Algebraic theory of integrable PDE with Alberto De Sole and - PDF document

Algebraic theory of integrable PDE with Alberto De Sole and collaborators (Wakimoto, Barakat, Carpentier, Valeri, Turhan) 1. Compatible evolution equations and integrability 2. Variational di ff erential forms 3. Local and non-local Poisson structure 4. Some non-commutative algebra: principal ideal rings 5. Local and non-local Poisson vertex algebras (PVA) 6. Lenard–Magri scheme of integrability of bi-Hamiltonian equations. 7. Hamiltonian reduction of PVA and generalized Drinfeld–Sokolov hierarchies 8. Dirac reduction of PVA 1

Evolution equation is a PDE of the form du dt = P ( u, u ′ , . . . , u ( n ) ) , (1)   u 1 . .  , u i = u i ( t, x ) is a function in one independent  where u = . u ℓ variable x , and t (time) is a parameter;   P 1 . .   ∈ V ℓ , V algebra of “di ff erential functions”. P = . P ℓ This equation is called compatible with another evolution equa- tion du = Q ( u, u ′ , . . . , u ( m ) ) dt 1 if “the corresponding flows commute”: d d d d u = dtu. dt dt 1 dt 1 2

Compute the LHS using the chain rule: � d d ∂ Q Q ( u, u ′ , . . . , u ( m ) ) = ∂ n P i = X P Q, dt dt 1 ∂ u ( n ) i ; n ∈ Z + i where ∂ = d dx is the total derivative , and � ∂ ( ∂ n P i ) X P = ∂ u ( n ) i ; n ∈ Z + i is the evolutionary vector field with characteristic P ∈ V ℓ . Hence, � d � dt , d u = [ X P , X Q ] = X [ P,Q ] , dt 1 where (2) [ P, Q ] = X P Q − X Q P is a Lie algebra bracket on V ℓ . Thus, equations dt du = P , du dt 1 = Q are compatible i ff the corre- sponding evolutionary vector fields commute. 3

Evolution equation is called integrable if it can be included in an infinite hierarchy of linearly independent compatible evolution equations: du = P n , [ P m , P n ] = 0 , m, n ∈ Z + , dt n called an integrable hierarchy . Thus, classification of integrable evolution equations = classifi- cation of infinite-dimensional (maximal) abelian subalgebras L in the Lie algebra of evolutionary vector fields V ℓ with the bracket (2). Trivial examples of integrable hierarchies: 1. linear: u t n = u ( n ) , since X u ( m ) u ( n ) = u ( m + n ) 2. dispersionless: u t f = f ( u ) u ′ , � � ′ 2 + fgu ′′ . f dg du + g d f since X f ( u ) u ′ ( g ( u ) u ′ ) = u du 4

Nontrivial examples of integrable hierarchies: u ′′ + uu ′ u t = (Burgers) u ′′′ + uu ′ u t = ( KdV ) u ′′′ + u 2 u ′ u t = ( mKdV ) u ′′′ + u ′ 2 u t = ( pKdV ) u ′′′ + u ′ 3 u t = ( LKdV ) u ′′′ − 3 u ′′′ 2 + h ( u ) u t = (Krichever–Novikov) u ′ 2 u ′ � �� � Schwarz KdV h ( u ) polynomial of degree at most 4 . Shabat, Sokolov, Mikhailov,..., Meshkov Theorem. Up to au- tomorphism of the algebra of di ff erential functions, there are only nine more integrable equations of the form u t = u ′′′ + f ( u, u ′ , u ′′ ). 5

Folklore Conjecture. Any order ≥ 7 integrable evolution equa- tion in one function u is contained in the hierarchy of a non-trivial integrable equation of order ≤ 5. In other words, any maximal infinite-dimensional subalgebra of V with bracket (2) contains a non-central element of order ≤ 5. There are partial classificational results on 2-component equa- tions, the most famous among them is the non-linear Schr¨ odinger: � u t = v ′′ + 2 v ( u 2 + v 2 ) v t = − u ′′ − 2 u ( u 2 + v 2 ) . I shall now discuss the other part of the problem: how to prove integrability. But first we have to answer the usually neglected question: 6

What is a di ff erential fuction f ∈ V ? An algebra of di ff erential functions is a di ff erential algebra V with the derivation ∂ (total derivative), endowed with commuting derivations ∂ , i = 1 , . . . , ℓ ; n ∈ Z + , ∂ u ( n ) i subject to two axioms: ∂ 1 f = 0 for all but finite number of i, n . ∂ u ( n ) i � � ∂ ∂ 2 , ∂ = (basic identity). ∂ u ( n ) ∂ u ( n − 1) i i Axiom 1 is needed, otherwise X P Q is divergent. Axiom 2 is satisfied for the main example, the algebra of di ff er- ential polynomials: V = F [ u ( n ) i | i = 1 , . . . , ℓ ; n ∈ Z + ] ∂ u ( n ) = u ( n +1) . i i Arbitrary V is its extension, for example, for KN we need to in- vert u ′ . ∂ − 1 cannot be defined if we want both axioms to hold! Note: 7

S.-S. Chern. In life both men and women are important. Like- wise in geometry both vector fields and di ff erential forms are im- portant. In our theory vector fields are evolutionary vector fields X P ( P ∈ V ℓ ) . They commute with ∂ = X u ′ . This tells us how to define varia- tional di ff erential forms . 8

