Jun 25, 2010 - ... the equation E. EÏ is a symmetry of E if EÏ(F)|E = lE(Ï) = 0, Sym(E) = kerlE ... âi Di(Ri)=0on E. Conservation laws of E are conserved currents mod. trivial ones ..... World Sci., 2007, pp. 100â108 ... Ï)dx1 â§Â·Â·Â·â§ dxn. E. 1,n. 1.
Nonlinear Physics VI 24 June – 2 July 2010, Gallipoli, Italy
Hamiltonian formalism for general PDEs Paul Kersten Joseph Krasil′ shchik Alexander Verbovetsky Raffaele Vitolo (speaker)
25 June 2010
Plan
1. Examples 2. Hamiltonian Operators as Variational Bivectors 3. Examples revisited
Example: KdV ut = uxxx + 6uux = Dx δ(u3 − u2x /2) = (Dxxx + 4uDx + 2ux )δ(u2 /2) ux = v,
vx = w,
wx = ut − 6uv
u 0 −1 0 v = 1 0 −6u δ uw − v 2 /2 + 2u3 w x 0 6u Dt 0 −2u −Dt − 2v u v = 2u Dt −12u2 − 2w δ −3u2 /2 − w/2 −Dt + 2v 12u2 + 2w 8uDt + 4ut w x S. P. Tsarev, The Hamilton property of stationary and inverse equations of condensed matter mechanics and mathematical physics, Math. Notes 46 (1989), 569–573
Example: Camassa-Holm equation ut − utxx − uuxxx − 2ux uxx + 3uux = 0
mt + umx + 2ux m = 0,
m − u + uxx = 0
mt = −umx − 2ux m = B1 δ(H1 ) = B2 δ(H2 ) where
Z 1 B1 = −(mDx + Dx m), H1 = mu dx 2 Z 1 3 B2 = D x − D x , H 2 = (u3 + uu2x ) dx. 2
H1 and H2 are viewed as functionals of m and not of u, with u = (1 − Dx2 )−1 m.
Example: Kupershmidt deformation B. Kupershmidt, KdV6 : An integrable system, Phys. Lett. A 372 (2008), 2634–2639
ut = f (t, x, u, ux , uxx , . . . ) A1 , A2 are compatible Hamiltonian operators H1 , H2 , . . . is a Magri hierarchy of conserved densities Dt (Hi ) = 0, A1 δ(Hi ) = A2 δ(Hi+1 ). ut = f − A1 (w),
A2 (w) = 0
(1)
The KdV6 equation (A. Karasu-Kalkanli, A. Karasu, A. Sakovich, S. Sakovich, and R. Turhan, A new integrable generalization of the Korteweg-de Vries equation, J. Math. Phys. 49 (2008) 073516, arXiv:0708.3247)
ut = uxxx + 6uux − wx ,
wxxx + 4uwx + 2ux w = 0
Theorem (Kupershmidt) H1 , H2 , . . . are conserved densities for (1).
Infinite jet space: notation The jet space J ∞ with coordinates xi , ujσ D i = ∂ xi + Eϕ =
P
P
j,σ
ujσi ∂ujσ are total derivatives
j ji Di (ϕ )∂uji + . . . is an evolutionary (ϕ1 , . . . , ϕm ) is a vector function on J ∞
j
ϕj ∂ u j +
P
ϕ=
P
ℓf = σ ∂ujσ (fi )Dσ is the linearization
of a vector function f on J ∞ , ℓf (ϕ) = Eϕ (f )
∆∗ = k
P
σ (−1)
σD
ji σ aσ k,
if ∆ = k
P
σ
the adjoint C-differential operator
aij σ Dσ k,
field,
Differential equations: notation Let Fk (xi , ujσ ) = 0, k = 1, . . . , l, be a system of equations Relations F = 0, Dσ (F ) = 0 define its infinite prolongation E ⊂ J ∞ ℓE = ℓF |E is the linearization of the equation E Eϕ is a symmetry of E if Eϕ (F )|E = ℓE (ϕ) = 0, Sym(E) = ker ℓE ϕ is its generating function Vector function R = (R1 , . . . , Rn ) on E is a conserved current if P i i Di (R ) = 0 on E Conservation laws of E are conserved currents mod. trivial ones Generating function of a conservation P law: ψ = (ψ1 , . . . , ψm ) = ∆∗ (1), where i Di (Ri ) = ∆(F ) on J ∞ ℓ∗E (ψ) = 0,
CL(E) ⊂ ker ℓ∗E
Analogy
Manifold M functions vector fields T ∗M TM De Rham complex multivectors Schouten bracket
Jet J ∞ functionals evolutionary vect. fields ˆ TJ∗∞ = Jh∞ (κ) TJ ∞ = Jh∞ (κ) E00,n−1 → E01,n−1 · · · variational multiv. variational Sch. br.
