SLIDE 1 Methods of tangent and cotangent coverings for Dubrovin-Novikov integrability operators
Joint work with E. Ferapontov, M. Pavlov arXiv:2017 Geometry of Integrable systems SISSA, 5-7 June 2017
SLIDE 2 Hamiltonian PDEs
An evolutionary system of PDEs F = ui
t − fi(t, x, uj, uj x, uj xx, . . .) = 0
admits a Hamiltonian formulation if ui
t = Aij
δH δuj
- where A = (Aij) is a Hamiltonian operator, i.e. a differential
- perator
Aij = aijσDσ such that A∗ = −A and [A, A] = 0 Dσ = Dx ◦ · · · ◦ Dx (total x-derivatives σ times). Finding Hamiltonian operators/PDEs is hard.
SLIDE 3 Symmetries
A Hamiltonian equation shows a correspondence between conservation laws and symmetries. Generalized symmetries are vector functions ϕi = ϕi(uj, uj
x, uj xx, . . .) such that
∂uj
σ Dσϕj = 0,
F k = 0 where ℓF is the Fr´ echet derivative of F.
SLIDE 4 Conservation laws
A conservation law is a one-form ω = Adx + Bdt which is closed modulo the equation: dω = ∇F where ∇ = aτσ
k DτσF k. The vector function
ψk = ψk(uj, uj
x, uj xx, . . .) = (−1)|τσ|Dτσaτσ k |F=0
represents uniquely the conservation law and fulfills the equation
F (ϕi) = −Dtψi + (−1)|σ|Dσ
∂ui
σ ψj
F k = 0 where ℓ∗
F is the formal adjoint of ℓF ;
SLIDE 5
A necessary condition
If an equation admits a Hamiltonian formulation, this implies that A maps conservation laws into symmetries: ℓF ◦ A = (A′)∗ ◦ ℓ∗
F
A: almost-Hamiltonian op. The condition can be extended to all integrability operators: ℓ∗
F ◦ S = S′ ◦ ℓF
S: almost-symplectic op. ℓF ◦ R = R′ ◦ ℓF R: recursion operator ℓ∗
F ◦ C = (C′)∗ ◦ ℓ∗ F
C: co-recursion operator Note that A′, S′, R′, C′ are arbitrary. Almost: it is a necessary condition . . .
SLIDE 6 Cotangent covering
Kersten, Krasil’shchik, Verbovetsky, JGP 2003. Introducing new variables pk, pkx, pkxx, . . . we can represent
- perators by linear functions:
A(ψ) = aijσDσψj ⇔ A = aijσpjσ Then a Hamiltonian operator fulfills the equations T ∗ :
F (p) = −pi,t + (−1)|σ|Dσ
∂ui
σ pj
F = ui
t − fi = 0
and ℓF (A) = 0. The system T ∗ is the cotangent covering. It is invariant.
SLIDE 7 Tangent covering
Introducing new variables qk, qk
x, qk xx, . . . we can represent
- perators by linear functions:
R(ϕ) = aiσ
j Dσϕj
⇔ R = aiσ
j qj σ
Then a recursion operator fulfills the equations T :
t − ∂fi ∂uj
σ qj
σ = 0
F = ui
t − fi = 0
and ℓF (R) = 0. The system T is the tangent covering. It is invariant.
SLIDE 8 Example: Hamiltonian operators for KdV
The KdV equation: ut = uux + uxxx The linearization: ℓF = Dt − uDx − ux − Dxxx The adjoint linearization: ℓ∗
F = −Dt + uDx + Dxxx
The cotangent covering for the KdV equation: pt = pxxx + upx ut = uxxx + uux The equation ℓF (A) = 0 has the two solutions: A1 = px
A1 = Dx A2 = 1 3(3p3x + 2upx + uxp)
A2 = 1 3(3Dxxx + 2uDx + ux) For example, ℓF (A1) = Dtpx − uDxpx − uxpx − Dxxxpx.
