classiﬁcation of homogeneous third order diﬀerential ..._pavlov_m.,_potemin_g.,_vitolo_… ·...

Classification of Homogeneous Third Order

Differential-Geometric Poisson Brackets

E.V. Ferapontov, M.V. Pavlov, G.V. Potemin, R. Vitolo

UK-Russia-Italian Collaboration

17.10.2013

EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 1 / 17

First Order Differential-Geometric Poisson Brackets

First order differential-geometric Poisson brackets

{ui (x), uj (x ′)} = [g ij (u(x))∂x + bijk (u(x))ukx ]δ(x − x ′)

satisfy the skew-symmetry and Jacobi identity iff g ij (u) is a

symmetric nondegenerate tensor and bijk (u) = −g isΓjsk , where Γ

jsk is

a Levi-Civita connection, while the metric gij (u) is flat.







jsk is


Corresponding Hamiltonian systems are

uit = [g ij∂x − g isΓjsku

kx ]

δH

δuj,

where the Hamiltonian functional H =∫

h(u,ux ,uxx ,uxxx , ...)dx .







jsk is


Corresponding Hamiltonian systems are

uit = [g ij∂x − g isΓjsku

kx ]

δH

δuj,

where the Hamiltonian functional H =∫

h(u,ux ,uxx ,uxxx , ...)dx .

In a particular case H =∫

h(u)dx , we have N componenthydrodynamic type system

uit = (∇i∇jh)ujx .


Third Differential-Geometric Poisson Brackets

{ui (x), uj (x ′)} = [g ij (u(x))∂3x + bijk (u(x))ukx ∂2x

+(c ijk (u(x))ukxx + c ijkm(u(x))u

kx u

mx )∂x

+d ijk (u(x))u

kxxx + d ij

km(u(x))ukx u

mxx + d ij

kmn(u(x))ukx u

mx u

nx ]δ(x − x ′).





kx u

mx )∂x

+d ijk (u(x))u

kxxx + d ij

km(u(x))ukx u

mxx + d ij

kmn(u(x))ukx u

mx u

nx ]δ(x − x ′).

In special coordinate system d ijk (a) = 0, d ij

km(a) = 0, d ijkmn(a) = 0. Then





kx u

mx )∂x

+d ijk (u(x))u

kxxx + d ij

km(u(x))ukx u

mxx + d ij

kmn(u(x))ukx u

mx u

nx ]δ(x − x ′).



{ai (x), aj (x ′)} = ∂x [gij (a(x))∂x + bijk (a(x))u

kx ]δ

′(x − x ′).

satisfy the skew-symmetry and Jacobi identity iff





kx u

mx )∂x

+d ijk (u(x))u

kxxx + d ij

km(u(x))ukx u

mxx + d ij

kmn(u(x))ukx u

mx u

nx ]δ(x − x ′).




kx ]δ

′(x − x ′).


gmk,n + gkn,m + gmn,k = 0,

gmn,kl =1

9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l

+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n),





kx u

mx )∂x

+d ijk (u(x))u

kxxx + d ij

km(u(x))ukx u

mxx + d ij

kmn(u(x))ukx u

mx u

nx ]δ(x − x ′).




kx ]δ

′(x − x ′).



gmn,kl =1


+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n),

where cnkm = 13 (gmn,k − gkn,m) and cijk = giqgjpb

pqk .


Metric. Nonlinear System


gmn,kl =1


+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n).

This system has a solution in the form

gik = g(0)ik + g

(1)ikma

m + g(2)ikmna

man.

The nonlinear part

gmn,kl =1


+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n)

can be parameterized in the form

gik = φβγψiβψkγ,

where ψiβ = ψiβmam + ξ iβ and ψiβk = −ψkβi .




gmn,kl =1



This system has a solution in the form

gik = g(0)ik + g

(1)ikma

m + g(2)ikmna

man.

The nonlinear part

gmn,kl =1


+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n)

can be parameterized in the form

gik = φβγψiβψkγ,

where ψiβ = ψiβmam + ξ iβ and ψiβk = −ψkβi .Lie Algebras.



The linear partgmk,n + gkn,m + gmn,k = 0

can be solved by virtue of the Monge parameterization:

ds2 = gik(a)daidak = ~dT Q̂~d ,

where ~d = (da1, da2, ..., daN , a1da2 − a2da1, a2da3 − a3da2, ...) andQ̂ is a constant nondegenerate symmetric matrix.



