The geometry of hydrodynamic integrability

Transcription

1 1 The geometry of hydrodynamic integrability David M. J. Calderbank University of Bath October 2017

2 2 What is hydrodynamic integrability? A test for integrability of dispersionless systems of PDEs. Introduced by E. Ferapontov and K. Khusnutdinova [FeKh]. Applies to systems which can be written in translation-invariant quasilinear first order form. Integrable means system has sufficiently many hydrodynamic reductions ( Lots of solutions given by nonlinear superpositions of plane waves.) Known to be equivalent to integrability by dispersionless Lax pair in some cases [BFT,DFKN1,DFKN2,FHK]. Computationally intensive: need symbolic computer algebra.

3 Quasilinear first order systems A (translation-invariant first order) quasilinear system is a PDE system of the form [Tsa] (1) A 1 (ϕ) x1 ϕ + + A n (ϕ) xn ϕ = 0 on maps ϕ: R n R s, where A j : R s M k s (R). Example. An N-component hydrodynamic system is a system of the form (2) xj R a = µ aj (R) x1 R a for j {2,... n}, a {1,... N} and functions µ aj of R = (R 1,... R N ) which satisfy the compatibility conditions b µ aj = γ ab (R)(µ bj µ aj ) for all a b and j {2,... n}. An N-component hydrodynamic reduction of (1) is an ansatz ϕ = F (R 1,... R N ) s.t. ϕ satisfies (1) if and only if R satisfies (2).

4 Example: dispersionless KP Dispersionless limit of the Kadomtsev Petviashvili equation: (dkp) (u t + uu x ) x = u yy. Put into quasilinear first order form: u y v x = 0 = u t + uu x v y. Substitute u = U(R 1,... R N ) and v = V (R 1,... R N ) with t R a = λ a (R 1,... R N ) x R a and y R a = µ a (R 1,... R N ) x R a, using u x = a ( au) x R a etc. to get ( (λa + U) a U µ a a V ) x R a a (µ a a U a V ) x R a = 0 = a so require µ a a U = a V and (λ a + U) a U = µ a a V for all a. In particular λ a + U = µ 2 a (the dispersion relation).

5 Method of hydrodynamic reductions In general, the condition for functions F (r 1,... r N ) and µ aj (r 1,... r N ) to define a hydrodynamic reduction of a quasilinear system (1) is itself a PDE system. For dkp, after eliminating V by a V = µ a a U and λ a = µ 2 a U, the PDE system for U and µ a to define a hydrodynamic reduction is that for all a b, b µ a = bu, a b U = 2 au b U µ b µ a (µ b µ a ) 2, Definition. A quasilinear system (1) is integrable by hydrodynamic reductions if the PDE system for N-component reductions is compatible for all N 2. (It then admits solutions depending on N functions of 1-variable.) In fact the 2-component system is always compatible and it is enough to check N = 3 [FeKh].

6 6 Hydrodynamic integrability of dkp In the dkp case, a tedious computation of the derivatives of the system yields (3) (4) c ( b µ a ) = bu cu (µ b +µ c 2µ a) (µ b µ c) 2 (µ b µ a)(µ c µ a), c ( a b U) = 4 au bu cu ((µ a) 2 +(µ b ) 2 +(µ c) 2 µ aµ b µ aµ c µ b µ c) (µ a µ b ) 2 (µ a µ c) 2 (µ b µ c) 2, for all distinct a, b, c. Since the RHS of (3) is symmetric in b, c and the RHS of (4) is totally symmetric in a, b, c, the system is compatible. This is how the method works for one specific PDE with just one quadratic nonlinearity. For anything remotely general, the computations are brutal.

7 7 What is going on? Two clues to some underlying geometric meaning. The dispersion relation. For dkp, this says [z 0, z 1, z 2 ] = [1, λ a, µ a ] is a point on z 0 z 1 + uz 2 0 = z2 2. This quadric is the characteristic variety of dkp. For general hydrodynamic reductions, the characteristic momenta ω a = n j=1 µ aj dx j are on the characteristic variety. Papers [BFT,DFKN1,DFKN2,FHK] showing that for three particular classes of systems, hydrodynamic reductions are nice submanifolds with respect to some interesting geometric structure on the codomain of ϕ. Also inspiring ideas of A. Smith [Smi1,Smi2].

8 8 Plan for rest of talk Explain geometry of hydrodynamic reductions using the characteristic correspondence of a quasilinear system. Use some algebraic geometry (projective embeddings) and differential geometry (nets) to give a fairly general result which unifies aforementioned observations of [BFT,DFKN1,DFKN2,FHK]. (But no progress yet on the harder, computationally intensive parts of these papers e.g. showing equivalence of hydrodynamic and Lax integrability.)

