The geometry of hydrodynamic integrability David M. J. Calderbank University of Bath October 2017 1
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. 2
Quasilinear first order systems A (translation-invariant first order) quasilinear system is a PDE system of the form [Tsa] (1) A 1 ( ϕ ) ∂ x 1 ϕ + · · · + A n ( ϕ ) ∂ x n 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) ∂ x j R a = µ aj ( R ) ∂ x 1 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). 3
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 ( ∂ a U ) ∂ x R a etc. to get � � ( µ a ∂ a U − ∂ a V ) ∂ x R a = 0 = � ( λ a + U ) ∂ a U − µ a ∂ a V � ∂ x R a a 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 ). 4
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 U ∂ a ∂ b U = 2 ∂ a U ∂ b U ∂ b µ a = , ( µ b − µ a ) 2 , µ b − µ a 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]. 5
Hydrodynamic integrability of dKP In the dKP case, a tedious computation of the derivatives of the system yields ∂ b U ∂ c U ( µ b + µ c − 2 µ a ) (3) ∂ c ( ∂ b µ a ) = ( µ b − µ c ) 2 ( µ b − µ a )( µ c − µ a ) , ∂ c ( ∂ a ∂ b U ) = 4 ∂ a U ∂ b U ∂ c U (( µ a ) 2 +( µ b ) 2 +( µ c ) 2 − µ a µ b − µ a µ c − µ b µ c ) (4) , ( 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. 6
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 = z 2 2 . This quadric is the characteristic variety of dKP. For general hydrodynamic reductions, the characteristic momenta ω a = � n j =1 µ aj d x 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]. 7
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.) 8
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 d ϕ = � ψ, d x � ∈ Ω 1 ( M , ϕ ∗ T Σ) where ◮ ψ ∈ C ∞ ( M , t ∗ ⊗ ϕ ∗ T Σ) and ◮ d x ∈ 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 ⊗ ∂ r a : a ∈ A . Equivalently, setting ω a = � µ a , d x � , the 2-forms ω a ∧ d r a pull back to zero by ( id , ϕ ): M → M × Σ. (A very simple exterior differential system whose compatibility condition is d ω a ∧ d r a = 0 ∀ a ∈ A .) 9
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: R Ψ π χ π C ✛ ✲ C Ψ ≤ P ( T Σ) Σ × P ( t ∗ ) ≥ χ Ψ (Assumed smooth double fibration.) 10
Examples ◮ Hydrodynamic system: χ Ψ = { [ µ a ] : a ∈ A} , C Ψ = { [ ∂ r a ] : a ∈ A} , R Ψ = { [ µ a ⊗ ∂ r a ] : 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 ( u , v ) = { ( λ 2 , λµ, µ 2 − u λ 2 ) ⊗ ( λ, µ ) : λ, µ ∈ R } . R Ψ 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) } p = { [ ξ ⊗ ξ ⊗ v ] ∈ P ( t ∗ ⊗ T p Σ) } . R Ψ Get many examples this way (including Ferapontov et al.). 11
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 } , d R a ∧ � µ a ( R ) , d x � = 0 i.e., d R a = f a ( R ) � µ a ( R ) , d x � for some functions f a . d ϕ = R ∗ d S ◦ d R = � Chain rule: d R a ⊗ ∂ a S ( R ) a ∈A � = f a ( R ) � µ a ( R ) , d x � ⊗ ∂ a S ( R ) = � ψ, d x � , 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. 12
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 13
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. 14
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