The linear partgmk,n + gkn,m + gmn,k = 0

can be solved by virtue of the Monge parameterization:

ds2 = gik(a)daidak = ~dT Q̂~d ,

where ~d = (da1, da2, ..., daN , a1da2 − a2da1, a2da3 − a3da2, ...) andQ̂ is a constant nondegenerate symmetric matrix.

Passive Form.gmk,n + gkn,m + gmn,k = 0,

gmn,kl =1




Particular Case

Theorem: In the particular case cijk = 0 (three distinct indices), a generalsolution of the above nonlinear system is parameterized by a solepolynomial function G of degree 4 such that

gkk = −2G,kk , gkm = G,km, ckkm = −ckmk = G,kkm, k 6= m,

while all other connection coefficients cijk = 0.


Particular Case

Theorem: In the particular case cijk = 0 (three distinct indices), a generalsolution of the above nonlinear system is parameterized by a solepolynomial function G of degree 4 such that

gkk = −2G,kk , gkm = G,km, ckkm = −ckmk = G,kkm, k 6= m,

while all other connection coefficients cijk = 0.

This leads to

gmm = − ∑p 6=m

Rmp(ap)2 − 2 ∑

p 6=m

Hmpap +Dm,

gkm = Rkmakam +Hkma

k +Hmkam + Fkm, k 6= m,

where Rkm = Rmk ,Fkm = Fmk , Dk are constants, and

cmmk = −cmkm = Rkmak +Hmk , k 6= m.


Two Component Case

Theorem: only two metrics in 2-component case:

g(1)ik =

(

1− (a2)2 1+ a1a2

1+ a1a2 1− (a1)2

)

, g(2)ik =

(

−2a2 a1

a1 0

)

.


Three Component Case

Using the ansatz cijk = 0 we are able to reduce the nonlinear system to:

R12F12 = H12H21, F13R13 = H13H31, F23R23 = H23H32,

D1R12R13 +H212R13 +H2

13R12 = 0,

D2R12R23 +H221R23 +H2

23R12 = 0,

D3R13R23 +H231R23 +H2

32R13 = 0.

In the generic case Rij 6= 0 we obtain the three-parameter family of metrics

gik =

−(a2 + β2)2 − (a3)2 a1(a2 + β2) a3(a1 + β1)

a1(a2 + β2) −(a1)2 − (a3 + β3)2 a2(a3 + β3)

a3(a1 + β1) a2(a3 + β3) −(a1 + β1)2 − (a2)2


Three Component Case

Using the ansatz cijk = 0 we are able to reduce the nonlinear system to:

R12F12 = H12H21, F13R13 = H13H31, F23R23 = H23H32,

D1R12R13 +H212R13 +H2

13R12 = 0,

D2R12R23 +H221R23 +H2

23R12 = 0,

D3R13R23 +H231R23 +H2

32R13 = 0.

In the generic case Rij 6= 0 we obtain the three-parameter family of metrics

gik =

−(a2 + β2)2 − (a3)2 a1(a2 + β2) a3(a1 + β1)

a1(a2 + β2) −(a1)2 − (a3 + β3)2 a2(a3 + β3)

a3(a1 + β1) a2(a3 + β3) −(a1 + β1)2 − (a2)2

The particular cases

1 R12 = 0, R13 6= 0, R23 6= 0,2 R12 = 0, R13 = 0, R23 6= 0,3 R12 = 0, R13 = 0, R23 = 0,

were solved separately, obtaining a complete classification of 53 metrics.EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 8 / 17

WDVV Associativity Equations

In three component case, WDVV associativity equations reduce to thesingle equation

fttt = f 2xxt − fxxx fxtt ,





which can be written as the hydrodynamic type system

at = bx , bt = cx , ct = (b2 − ac)x ,

where a = fxxx , b = fxxt , c = fxtt .





which can be written as the hydrodynamic type system

at = bx , bt = cx , ct = (b2 − ac)x ,

where a = fxxx , b = fxxt , c = fxtt .

This system is bi-Hamiltonian, i.e.

abc

t

= J0δH1 = J1δH0,

where

J0 =

− 32D

12Da Db

12aD

12 (bD +Db) 3

2cD + cxbD 3

2Dc − cx (b2 − ac)D +D(b2 − ac)



and

J1 =

0 0 D3

0 D3 −D2aDD3 −DaD2 D2bD +DbD2 +DaDaD

.



and

J1 =

0 0 D3

0 D3 −D2aDD3 −DaD2 D2bD +DbD2 +DaDaD

.