9 Quasilinear systems revisited Natural context for quasilinear systems (QLS): Maps ϕ: M Σ where M is an affine space modelled on an n-dimensional vector space t and Σ is an s-manifold. Have dϕ = ψ, dx Ω 1 (M, ϕ T Σ) where ψ C (M, t ϕ T Σ) and dx Ω 1 (M, t) is the tautological isomorphism TM = M t. QLS is ψ C (M, ϕ Ψ) for a vector subbundle Ψ t T Σ over Σ (locally defined as kernel of some A: t T Σ R k ). Hydrodynamic case: Σ has coordinates r a : a A = {1,... s} and functions µ a : Σ t s.t. Ψ is spanned by µ a ra : a A. Equivalently, setting ω a = µ a, dx, the 2-forms ω a dr a pull back to zero by (id, ϕ): M M Σ. (A very simple exterior differential system whose compatibility condition is dω a dr a = 0 a A.)

10 10 The characteristic correspondence Projective bundle P(t T Σ) Σ has subbundle R with fibre R p := {[ξ Z] : ξ t, Z T p Σ} i.e., rank one tensors Segre image of P(t ) P(T p Σ). Definition. Let Ψ t T Σ be a QLS. Rank one variety of Ψ is R Ψ := R P(Ψ). Characteristic and cocharacteristic varieties of Ψ are projections χ Ψ and C Ψ of R Ψ to Σ P(t ) and P(T Σ) resp. Characteristic correspondence of Ψ: Σ P(t ) χ Ψ π χ R Ψ π C C Ψ P(T Σ) (Assumed smooth double fibration.)

11 11 Examples Hydrodynamic system: χ Ψ = {[µ a ] : a A}, C Ψ = {[ ra ] : a A}, R Ψ = {[µ a ra ] : a A}. dkp: ϕ = (u, v): M = R 3 Σ = R 2. Then Ψ (u,v) is {(u x, u y, u t ) (1, 0) + (u y, u t + uu x, v t ) (0, 1)} and rank one elts have (u x, u y, u t ) and (u y, u t + uu x, v t ) lin. dep., giving R Ψ (u,v) = {(λ2, λµ, µ 2 uλ 2 ) (λ, µ) : λ, µ R}. Then C Ψ = P 1, and χ Ψ is a u-dependent conic in P 2. For Σ t V Gr n (t V ), ϕ: M Σ is derivative of u : M V iff ψ(x) Ψ ϕ(x) with Ψ p := t T p Σ S 2 t V t t V. Then χ Ψ p = {[ξ] P(t ) : ξ v T p Σ for some v V } C Ψ p = {[ξ v] P(T p Σ)} R Ψ p = {[ξ ξ v] P(t T p Σ)}. Get many examples this way (including Ferapontov et al.).

12 Hydrodynamic reductions revisited Seek to write ϕ = S R with R : M R N and S : R N Σ so that ϕ solves Ψ iff a A = {1,... N}, dr a µ a (R), dx = 0 i.e., dr a = f a (R) µ a (R), dx for some functions f a. Chain rule: dϕ = R ds dr = a A dr a a S(R) = f a (R) µ a (R), dx a S(R) = ψ, dx, a A where ψ = f a (R)µ a (R) a S(R). a A Want many solns: µ a a S Ψ Definition. An N-component hydrodynamic reduction of a QLS Ψ t T Σ is a map (S, [µ 1 ],... [µ N ]): R N χ Ψ Σ Σ χ Ψ (N-fold fibre product) s.t. µ a a S is in Ψ for all a, and the hydrodynamic system defined by µ a is compatible.

13 Main result So far: turned simple-minded but fearsome calculus into abstract nonsense geometry. No PDE person would call this progress. So do we win anything? Theorem. Let Ψ t T Σ be a compliant QLS. Then modulo natural equivalences, generic N-component hydrodynamic reductions of Ψ, with N dim Σ, correspond bijectively to N-dimensional cocharacteristic nets in Σ. Remaining business: Explain what is a compliant QLS (alg. geom.) Explain what is a cocharacteristic net (diff. geom.) Prove the theorem

14 14 Algebraic geometry: projective embeddings χ Ψ and C Ψ are fibrewise projective varieties in projectivized vector bundles, and the corresponding dual tautological line bundles pull back to line bundles L χ χ Ψ and L C C Ψ. For a line bundle L over a bundle of projective varieties over Σ, let H 0 (L) Σ be the bundle of fibrewise regular sections. Have canonical maps Σ t H 0 (L χ ) and T Σ H 0 (L C ) given by restricting fibrewise sections of the dual tautological line bundles to χ Ψ and C Ψ. If χ Ψ and C Ψ are not contained (fibrewise) in any hyperplane, these maps are injective, hence fibrewise linear systems, and surjectivity means that these linear systems are complete.

15 Compliant QLS A QLS is compliant if the following conditions hold: 1. the characteristic correspondence maps are isomorphisms, and we let ζ Ψ = π χ π 1 C be the induced isomorphism C Ψ χ Ψ ; 2. the canonical maps Σ t H 0 (L χ ) and T Σ H 0 (L C ) are isomorphisms; 3. V Ψ := H 0 (L C (ζ Ψ ) L χ) Σ is a nonzero vector bundle, and the canonical vector bundle map T Σ t V Ψ induced by the transpose of the tensor product map H 0 ((ζ Ψ ) L χ ) H 0 (L C (ζ Ψ ) L χ) H 0 (L C ) is an embedding; 4. if rank(v Ψ ) 2, no 2-dimensional submanifold of Σ has rank one tangent space in t V Ψ. Key point: under isomorphism in 1., L C is at least as ample as L χ by 3., so T Σ has a tensor product decomposition using 2.

