Paolo Lorenzoni Based on a joint work with Jenya Ferapontov and Andrea Savoldi

Transcription

1 Hamiltonian operators of Dubrovin-Novikov type in 2D Paolo Lorenzoni Based on a joint work with Jenya Ferapontov and Andrea Savoldi June 14, 2015 Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

2 Problem (Dubrovin-Novikov 1984) To classify multidimensional Hamiltonian operators of hydrodynamic type. P ij = g ij (1) (u) d dx 1 + b ij (1)k (u)uk x g ij (D) (u) d dx D + b ij (D)k (u)uk x D, (1) where u = (u 1,..., u n ) and D is the number of independent variables (the dimension). Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

3 Examples of 1D operators KdV u t = uu x + ɛ 2 uxxx 12 u t = x δh δu, H = can be written as S 1 [ ] u ɛ2 24 (u2 x + 2uu xx ) NLS iψ t = 1 2 ψ xx + ψ 2 ψ in the variables u = ɛ 2i (ln ψ/ ψ) x, v = ψ 2 [ ut v t ] = [ 0 x x 0 dx ] [ δh ] δu δh, H = [v u2 δv S v ɛ2 v 2 ] x dx 8 v Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

4 Examples of 2D operators [ ut v t ] [ ] [ 0 δh ] x = δu x δh, H = h(u, v) dx y δv S 1 the choice h(u, v) = v eu is equivalent to Boyer-Finley equation. the choice h(u, v) = u2 equation v 3/2 is equivalent to disperionless KP Classidication of 2-component integrable 2+1 Hamiltonian PDEs has been given by Ferapontov, Odesskii and Stoilov. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

5 1D DN Hamiltonian operators In 1D, the operators are defined by P ij = g ij (u) x + b ij l (u)ul x. For any pair of local functionals F = f (u, u x,...) dx, S 1 G = g(u, u x,...) dx. S 1 the Poisson bracket is defined by {F, G} = S 1 δf δg Pij δui δu j dx Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

6 Dubrovin-Novikov theorem Theorem if g is not degenerate, then P is a Hamiltonian operator iff g is a flat (pseudo)-metric. Γ i jk are the Christoffel symbols of the associated Levi-Civita connection. Consequences Darboux theorem : in flat coordinates g ij = constants and b ij k = 0: P ij = g ij x. P is degenerate Poisson bracket: the Casimirs are C i = S 1 u i dx where u i are flat coordinates. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

7 2D DN Hamiltonian operators In 2D, the operators are defined by P ij = g ij (u) d dx + bij k (u)uk x + g ij (u) d dy + b ij k (u)uk y, (2) where u = (u 1,..., u n ) are dependent variables, i, j, k = 1..., n. We assume that the tensors g and g are not degenerate: det g 0 and detg 0. As in one-dimensional case g and g are flat metrics and Γ i jk = g jsb si k, Γi jk = g js bsi k. are the Christoffel symbols of the associated Levi-Civita connections. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

8 The obstruction tensor In the non-degenerate case we introduce the obstruction tensor where Γ i jk = g jsb si k, T i jk = Γ i jk Γi jk. Γ i jk = g si js b k. Remark (Dubrovin-Novikov) A 2D Hamiltonian operator can be reduced to a constant form if and only if the obstruction tensor vanishes. Theorem (Mokhov) If the eingenvalues of the affinor L = gg 1 are pairwise distinct then the obstruction tensor vanishes. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

9 Mokhov s conditions on the obstruction tensor Theorem Flat non-degenerate metrics g ij and g ij define a 2D Hamiltonian operator of the form (4) if and only if the following conditions are fulfilled: T ijk = T kji (3a) T ijk + T jki + T kij = 0 T ijs T r st = T irs T j st r T ijk = r T ijk = 0 (3b) (3c) (3d) where T ijk = g ks g ir T j rs and r and r are the covariant derivatives given respectively by the connections Γ i jk and Γ i jk. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

10 New formulation of Mokhov s conditions Theorem Flatness of g and g and Mokhov s conditions (3a) (3d) are equivalent to the following conditions: 1 Flatness of g. 2 Linearity of g jk in the flat coordinates of g. 3 Vanishing of the Nijenhuis torsion of the affinor L i j = g il g lj. 4 The Killing condition: i g kj + k g ij + j g ik = 0. Remarks I. Notice that the flatness of g and the above conditions 1,2,3 imply the flatness of the second metric g. II. A Killing bivector in flat space is the sum of symmetrised tensor products of Killing vectors. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

11 Classification P ij = g ij (u) d dx + bij k (u)uk x + g ij (u) d dy Due to Mokhov s conditions, in flat coordinates for g: P ij = g ij d dx + b ij k (u)uk y, (4) + [( b ij k + b ji k )uk + g ij 0 ) d dy + b ij k uk y (5) We assume that the matrix L 0 = g 0 g 1 has a certain Jordan structure. Fixed this structure we use the fact that a pair of symmetric matrices (g and g 0 ) can be brought to the Segre normal form Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

