HYDRODYNAMIC INTEGRABILITY VIA GEOMETRY

DAVID M.J. CALDERBANK

Abstract. This paper develops a geometric approach to the theory of integrability by hydrodynamic reductions to establish an equivalence, for a large class of quasilinear systems, between hydrodynamic integrability and the existence of nets compatible with the geometry induced on the codomain of the system. This unifies and extends known results for three subclasses of such systems. The generalization is obtained by studying the algebraic geometry of the characteristic correspondence of the system, and by introducing a generalized notion of conjugate nets.

1. Introduction

The method of hydrodynamic reductions is a test for the integrability of dispersionless systems introduced by E. Ferapontov and K. Khusnutdinova [9, 11]. It applies to PDE systems which can be written in the following first order quasilinear form:

(1.1)  A_{1}(u)∂_{t_{1}}u + ⋯ + A_{n}(u)∂_{t_{n}}u = 0,

where the unknown is a function u: ℳ → 𝒰 , t = (t_{1}, … t_{n}) are standard coordinates on the open domain ℳ ⊆ 𝔱 ≅ ℝ^{n} (with n ⩾ 3), the codomain 𝒰 is (for simplicity here) an open subset of ℝ^{m}, and each A_{j}: 𝒰 → M_{k×m}(ℝ) is a given matrix-valued function (so k is the number of equations in the system). Note that A_{1}, … A_{n} do not depend explicitly on t, so the system is invariant under local translations in ℳ.

Fundamental examples include N-component hydrodynamic systems of the form

(1.2)  ∂_{t_{j}}R^{a} = κ_{aj}(R)∂_{t_{1}}R^{a},  j ∈ {2, … n}, a ∈ {1, … N},

for R = (R^{1}, … R^{N}): ℳ → 𝒱 , with 𝒱 open in ℝ^{N}, where κ_{aj} are given functions of r = (r_{1}, … r_{N}) ∈ 𝒱 which satisfy the compatibility conditions that

for all a ≠ b ∈ {1, … N} and j ∈ {2, … n}, ∂_{b}κ_{aj} = γ_{ab}(r) (κ_{bj} − κ_{aj}),

with γ_{ab}(r) independent of j and ∂_{b} := ∂_{r_{b}}. Integrability for such systems was shown by Tsarev [23], who called them semi-hamiltonian systems of hydrodynamic type.

An N-component hydrodynamic reduction of (1.1) consists of functions U: 𝒱 → 𝒰 and κ_{aj}: 𝒱 → ℝ, for a ∈ {1, … N} and j ∈ {2, … n}, such that (1.2) is compatible and u = U ∘ R satisfies (1.1) if R = (R^{1}, … R^{N}) satisfies (1.2). These requirements impose a PDE system on the functions U and κ_{aj}, and (1.1) is hydrodynamically integrable if this PDE system is compatible for all N ⩾ 2 (although it suffices to check N = 3 [9, 11]).

Hydrodynamic integrability is known to be equivalent to the existence of a dispersionless Lax pair in some cases [3, 6, 7, 8, 10], but it does not require one to find the Lax pair. Thus it has the benefit that it is an algorithmic test of integrability (for this class of systems). However, it is computationally intensive, and for all but the simplest cases, carrying out the test requires symbolic computer algebra. Other algorithmic tests are available when n = 3 or 4 using Einstein–Weyl or self-dual conformal geometry respectively (called “integrable background geometries” in [4]), but these only apply when the characteristic variety of the system (see below) is a quadric—see [2, 5] for the general relation to Lax pairs in this setting. In some overlapping cases, Einstein–Weyl/self-dual integrability and hydrodynamic integrability are known to be equivalent [6, 7, 12], but this is established by direct computation, and a conceptual explanation is lacking.

Example 1.1. To illustrate the process in a case amenable to hand computation, let n = 3, (t_{1}, t_{2}, t_{3}) = (t, x, y) and consider an equation of the form

(1.3)  (u_{x} + τ(u)u_{t})_{t} = u_{yy},

for a scalar function u, where τ is a given function of one variable, and t, x, y subscripts denote partial derivatives. Then (1.3) is equivalent to the first order quasilinear system

(1.4)  u_{y} − v_{t} = 0,  u_{x} + τ(u)u_{t} − v_{y} = 0,

for u = (u, v). A hydrodynamic reduction in this case consists of functions U, V, μ_{a} = κ_{a2} and λ_{a} = κ_{a3} on an open subset 𝒱 of ℝ^{N}, and the compatibility of (1.2) requires that

(1.5)  \frac{∂_{b}μ_{a}}{μ_{b} − μ_{a}} = \frac{∂_{b}λ_{a}}{λ_{b} − λ_{a}}  for all a ≠ b.