SLIDE 9 Example: recursion operator for KdV
The tangent covering of KdV: T : qt = uxq + uqx + qxxx ut = uxxx + uux Unfortunately, the equation for recursion operators ℓF (R) = 0 has the only trivial solution R = q. However, there is a conservation law on T : ω = qdx + (uq + qxx)dt. We can introduce a new non-local variable w such that wx = q, wt = uq + qxx. Then we have the non-local recursion operator R = qxx + 2 3uq + 1 3uxw
R = Dxx + 2 3u + 1 3uxD−1
x
SLIDE 10 Applications to Dubrovin–Novikov operators
The cotangent covering of a hydrodynamic-type system is:
i,j − V k j,i)uj xpk + V k i pk,x
ui
t = V i j (u)uj x
A first-order Dubrovin–Novikov Hamiltonian operator: Ai = gijpjx + Γij
k uk xpj.
Tsarev’s compatibility conditions are the coefficients of the linear equation in pk,σ, ℓF (A) = 0: DtAi − V i
j,kuj xAk − V i j DxAj = 0
⇔
k = gjkV i k
∇iV k
j = ∇jV k i
SLIDE 11
Further applications
◮ Higher order Dubrovin–Novikov Hamiltonian operators. ◮ Symplectic Dubrovin–Novikov operators. ◮ Recursion operators for cosymmetries. ◮ Nonlocal Dubrovin–Novikov first-order operators, also
known as Ferapontov–Mokhov operators. Higher order analogue.
SLIDE 12 Application to third-order DN operators
Dubrovin–Novikov operators can be defined for arbitrary
- rders. Here we consider the third order ones:
Aij
3 =gij(u)D3 x + bij k (u)uk xD2 x
+ [cij
k (u)uk xx + cij km(u)uk xum x ]Dx
+ dij
k (u)uk xxx + dij km(u)uk xum xx + dij kmn(u)uk xum x un x,
Potemin’s canonical form in Casimirs: Aij
3 = Dx(gijDx + cij k uk x)Dx
Remark: gij is the Monge form of a quadratic line complex (Ferapontov, Pavlov, V. JGP 2014, IMRN 2016). We restrict our consideration to hydrodynamic-type systems in these Casimirs. Then they can be written in conservative form: V i
j = (V i),j
SLIDE 13 Compatibility conditions (Ferapontov, Pavlov, V. 2017)
Theorem Let A3 be a Hamiltonian operator. Then ui
t = V i j uj x = (V i),juj x admits a Hamiltonian formulation with
A3 if and only if gimV m
j
= gjmV m
i
cmkjV m
i
+ cmikV m
j
+ cmjiV m
k
= 0, V k
i,j = gkscsmjV m i
+ gkscsmiV m
j
(1)
- Theorem. The above system is in involution. Its solution
depends on at most (1/2)n(n + 3) parameters. The solution is reduced to a linear algebra problem either if the unknown is gij or if the unknown is V i.
- Remark. No Hamiltonian needed at this stage!
SLIDE 14
Properties of the systems of conservation laws
Following a construction of Agafonov and Ferapontov (1996-2001) we associate to each system ui
t = (V i),juj x a
congruence of lines in Pn+1 with coordinates [y1, . . . , yn+2] yi = uiyn+1 + V iyn+2 Theorem.
◮ The congruence is linear: there are n linear relations
between ui, V i, uiV j − ujV i.
◮ The system is linearly degenerate, and non diagonalizable. ◮ V i = ψi αwα, where ψi α is determined by gij = ϕαβψα i ψβ j and
wα are linear functions. This means that V i = p(uj)/q(uj) where p(uj), q(uj) are polynomials of degree n, n − 1.
SLIDE 15 Hamiltonian, momentum and more
The above systems of conservation laws all admit non-local Hamiltonian, momentum and Casimirs. They all are new non-local conserved quantities. Let us set ψγ
k = ψγ kmum + ωγ k, and wγ = ηγ mum + ξγ.
Let us set ui = bi
x; the system becomes bi t = V i(bx).
Theorem.