Here H1 =∫

cdx ,H0 = − 12a(D

−1b)2 − (D−1b)(D−1c).


Lorentzian Metric

The third order Hamiltonian structure is associated with the action

S =1

2

∫

(f 2xt fxxx + fxt ftt)dxdt.


Riemann Curvature Tensor

The Riemann tensor of curvature has the form

Rjikl = gjsRsikl = gjs [∂kΓs

il − ∂lΓsik + Γs

knΓnil − Γs

lnΓnik ],

where

Γijk =

1

2g im(gmj ,k + gmk,j − gjk,m).

However, taking into account


we obtainΓijk = −g imgjk,m.

ThenRjikl = gik,jl − gil ,jk + g sm(gik,mgjl ,s − gil ,mgjk,s).


References

beamericonarticleA.V. Balandin, G.V. Potemin,On non-degenerate differential-geometric Poisson brackets of thirdorder, Russian Mathematical Surveys 56 No. 5 (2001) 976-977.

beamericonarticleP.W. Doyle,Differential geometric Poisson bivectors in one space variable, J.Math. Phys. 34 No. 4 (1993) 1314-1338.

beamericonarticleB.A. Dubrovin and S.P. Novikov,Hamiltonian formalism of one-dimensional systems of hydrodynamictype and the Bogolyubov-Whitham averaging method, Soviet Math.Dokl., 27 (1983) 665–669.

beamericonarticleE.V. Ferapontov,On integrability of 3× 3 semi-Hamiltonian systems of hydrodynamictype which do not possess Riemann invariants, Physica D 63 (1993)50-70.


References

beamericonarticleE.V. Ferapontov,On the matrix Hopf equation and integrable Hamiltonian systems ofhydrodynamic type which do not possess Riemann invariants, Phys.lett. A 179 (1993) 391-397.

beamericonarticleE.V. Ferapontov,Several conjectures and results in the theory of integrable Hamiltoniansystems of hydrodynamic type, which do not possess Riemanninvariants, Teor. Math. Phys., 99 No. 2 (1994) 257-262.

beamericonarticleE.V. Ferapontov,Dupin hypersurfaces and integrable Hamiltonian systems ofhydrodynamic type, which do not possess Riemann invariants, Diff.Geom. Appl., 5 (1995) 121-152.


References

beamericonarticleE.V. Ferapontov,Isoparametric hypersurfaces in spheres, integrable nondiagonalizablesystems of hydrodynamic type, and N-wave systems, Diff. Geom.Appl., 5 (1995) 335-369.

beamericonarticleE.V. Ferapontov, C.A.P. Galvao, O. Mokhov, Y. Nutku,Bi-Hamiltonian structure of equations of associativity in 2-dtopological field theory, Comm. Math. Phys. 186 (1997) 649-669.

beamericonarticleE.V. Ferapontov, O.I. Mokhov,Equations of associativity of two-dimensional topological field theoryas integrable Hamiltonian nondiagonalisable systems of hydrodynamictype, Func. Anal. Appl. 30 No. 3 (1996) 195-203.


References

beamericonarticleE.V. Ferapontov, R. Sharipov,On the first order conservation laws of hydrodynamic type systems,Teor. Math. Phys. 108 No. 1 (1996) 937-952.

beamericonarticleO.I. Mokhov,Symplectic and Poisson structures on loop spaces of smoothmanifolds, and integrable systems, Russian Math. Surveys 53 No. 3(1998) 85–192.

beamericonarticleS.P. Novikov,The geometry of conservative systems of hydrodynamic type. Themethod of averaging for field-theoretical systems, Russian Math.Surveys 40 No. 4 (1985) 85–98.


References

beamericonarticleG.V. Potemin,On Poisson brackets of differential-geometric type, Dokl. Akad. NaukSSSR, 286 (1986) 39–42; Soviet Math. Dokl. 33 (1986) 30–33.

beamericonarticleG.V. Potemin,On third-order Poisson brackets of differential geometry, Russ. Math.Surv. 52 (1997) 617-618.

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classiﬁcation of homogeneous third order diﬀerential ..._pavlov_m.,_potemin_g.,_vitolo_… ·...

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