equation. Computing first integrals large linear systems.

80 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.

81 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.

82 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.

83 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.

84 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.

85 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.

86 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.

87 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.

88 Direct Symmetry Computations, the Conditions: N v e 1 t 1 e 2 t 2 p ,412 3, ,294 1,904 4,448 12, ,914 3,886 5,784 13,878 44, ,746 11,662 17,440 43, , ,242 34,990 52, , , , , , ,470 1,623, , , ,312 1,258,526 5,304, , ,782 1,417,088 3,862,086 17,212, ,125,762 2,834,350 4,251,432 11,835,758 55,535, ,377,290 8,503,054 12,754,480 36,228, ,298, ,131,874 25,509,166 38,263, ,777, ,970, ,395,626 76,527,502 > > > > > > > > Table: For a symmetry ansatz of degree N are listed the # v of variables, the # e 1 of equations and # t 1 of terms of system D τ (I ) = 0 and the # e 2 of equations and # t 2 of terms of system [D t, D τ ](u, v) = 0 and the # p of free parameters of the solution.

89 Quantitative Progress 1-term equ. (A), 1-term equ. (B), N sorting by size, sorting by size, sorting by size, stream solve stream solve stream solve stream solve solve subst. solve subst. solve subst. solve subst , , , , , , , , , Table: Times in sec for solving the symmetry conditions [D t, D τ ](u, v) = 0 for each degree N by different methods

90 Comparison with other Programs Maple 14 LinBox LinSolve N solve total default sparse (Reduce) Sym+FI Sym Sym+FI Sym Sym+FI Sym Sym+FI Sym Sym+FI Table: Times in sec for solving the symmetry conditions [D t, D τ ](u, v) = 0 for each degree N by different programs

91 Iterating the Formulation and Solution of Systems I [D t, D τ ](u, v) = 0 [D t, D τ ](u, v) = 0 D τ (I ) = 0 Iteration at once in two stages first first formulation of ansatz for D τ computation of Dτ computing + splitting D τ (I ) = 0, [D t, D τ ](u) = 0, D τ (I ) = 0, [D t, D τ ](v) = 0, 587 each time only extracting and using 1-term conditions computation and splitting of D τ (I ) = 0, extracting and using 1-term conditions computation and splitting of [D t, D τ ](u) = extracting and using 1-term conditions, keeping others computation and splitting of [D t, D τ ](v) = complete solution of all remaining equations Table: Times in sec of the whole symmetry computation for N = 13

92 Iterating the Formulation and Solution of Systems II (continuation of the table above:) [D t, D τ ](u, v) = 0 [D t, D τ ](u, v) = 0 D τ (I ) = 0 Iteration at once in two stages first first substitution of solution in D τ (u, v) total CPU time total garbage collection time overall time Table: Times in sec of the whole symmetry computation for N = 13

93 Outline Non-commutative ODEs First Integrals and Lax Pairs Symmetries Pre-Hamiltonian Operators Recursion Operators Classifications Computer Runs References

94 References I Integrability, non-abelian ODE Systems A.V. Mikhailov and V.V. Sokolov, Integrable ODEs on Associative Algebras, Comm in Math Phys. 211 (2000), O.V. Efimovskaya, Integrable cubic ODEs on Associative Algebras, Fundamentalnaya i Prikladnaya Matematika 8, no. 3 (2002), 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 , DOI: /s P.J. Olver and V.V. Sokolov, V. V., Integrable evolution equations on associative algebras, Comm. in Math. Phys., 1998, 193, no.2, P.J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, 107, Springer Verlag, New York, M. Kontsevich Noncommutative identities arxiv: v1

95 References II Computer Algebra T. Wolf, E. Schrüfer, K. Webster, Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs, Programming and Computer Software DOI: /S (2012). T. Wolf, An online tutorial for the package Crack, (2004) T. Wolf, Applications of Crack in the Classification of Integrable Systems, CRM Proceedings and Lecture Notes, Centre de Recherches Mathematiques/Montreal, 37, (2004), , also preprint on arxiv: nlin.si/ Compatible Poisson Brackets A. Odesskii and T. Wolf, Compatible quadratic Poisson brackets related to a family of elliptic curves, preprint, 17 pages, arxiv (2012)

96 The End Thank you!