◮ Hamiltonian op. A3 = −gij(bx)Dx − cij k (bx)bk xx ◮ Hamiltonian H = −
3ηγ pψβ qmbm x + 1 2ωβ p ηγ q
x
2ψβ pqξγbpbq x + ξγωβ q bq
]dx
◮ n Casimirs Cα =
1
2ψα mkbk x + ωα m
◮ momentum P = − 1 3ϕβγωβ q ψγ pmbm x + 1 2ϕβγωβ p ωγ q
SLIDE 16 Invariance of the hydrodynamic-type system
- Theorem. The class of conservative systems of hydrodynamic
type possessing third-order Hamiltonian formulation is invariant under reciprocal transformations of the form d˜ x = (aiui + a)dx + (aiV i + b)dt d˜ t = (biui + c)dx + (biV i + d)dt
SLIDE 17 Classification results
t = (V i)x be a hydrodynamic-type system,
and suppose that it admits a Hamiltonian formulation via a third-order Dubrovin-Novikov operator whose Casimirs are ui. Then: n = 2 The system is linearisable. n = 3 The system is either linearisable, or equivalent to the system of WDVV equations (to be discussed); from Castelnuovo’s classification of linear line congruences. n = 4 Far more complicated: there exists no classification of linear congruences in P5. There exist one generic nontrivial integrable example.
SLIDE 18 Example: WDVV equations in 3-comp.
From fttt = f2
xxt − fxxxfxtt setting u1 = fxxx, u2 = fxxt,
u3 = fxtt we have u1
t = u2 x,
u2
t = u3 x,
u3
t = ((u2)2 − u1u3)x,
endowed with a third-order Hamiltonian operators with nonlocal Hamiltonian H = − 1 2u1 ∂x
−1u22 + ∂x −1u2∂x −1u3
(Ferapontov, Galvao, Mokhov, Nutku, 1995). It is bi-Hamiltonian and up to a non-trivial transformation is the 3-wave equation (Zakharov, Manakov, ∼1970).
SLIDE 19
Example: WDVV system in 6-comp.
Dubrovin 1996; Ferapontov-Mokhov 1998; Pavlov-V. 2015. We have a pair of hydrodynamic type systems in conservative form: ai
y = (vi(a))x,
ai
z = (wi(a))x,
where
v1 = a2, w1 = a3, v2 = a4, v3 = w2 = a5, w3 = a6, v4 = fyyy = 2a5 + a2a4 a1 , v5 = w4 = fyyz = a3a4 + a6 a1 , v6 = w5 = fyzz = 2a3a5 − a2a6 a1 , w6 = fzzz = (a5)2 − a4a6 + (a3)2a4 + a3a6 − 2a2a3a5 + (a2)2a6 a1 .
SLIDE 20
Monge metric for 6-components WDVV
gik(a) = (a4)2 −2a5 2a4 −(a1a4 + a3) a2 1 −2a5 −2a3 a2 a1 2a4 a2 2 −a1 −(a1a4 + a3) −a1 (a1)2 a2 a1 1 Remark: the metric can be found in few seconds by computer.
SLIDE 21 Example: generic value of n
The system of conservation laws: u1
t = u2 x,
u2
t = u3 x, ...,
un−1
t
= un
x,
un
t = [u1u3 − (u2)2]x.
The third-order Hamiltonian operator’s Monge metric: gij = 2a2 −a1 1 −a1 1 1 1 1 and the Hamiltonian is H = −1 2a1(D−1a2)2 + 1 2
N
(D−1am)(D−1aN+2−m). Problem: integrability for n 4?
SLIDE 22
Open problems
◮ Integrability for n 4 of the systems of conservation laws. ◮ Geometry of WDVV equations. All of them have a
third-order H.o. and all of them are linear line congruences.
◮ What happens when the fluxes are functions of first or
second order derivatives?
◮ Non-local Hamiltonian operators of second and third order,
and their compatibility with hydrodynamic-type systems.
◮ Extension to symplectic operators, local and non-local.
SLIDE 23
Symbolic computations
Within the REDUCE CAS (now free software) we use the packages CDIFF and CDE, freely available at http://gdeq.org. CDE (by RV) can compute symmetries and conservation laws, local and nonlocal Hamiltonian operators, Schouten brackets of local multivectors, Fr´ echet derivatives (or linearization of a system of PDEs), formal adjoints, Lie derivatives of Hamiltonian operators. Cooperation with AC Norman (Trinity College, Cambridge) to improvements and documentation of REDUCE’s kernel. Forthcoming book, in cooperation with JS Krasil’shchik and AM Verbovetsky: The symbolic computation of integrability structures for partial differential equations, to appear in the series Texts and Monographs in Symbolic Computation, Springer, 2017.
SLIDE 24
Thank you!
Contacts: raffaele.vitolo@unisalento.it