PDE E conservation laws symmetries L∗ (E) L(E) 0,n−1 → E11,n−1 · · · E1 variational multiv. variational Sch. br.
The analogy can be extended to the Liouville one-form θ0 ∈ Ω1 (T ∗ M ) and the symplectic form ω0 = dθ0 .
Differential equations: the model D(E) = Sym(E) = the Lie algebra of symmetries of E Λq (E) ⊃ CΛq (E) ⊃ C 2 Λq (E) ⊃ C 3 Λq (E) ⊃ · · · E10,n
d0,n
1 −− −−→ E11,n
d0,n−1
d1,n
1 −− −−→ E12,n
d1,n−1
d2,n
1 −− −−→ E13,n
d2,n−1
d3,n
1 −− −−→ · · ·
d3,n−1
1 1 1 1 E10,n−1 −− −−→ E11,n−1 −− −−→ E12,n−1 −− −−→ E13,n−1 −− −−→ · · ·
E10,n−2 .. . E10,0 E10,n−1 = space of conservation laws E11,n−1 = Cosym E = ker ℓ∗E .
E12,n−1 = { ∆ | ℓ∗E ∆ = ∆∗ ℓE } { ∇ ℓE | ∇∗ = ∇ }
Differential equations: the cotangent space TE∗ :
F = 0,
L = hF, pi
ℓ∗E (p) = 0 ℓ∗T ∗ = ℓTE∗ E
Variational multivectors on E are conservation laws on TE∗ .
Theorem A variational bivector on E can be identified with the equivalence class of operators A on E that satisfy the condition ℓE A = A∗ ℓ∗E , with two operators being equivalent if they differ by an operator of the form ℓ∗E . If A is a bivector and E is written in evolution form then A∗ = −A.
Differential equations: the Schouten bracket of bivectors
[[A1 , A2 ]](ψ1 , ψ2 ) = ℓA1 ,ψ1 (A2 (ψ2 )) − ℓA1 ,ψ2 (A2 (ψ1 )) + ℓA2 ,ψ1 (A1 (ψ2 )) − ℓA2 ,ψ2 (A1 (ψ1 )) − A1 (B2∗ (ψ1 , ψ2 )) − A2 (B1∗ (ψ1 , ψ2 )), where ℓF Ai − A∗i ℓ∗F = Bi (F, ·) on J ∞ , Bi∗ (ψ1 , ψ2 ) = Bi∗1 (ψ1 , ψ2 )|E . Bi∗ are skew-symmetric and skew-adjoint in each argument. If E is in evolution form then Bi∗ (ψ1 , ψ2 ) = ℓ∗Ai ,ψ2 (ψ1 )
Differential equations: Poisson bracket Definition A variational bivector is called Hamiltonian if [[A, A]] = 0 S1 , S2 ∈ CL(E), ψ1 , ψ2 are the generating functions {S1 , S2 }A = EA(ψ1 ) (S2 )
Definition The Magri hierarchy on a bihamiltonian equation E is the infinite sequence S1 , S2 , . . . of conservation laws of E such that A1 (ψi ) = A2 (ψi+1 ).
Proposition For Magri hierarchy we have {Si , Sj }A1 = {Si , Sj }A2 = {Eϕi , Eϕj } = 0, where ϕi = A1 (ψi ) = A2 (ψi+1 ).
