Methods of tangent and cotangent coverings for Dubrovin-Novikov - PowerPoint PPT Presentation

Methods of tangent and cotangent coverings for Dubrovin-Novikov integrability operators R. Vitolo Joint work with E. Ferapontov, M. Pavlov arXiv:2017 Geometry of Integrable systems SISSA, 5-7 June 2017

Hamiltonian PDEs An evolutionary system of PDEs F = u i t − f i ( t, x, u j , u j x , u j xx , . . . ) = 0 admits a Hamiltonian formulation if � δH � u i t = A ij δu j where A = ( A ij ) is a Hamiltonian operator, i.e. a differential operator A ij = a ijσ D σ A ∗ = − A such that and [ A, A ] = 0 D σ = D x ◦ · · · ◦ D x (total x -derivatives σ times). Finding Hamiltonian operators/PDEs is hard.

Symmetries A Hamiltonian equation shows a correspondence between conservation laws and symmetries. Generalized symmetries are vector functions ϕ i = ϕ i ( u j , u j x , u j xx , . . . ) such that � ℓ F ( ϕ i ) = D t ϕ i − ∂f i σ D σ ϕ j = 0 , ∂u j F k = 0 where ℓ F is the Fr´ echet derivative of F .

Conservation laws A conservation law is a one-form ω = Adx + Bdt which is closed modulo the equation: dω = ∇ F where ∇ = a τσ k D τσ F k . The vector function xx , . . . ) = ( − 1) | τσ | D τσ a τσ ψ k = ψ k ( u j , u j x , u j k | F =0 represents uniquely the conservation law and fulfills the equation � � � ∂f j ℓ ∗ F ( ϕ i ) = − D t ψ i + ( − 1) | σ | D σ σ ψ j = 0 ∂u i F k = 0 where ℓ ∗ F is the formal adjoint of ℓ F ;

A necessary condition If an equation admits a Hamiltonian formulation, this implies that A maps conservation laws into symmetries: ℓ F ◦ A = ( A ′ ) ∗ ◦ ℓ ∗ A : almost-Hamiltonian op. F The condition can be extended to all integrability operators : F ◦ S = S ′ ◦ ℓ F ℓ ∗ S : almost-symplectic op. ℓ F ◦ R = R ′ ◦ ℓ F R : recursion operator F ◦ C = ( C ′ ) ∗ ◦ ℓ ∗ ℓ ∗ C : co-recursion operator F Note that A ′ , S ′ , R ′ , C ′ are arbitrary. Almost : it is a necessary condition . . .

Cotangent covering Kersten, Krasil’shchik, Verbovetsky, JGP 2003. Introducing new variables p k , p kx , p kxx , . . . we can represent operators by linear functions: A ( ψ ) = a ijσ D σ ψ j A = a ijσ p jσ ⇔ Then a Hamiltonian operator fulfills the equations � � ∂f j � ℓ ∗ F ( p ) = − p i,t + ( − 1) | σ | D σ σ p j = 0 T ∗ : ∂u i and ℓ F ( A ) = 0 . t − f i = 0 F = u i The system T ∗ is the cotangent covering . It is invariant .

Tangent covering Introducing new variables q k , q k x , q k xx , . . . we can represent operators by linear functions: R ( ϕ ) = a iσ j D σ ϕ j R = a iσ j q j ⇔ σ Then a recursion operator fulfills the equations t − ∂f i σ q j � ℓ F ( q ) = q i σ = 0 ∂u j T : and ℓ F ( R ) = 0 . t − f i = 0 F = u i The system T is the tangent covering . It is invariant .

Example: Hamiltonian operators for KdV The KdV equation: u t = uu x + u xxx The linearization: ℓ F = D t − uD x − u x − D xxx The adjoint linearization: ℓ ∗ F = − D t + uD x + D xxx The cotangent covering for the KdV equation: � p t = p xxx + up x u t = u xxx + uu x The equation ℓ F ( A ) = 0 has the two solutions: A 1 = p x or A 1 = D x A 2 = 1 A 2 = 1 3(3 p 3 x + 2 up x + u x p ) or 3(3 D xxx + 2 uD x + u x ) For example, ℓ F ( A 1 ) = D t p x − uD x p x − u x p x − D xxx p x .

Example: recursion operator for KdV The tangent covering of KdV: � q t = u x q + uq x + q xxx T : u t = u xxx + uu x Unfortunately, the equation for recursion operators ℓ F ( R ) = 0 has the only trivial solution R = q . However, there is a conservation law on T : ω = qdx + ( uq + q xx ) dt . We can introduce a new non-local variable w such that w x = q, w t = uq + q xx . Then we have the non-local recursion operator R = q xx + 2 3 uq + 1 R = D xx + 2 3 u + 1 3 u x D − 1 3 u x w or x

Applications to Dubrovin–Novikov operators The cotangent covering of a hydrodynamic-type system is: � j,i ) u j p i,t = ( V k i,j − V k x p k + V k i p k,x j ( u ) u j u i t = V i x A first-order Dubrovin–Novikov Hamiltonian operator: A i = g ij p jx + Γ ij k u k x p j . Tsarev’s compatibility conditions are the coefficients of the linear equation in p k,σ , ℓ F ( A ) = 0: g ik V j � k = g jk V i D t A i − V i x A k − V i j D x A j = 0 j,k u j k ⇔ ∇ i V k j = ∇ j V k i

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.