Ordinary di ff erential forms (dual to all vector fields) are � i 1 ,...,i k du ( u 1 ) ∧ · · · ∧ du ( n k ) f n,...,n k ω = i 1 i k with the usual de Rham di ff erential d : Ω 0 = V d d Ω 2 → · · · � → � Ω 1 → � and derivation ∂ : ∂ ( du ( n ) i ) = du ( n +1) . i Axiom 2. of V (the basic identity) is equivalent to the property that ∂ commutes with d . Therefore we can define the variational complex by letting Ω k = � Ω k : Ω k / ∂ � d d d → � Ω 1 / ∂ � → � Ω 2 / ∂ � Ω 1 Ω 2 V/ ∂ V → ... Here V/ ∂ V is the space (not algebra any more) of local func- tionals , the universal space where we can perform integration by parts. Now we can describe the variational complex more ex- plicitely: 9

V/ ∂ V → V ⊕ ℓ → skew-adjoint matrix di ff erential operators on V ℓ → ... � � f f �→ δ δ u F �→ D F − D ∗ F where � δ � � δ δ f ( − ∂ ) n ∂ f δ u = , = ∂ u ( n ) δ u j δ u i j n ∈ Z + i is the variational derivative; � ∂ F i ∂ n ( D F ) ij = ∂ u ( n ) n ∈ Z + j is the Frechet derivative. Note that δ (a) δ u ◦ ∂ = 0 ( ⇔ axiom 2) (Euler) δ u is self-adjoint (Helmholtz), is the condition on F ∈ V ⊕ ℓ (b) D δ f to be a variational derivative (exact 1-form is closed) 10

Theorem. Let ∂ V m,i = { f ∈ V | f = 0 , ( n, j ) > ( m, i ) } ∂ u ( n ) j ∂ and suppose that V m,i = V m,i . Then the variational complex ∂ u ( m ) i is exact. One can always embed V in a larger algebra of di ff erential functions � V s.t. the variational complex becomes exact. Note that we a have a non-degenerate pairing between the space of evolutionary vector fields = V ℓ and the space of varia- tional 1-forms Ω 1 = V ⊕ ℓ , induced from the usual pairing of vector fields with di ff erential 1-forms: � (3) ( X P | ω Q ) = ( P | Q ) := P · Q ∈ V/ ∂ V . 11

An e ff ective way of constructing an integrable equation is to use Poisson structures. What is a local (or non-local) Poisson structure on V ? Physicists define it by the following formula: (4) { u i ( x ) , u j ( y ) } = H ij ( u ( y ) , u ′ ( y ) , . . . , u ( n ) ( y ); ∂ / ∂ y ) δ ( x − y ) , � where f ( y ) δ ( x − y ) = f ( x ) and H = ( H ij ) is an ℓ × ℓ matrix di ff erential (or pseudo-di ff erential) operator, whose coe ffi cients are functions in u, u ′ , . . . , u ( n ) . Extending this formula (4) by Leibniz’s rule and bilinearity to f, g ∈ V , we obtain (5) � � ∂ f ( x ) ∂ g ( y ) ∂ m x ∂ n { f ( x ) , g ( y ) } = y { u i ( x ) , u j ( y ) } . ∂ u ( m ) ∂ u ( n ) i,j m,n ∈ Z + i j Integrating (5) by parts in x , we obtain (for g = u j ): � � f, u } H = H δ (6) { f . δ u Integrating (5) by parts in x and in y , we obtain: � � � � � δ δ u · H ( ∂ ) δ g f (7) { f, g } H = δ u . 12

Definition (a) An ℓ × ℓ matrix di ff erential operator H is called a (local) Poisson structure on V if (7) is a Lie algebra bracket on V/ ∂ V . This happens i ff H ∗ = − H and [ H, H ] (Schouten bracket) = 0. (b) Given a Poisson structure H on an algebra of di ff erential � functions V and a local functional h (Hamiltonian), the corre- sponding Hamiltonian evolution equation is � du (8) dt = { h, u } H (the corresponding evolutionary vector field is X H δ δ u ). � h (c) Two local functionals are in involution if their commutator (7) is zero. Remark. The map V/ ∂ V → Lie algebra of evolutionary vector fields V ℓ given by � f �→ X H δ � f δ u is a Lie algebra homomorphism. In particular, local functionals in involution correspond to commuting evolutionary vector fields. � Corollary. If h is contained in an infinite-dimensional abelian subalgebra of the Lie algebra ( V/ ∂ V, { , } H ) and dim Ker H < ∞ (i.e. H non-degenerate), then equation (8) is integrable. 13

� dxe λ ( x − y ) . An alternative approach is to apply the Fourier transform � dxe λ ( x − y ) { f ( x ) , f ( y ) } , to both sides of (5). Denoting { f λ g } = we get the Master Formula: ℓ � � ∂ g ( λ + ∂ ) n H ji ( − λ − ∂ ) m ∂ f (9) { f λ g } = . ∂ u ( n ) ∂ u ( m ) i,j =1 m,n ∈ Z + j i This λ -bracket satisfies: (i) (Leibniz rules) { f λ gh } = g { f λ h } + h { f λ g } ; { fg λ h } = { f λ + ∂ g } → h + { f λ + ∂ h } → g ; (ii) (sesquilinearity) { ∂ f λ g } = − λ { f λ g } , { f λ ∂ g } = ( λ + ∂ ) { f λ g } . Theorem. (a) The bracket (7) is a Lie algebra bracket i ff : (iii) (skewcommutativity) { g λ f } = − { f − λ − ∂ g } , (iv) (Jacobi identity) { f λ { g µ h }} − { g µ { f λ h }} = {{ f λ g } λ + µ h } . (b) It su ffi ces to check skewcommutativity of any pair ( u i , u j ) and Jacobi identity for any triple ( u i , u j , u k ). 14