12 Segre normal form For instance, for n = 2, 3: Segre [2] Segre [2,1] Segre [3] L 0 g g ( ) ( ) ( 0 ) λ λ 0 λ 1 0 λ 0 λ λ 0 0 λ λ λ λ λ λ 0 0 ± ±λ 0 1 λ 1 λ 0 λ 0 0 Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

13 Splitting lemma Lemma (Splitting Lemma, Bolsinov-Matveev) Let L be an affinor with zero Nijenhuis torsion on a manifold M, dimm = n. Suppose there exists a (non-holonomic) frame in which L takes block diagonal form, ( ) A 0 L =, 0 B where Spec(A) Spec(B) =. Then there exists a local coordinate system (u 1,..., u m, v m+1,..., v n ) such that ( ) A(u) 0 L =. 0 B(v) Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

14 Splitting Lemma (operators) Theorem If the Killing condition holds, then ( ) g1 (u) 0 g =, g = 0 g 2 (v) ) ( g1 (u) 0 0 g 2 (v) and P decouples into a direct sum of two Hamiltonian operators: ( ) P1 (u) 0 P = 0 P 2 (v) (6) Operators of the form (6) are called reducible. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

15 Two-component case (Dubrovin-Novikov, Mokhov) Theorem Theorem Every irreducible non-constant two-component Hamiltonian operator in 2D can be reduced to the following canonical form P = ( ) 0 1 d 1 0 dx + ( 2u 1 u 2 u 2 0 ) d dy + ( u 1 y 2uy 2 uy 2 0 ). (7) Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

16 Three-component case Theorem Every irreducible non-constant three-component Hamiltonian operator in 2D can be brought to one of the following canonical forms: Jordan block with constant eigenvalue P = d 2u 2 u 3 λ dx + u 3 λ 0 d uy 2 2u 3 y 0 dy + u y 3 0 0, λ Jordan block with non-constant eigenvalue P = d 2u 1 1 dx + 2 u2 u u2 u 3 0 d u 1 1 y u 3 dy + 2 u2 y 2u 3 y u 2 1 y 2 u3 y uy The complete list in the 4-component case is given in our paper. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

17 Single Jordan block case The affinor L and the metric g are given by L i j = c i jk uk + (L 0 ) i j = c i jk uk + g il 0 g lj, g ij = L i l g lj = c i n+1 j,k uk + g ij 0. We will work in coordinates where g and g 0 take canonical form λ 1 L 0 = λ , g = ± λ λ, g 0 = ±... λ λ Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

18 These have to satisfy a set of constraints: Linear part of the condition N (L) = 0 reads c k j,i 1 c k i,j 1 + c k+1 ij c k+1 ji = 0; (8) Quadratic part of the condition N (L) = 0 reads c s il cm js c s jl cm is + c m sl cs ij c m sl cs ji = 0; Symmetry of g gives The Killing condition gives c n+1 i jk = c n+1 j ik ; (9) c n+1 i jk + c n+1 k ij + c n+1 j ki = 0. (10) Remarkably, the linear system (8)-(10) can be solved explicitly and the solution automatically satisfies the quadratic part of the Nijenhuis condition. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

19 General solution (non constant eigenvalue) The general solution of Killing s and linear-nijenhuis s conditions for the n n Jordan block is given by n 2 L k j = c l (M n,l ) k j + (L 0 ) i j l=0 where L 0 = g 0 g 1 and M n,l are the n n affinors s.t 1 M n,0 (Mokhov s affinor) is defined as (M n,0 ) i j = (n+3i 3j 1) n 1 u n+i j. 2 (M n,k ) i j (for i=1,...,n-k j=k+1,...,n) coincide with the (n k) (n k) Mokhov affinor with u i replaced by u i+k (i = 1,..., n k). 3 (M n,k ) i j = 0 if if i > n k or j < k + 1 Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

20 An example: n = 4 2 L k j = c l (M 4,l ) k j + (L 0 ) i j l=0 +u 4 0 u 2 2 u 1 0 u 4 1/2 u 3 2 u 2 0 u 4 0 u u 4 1/2 u 3 M 4,0 =, M 4,1 = 0 0 u u u u 4 2 u 3 λ u 4 0 λ 1 0 M 4,2 =, L 0 = λ λ Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

21 Theorem The family of solutions n 2 c l M n,l + L 0 l=0 can be reduced to the following canonical forms: M n,0 if n 1 mod 3, g = (±) M n,0 + κm n,1 + L 0 if n = 4, M n,0 + κm n, n 1 if n 7 mod 3. 3 An analogous result can be proved for the constant eigenvalue case. Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22

22 The talk is based on E.V. Ferapontov, P. Lorenzoni and A. Savoldi Hamiltonian operators of Dubrovin-Novikov type in 2D, Lett. Math. Phys. 105 (2015), pp Paolo Lorenzoni (Milano-Bicocca) Hamiltonian operators of D-N type in 2D June 14, / 22