Application to third-order DN operators Dubrovin–Novikov operators can be defined for arbitrary orders. Here we consider the third order ones: A ij x + b ij 3 = g ij ( u ) D 3 k ( u ) u k x D 2 x + [ c ij xx + c ij k ( u ) u k km ( u ) u k x u m x ] D x + d ij k ( u ) u k xxx + d ij km ( u ) u k x u m xx + d ij kmn ( u ) u k x u m x u n x , Potemin’s canonical form in Casimirs: A ij 3 = D x ( g ij D x + c ij k u k x ) D x Remark: g ij 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

Compatibility conditions (Ferapontov, Pavlov, V. 2017) Theorem Let A 3 be a Hamiltonian operator. Then j u j x = ( V i ) ,j u j u i t = V i x admits a Hamiltonian formulation with A 3 if and only if g im V m = g jm V m  j i  c mkj V m + c mik V m + c mji V m = 0 , (1) i j k V k i,j = g ks c smj V m + g ks c smi V m  i j 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 g ij or if the unknown is V i . Remark. No Hamiltonian needed at this stage!

Properties of the systems of conservation laws Following a construction of Agafonov and Ferapontov t = ( V i ) ,j u j (1996-2001) we associate to each system u i x a congruence of lines in P n +1 with coordinates [ y 1 , . . . , y n +2 ] y i = u i y n +1 + V i y n +2 Theorem. ◮ The congruence is linear : there are n linear relations between u i , V i , u i V j − u j V i . ◮ The system is linearly degenerate , and non diagonalizable . ◮ V i = ψ i i ψ β α w α , where ψ i α is determined by g ij = ϕ αβ ψ α j and w α are linear functions. This means that V i = p ( u j ) /q ( u j ) where p ( u j ), q ( u j ) are polynomials of degree n , n − 1.

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. km u m + ω γ k , and w γ = η γ m u m + ξ γ . Let us set ψ γ k = ψ γ Let us set u i = b i x ; the system becomes b i t = V i ( b x ). Theorem. ◮ Hamiltonian op. A 3 = − g ij ( b x ) D x − c ij k ( b x ) b k xx � � b p b q + 3 η γ p ψ β 2 ω β p η γ ◮ Hamiltonian H = − � 1 qm b m x + 1 ϕ βγ [ q � q b q � 2 ψ β pq ξ γ b p b q x + ξ γ ω β 1 x ] dx � � 1 ◮ n Casimirs C α = 2 ψ α mk b k x + ω α � b m dx m � � � 3 ϕ βγ ω β q ψ γ 2 ϕ βγ ω β p ω γ ◮ momentum P = − 1 pm b m x + 1 b p b q dx q

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 x = ( a i u i + a ) dx + ( a i V i + b ) dt d ˜ t = ( b i u i + c ) dx + ( b i V i + d ) dt d ˜

Classification results Theorem. Let u i 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 u i . 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 P 5 . There exist one generic nontrivial integrable example.

Example: WDVV equations in 3-comp. xxt − f xxx f xtt setting u 1 = f xxx , u 2 = f xxt , From f ttt = f 2 u 3 = f xtt we have u 1 t = u 2 x , u 2 t = u 3 x , t = (( u 2 ) 2 − u 1 u 3 ) x , u 3 endowed with a third-order Hamiltonian operators with nonlocal Hamiltonian � � 1 � − 1 u 2 � 2 + ∂ x − 1 u 2 ∂ x − 1 u 3 2 u 1 � H = − ∂ x dx. (Ferapontov, Galvao, Mokhov, Nutku, 1995). It is bi-Hamiltonian and up to a non-trivial transformation is the 3-wave equation (Zakharov, Manakov, ∼ 1970).

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: a i y = ( v i ( a )) x , a i z = ( w i ( a )) x , where v 1 = a 2 , w 1 = a 3 , v 2 = a 4 , v 3 = w 2 = a 5 , w 3 = a 6 , v 4 = f yyy = 2 a 5 + a 2 a 4 v 5 = w 4 = f yyz = a 3 a 4 + a 6 , , a 1 a 1 v 6 = w 5 = f yzz = 2 a 3 a 5 − a 2 a 6 , a 1 w 6 = f zzz = ( a 5 ) 2 − a 4 a 6 + ( a 3 ) 2 a 4 + a 3 a 6 − 2 a 2 a 3 a 5 + ( a 2 ) 2 a 6 . a 1

Monge metric for 6-components WDVV − ( a 1 a 4 + a 3 ) ( a 4 ) 2 − 2 a 5 2 a 4 a 2  1  − 2 a 5 − 2 a 3 a 2 a 1 0 0     2 a 4 a 2 − a 1 2 0 0   g ik ( a ) = − ( a 1 a 4 + a 3 )   − a 1 ( a 1 ) 2 0 0 0     a 2 a 1 0 0 0 0   1 0 0 0 0 0 Remark : the metric can be found in few seconds by computer.