16 Differential geometry: nets A pre-net on an N-manifold Q is a direct sum decomposition TQ = j J D j into rank one distributions D j TQ for j J := {1,... N}. A pre-net D j : j J on Q is integrable if for every subset I J, D I := i I D i is an integrable distribution (i.e., tangent to a foliation with #I dimensional leaves); an integrable pre-net is called a net. Frobenius theorem gives characterizations of integrability. Also need a special class of nets. If D j : j J is a pre-net on Q, and TQ V t for a line bundle V Q and a vector space t, then each D i defines a line subbundle M i of Q t. May then require that for any section X i of D i, have d Xi M j M i M j. If this holds then D j : j J is a net and will be called a conjugate net. (Well known when Q is an affine space with translation group t.) 16

17 Cocharacteristic nets Let Ψ t T Σ be a compliant QLS with T Σ t V Ψ. An N-dimensional cocharacteristic net in Σ is an N-dimensional submanifold S : R N Σ such that: 1. the net spanned by a S : a A satisfies [ a S] C Ψ ; and 2. if V Ψ has rank one, the net is conjugate. Clearly a hydrodynamic reduction defines a net satisfying 1. Conversely, given such a net, the embedding of C Ψ into P(t V Ψ ) gives a S = µ a v a for some local sections v a of S V Ψ. The main point is to show that the compatibility of the hydrodynamic system with characteristic momenta µ a, dx is equivalent to 2.

18 Proof of Theorem Choose a basis for t and rescale characteristic momenta s.t. µ a1 = 1. Then have b S k = µ bk b S 1 = µ bk v b for k {1,... n}. Differentiate by a and commute partial derivatives to obtain (5) ( a µ bk ) b S 1 ( b µ ak ) a S 1 = (µ ak µ bk ) a b S 1. Dividing by µ ak µ bk, RHS is independent of k so ( a µ bk ) ( aµ bl b µ ak v b = ) bµ al v a. µ ak µ bk µ al µ bl µ ak µ bk µ al µ bl Both sides are zero unless v a and v b are lin. dep., i.e., multiples of some v V Ψ, say. But then span of a S = µ a v a and b S = µ b v b is span{µ a, µ b } span{v}, i.e., entirely rank one. For rank(v Ψ ) > 1, the set where this holds has empty interior by compliancy, so hydrodynamic compatibility condition is satisfied on dense complement, hence everywhere by continuity.

19 The rank one case If a µ bk = γ ba (µ ak µ bk ) for a b, have a b S k = ( a µ bk ) b S 1 + µ bk a b S 1 = γ ba (µ ak µ bk ) b S 1 + µ bk (γ ab a S 1 + γ ba b S 1 ) by (5) so S is conjugate. = γ ab (v a /v b ) b S k + γ ba (v b /v a ) a S k Conversely, if S is conjugate with a b S k = α ab b S k + β ab a S k for a b, then taking k = 1, have On the other hand a b S 1 = α ab b S 1 + β ab a S 1 = α ab v b + β ab v a. µ bk a b S 1 = a b S k ( a µ bk ) b S 1 = α ab µ bk v b +β ab µ ak v a ( a µ bk )v b Now eliminate a b S 1 to obtain α ab µ bk v b + β ab µ bk v a = α ab µ bk v b + β ab µ ak v a ( a µ bk )v b and hence a µ bk = β ab (v a /v b )(µ ak µ bk ).

20 References [BFT] P. A. Burovskii, E. V. Ferapontov and S. P. Tsarev, Second order quasilinear PDEs and conformal structures in projective space, Int. J. Math. 21 (2010) [DFKN1] B. Doubrov, E. V. Ferapontov, B. Kruglikov and V. Novikov, On the integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5), arxiv: [DFKN2] B. Doubrov, E. V. Ferapontov, B. Kruglikov and V. Novikov, Integrable systems in 4D associated with sixfolds in Gr(4,6), arxiv: [FHK] E. V. Ferapontov, L. Hadjikos and K. R. Khusnutdinova, Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, Int. Math. Res. Notices (2010) [FeKh] E. V. Ferapontov and K. R. Khusnutdinova, On the Integrability of (2+1)-Dimensional Quasilinear Systems, Comm. Math. Phys. 248 (2004) [Smi1] A. D. Smith, Integrable GL(2) geometry and hydrodynamic partial differential equations, Comm. Anal. Geom. 18 (2010) [Smi2] A. D. Smith, Towards generalized hydrodynamic integrability via the characteristic variety, Fields Institute, Toronto (2013). [Tsa] S. P. Tsarev, Geometry of hamiltonian systems of hydrodynamic type. Generalized hodograph method, Izv. AN USSR Math. 54 (1990)