Invariance of the cotangent equation
v: vv E HH H$
J1∞ J2∞
Each two resolutions of the module of Cartan forms CΛ1 are homotopic. In particular, we consider normal equations, for which CΛ1 admits resolutions of length 1: 0
/ C(P1 , F) O
1
β ′+
α′+
0
ℓ¯+ F
/ C(P2 , F)
ℓ¯+ F
2
/ C(κ1 , F) O
r1
β+
α+
/ C(κ2 , F)
/ CΛ1
/0
id r2
/ CΛ1
/0
Invariance of the cotangent equation Theorem Let E be a normal equation. Then: ℓ1E is homotopically equivalent to ℓ2E ⇒ 1∗ ℓE is homotopically equivalent to ℓ2∗ E . It follows that the cotangent space to E does not depend on the inclusion of E into J ∞ . We have the change of coordinate formula for bivectors: A2 = α A1 α′∗ A1 = β A2 β ′∗
Example: KdV s1
x
•O
β′
α
β
ℓ2E
•
/• O
ℓ1E
α′
/•
f
F1 = ut − uxxx − 6uux = 0 ux − v =0 vx − w F2 = wx − ut + 6uv
Dx −1 0 0 Dx −1 ℓ1E = Dt − Dxxx − 6uDx − 6ux ℓ2E = −Dt + 6v 6u Dx 1 0 β= 1 0 0 ′ α = Dx α = 0 ′ −D − 6u −D −1 β = xx x D −1
s2
xx
0 0 0 s1 = 0 s2 = 1 0 0 Dx 1 0
Example: Camassa-Holm equation ut − utxx − uuxxx − 2ux uxx + 3uux = 0 A1 = Dx
A2 = −Dt − uDx + ux .
mt + umx + 2ux m = 0, m − u + uxx = 0 hhh (( 2( −1 h h h u( =( (1( −( D( m x )hh h (
A′1
=
Dx 0 Dx − Dx3 0
A′2
=
0
2mDx + mx
−1 0
Example: Kupershmidt deformation Let E be a bi-Hamiltonian equation given by F = 0
Definition
The Kupershmidt deformation E˜ has the form F + A∗1 (w) = 0,
A∗2 (w) = 0,
where w = (w1 , . . . , wl ) are new dependent variables
Theorem
The Kupershmidt deformation E˜ is bi-Hamiltonian.
Proof. The following two bivectors define a bi-Hamiltonian structures: A2 −A2 A1 −A1 ˜ ˜ A2 = A1 = −ℓF +A∗1 (w)+A∗2 (w) 0 0 ℓF +A∗1 (w)+A∗2 (w)
More examples ◮
H. Baran and M. Marvan, On integrability of Weingarten surfaces: a forgotten class, J. Phys. A: Math. Theor. 42 (2009), 404007 zyy + (1/z)xx + 2 = 0 Dxx ,
◮
2zDxy − zy Dx + zx Dy .
F. Neyzi, Y. Nutku, M.B. Sheftel, Multi-Hamiltonian structure of Plebanski’s second heavenly equation arxiv:nlin/0505030 utt uxx − u2tx + uxz + uty = 0 It is Lagrangian, hence the identity operator is a Hamiltonian bivector. This is rewritten in the above paper in evolutionary coordinates.
Symbolic computations Hamiltonian operators, recursion operators, symplectic operators, etc. can be computed as (generalized or higher) symmetries or cosymmetries in the cotangent space of the given PDE. We use a set of packages for Reduce developed by Kersten et al. at the Twente University (Holland). This is available at the Geometry of Differential Equations website http://gdeq.org/ together with documentation, a tutorial (by R.V.) and examples. We are currently extending it to work for non-evolutionary equations.
References ◮
P. Kersten, I. Krasil′ shchik, and A. Verbovetsky, Hamiltonian operators and ℓ∗ -coverings, J. Geom. Phys. 50 (2004), 273–302
◮
P. Kersten, I. Krasil′ shchik, and A. Verbovetsky, (Non)local Hamiltonian and symplectic structures, recursions, and hierarchies: a new approach and applications to the N = 1 supersymmetric KdV equation, J. Phys. A: Math. Gen. 37 (2004), 5003–5019
◮
P. Kersten, I. Krasil′ shchik, and A. Verbovetsky, The Monge-Amp`ere equation: Hamiltonian and symplectic structures, recursions, and hierarchies, Memorandum of the Twente University 1727 (2004)
◮
P. Kersten, I. Krasil′ shchik, and A. Verbovetsky, A geometric study of the dispersionless Boussinesq type equation, Acta Appl. Math. 90 (2006), 143–178
◮
J. Krasil′ shchik, Nonlocal geometry of PDEs and integrability, in Symmetry and perturbation theory (G. Gaeta, R. Vitolo, and S. Walcher, eds.), World Sci., 2007, pp. 100–108
References ◮
V. A. Golovko, I. S. Krasil′ shchik, and A. M. Verbovetsky, Variational Poisson-Nijenhuis structures for partial differential equations, Theor. Math. Phys. 154 (2008), 227–239
◮
V. A. Golovko, I. S. Krasil′ shchik, and A. M. Verbovetsky, On integrability of the Camassa-Holm equation and its invariants, Acta Appl. Math. 101 (2008), 59–83
◮
P. Kersten, I. Krasil′ shchik, A. Verbovetsky, R. Vitolo, Integrability of Kupershmidt deformations, Acta Appl. Math. 109 (2010), 75-86
◮
P. Kersten, I. Krasil′ shchik, A. Verbovetsky, R. Vitolo, Hamiltonian structures for general PDEs, in Differential Equations—Geometry, Symmetries and Integrability: The Abel Symposium 2008 (B. Kruglikov, V. Lychagin, and E. Straume, eds.), Abel Symposia 5, Springer, 2009, pp. 187–198
◮
J. Krasil’shchik and A. Verbovetsky, Geometry of jet spaces and integrable systems, arXiv:1002.0077
◮
S. Igonin, P. Kersten, J. Krasil′ shchik, A. Verbovetsky, R. Vitolo, Variational brackets in geometry of PDEs, 2010, to appear
Infinite jet space: the model D(J ∞ ) = κ = the Lie algebra of evolutionary fields Λq (J ∞ ) ⊃ CΛq (J ∞ ) ⊃ C 2 Λq (J ∞ ) ⊃ C 3 Λq (J ∞ ) ⊃ · · · d0,n
d1,n
d2,n
d3,n
1 1 1 1 E10,n −− → E11,n −− → E12,n −− → E13,n −− → ···
E10,n−1 .. . E10,0 n is number of x’s R E10,n consists of all “actions” L(xi , ujσ ) dx1 ∧ · · · ∧ dxn ˆ κˆ = HomC ∞ (J ∞ ) (κ, Λn (J ∞ ) CΛn (J ∞ )) E11,n = κ, d0,n 1 is the Euler operator ˆ E12,n = C skew (κ, κ) ∗ d1,n (ψ) = ℓ − ℓ ψ 1 ψ
Infinite jet space: the cotangent space B. A. Kupershmidt, Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms, Lect. Notes Math. 775, 1980, pp. 162–218 ˆ TJ∗∞ = Jh∞ (κ) ˆ κ ⊕ κ) ˆ S ∈ Ω2 (TJ∗∞ ) = C(κ ⊕ κ, D2 (J ∞ ) = C skew (κ, ˆ κ)
S(ϕ, ψ) = (−ψ, ϕ)
A1 , A2 ∈ D2 (J ∞ )
[[A1 , A2 ]](ψ1 , ψ2 ) = ℓA1 ,ψ1 (A2 (ψ2 )) − ℓA1 ,ψ2 (A2 (ψ1 )) + ℓA2 ,ψ1 (A1 (ψ2 )) − ℓA2 ,ψ2 (A1 (ψ1 )) − A1 (ℓ∗A2 ,ψ2 (ψ1 )) − A2 (ℓ∗A1 ,ψ2 (ψ1 )), where ℓA,ψ = ℓA(ψ) − Aℓψ