Using the chain rule for u = U ∘ R and v = V ∘ R, the system

(1.6)  R_{x}^{a} = μ_{a}(R)R_{t}^{a},  R_{y}^{a} = λ_{a}(R)R_{t}^{a},

yields

u_{y} − v_{t} = ∑_{a}(λ_{a} ∂_{a}U − ∂_{a}V)R_{t}^{a}, u_{x} + τ(u)u_{t} − v_{y} = ∑_{a}((μ_{a} + τ(U)) ∂_{a}U − λ_{a} ∂_{a}V )R_{t}^{a}.

Hence to satisfy (1.4) for any solution R of (1.6), the ansatz requires that

for all a ∈ {1, … N}, λ_{a} ∂_{a}U = ∂_{a}V and (μ_{a} + τ(U)) ∂_{a}U = λ_{a} ∂_{a}V.

In particular

μ_{a} + τ(U) = λ_{a}^{2},

which is called the dispersion relation and can be used to eliminate μ_{a}. Then (1.5) becomes

\frac{2λ_{a}∂_{b}λ_{a} − τ′(U)∂_{b}U}{λ_{b}^{2} − λ_{a}^{2}} = \frac{∂_{b}λ_{a}}{λ_{b} − λ_{a}} , i.e., ∂_{b}λ_{a} = −\frac{τ′(U) ∂_{b}U}{λ_{b} − λ_{a}} ,

for a ≠ b, while the symmetry of ∂_{b}∂_{a}V = ∂_{b}(λ_{a} ∂_{a}U) in a, b implies

∂_{b}∂_{a}U = −2\frac{τ′(U) ∂_{a}U ∂_{b}U}{(λ_{b} − λ_{a})^{2}}

(and then V may also be eliminated). Hydrodynamic integrability requires that no further equations for U, λ_{a} arise in this way. In particular, the symmetry of ∂_{c}∂_{b}λ_{a} in b, c (for a, b, c distinct) implies that τ′′ = 0, and the symmetry of ∂_{c}∂_{b}∂_{a}U is then consistent with the system. Thus (1.3) is hydrodynamically integrable if and only if either τ is constant, so that (1.3) is linear, or τ has degree 1, so that (1.3) is equivalent by an affine transformation of u to the dispersionless Kadomtsev–Petviashvilli (dKP) equation (u_{x} + uu_{t})_{t} = u_{yy}.

The computational intensiveness motivates the search for a geometric underpinning to hydrodynamic integrability, and there is much that is already understood. First the dispersion relation has a geometric interpretation. In the case of equation (1.3), it states that for each a, [1, μ_{a}(r), λ_{a}(r)] is a point on the (u-dependent) projective variety

{[ξ_{1}, ξ_{2}, ξ_{3}] ∈ P(ℝ^{3}) : ξ_{1}ξ_{2} + τ(u)ξ_{1}^{2} = ξ_{3}^{2}},

at u = U(r). This is the characteristic variety of (1.3), and for hydrodynamic reductions of general quasilinear systems (1.1), the characteristic momenta θ_{a} = ∑_{j=1}^{n} κ_{aj} dt_{j}, with κ_{a1} = 1, always define points [θ_{a}] on the characteristic variety of the equation.

In the important case n = 3, A. Odeskii and V. Sokolov [17, 18] have axiomatised and studied the systems of Gibbons–Tsarev type [14] arising from hydrodynamic reductions. Here the characteristic variety is a bundle of curves over 𝒰 and so the “Gibbons–Tsarev structure” may be described using 1-parameter families of vector fields on 𝒰 .

For arbitrary n, there are now several works [3, 6, 8] showing that for particular classes of quasilinear systems (1.1), hydrodynamic reductions correspond to nice submanifolds with respect to some interesting geometric structure on the codomain 𝒰 of u.

The main result of this paper places these latter classes in common framework. One of the central ingredients is inspired by work of A. Smith [19, 20, 21, 22], who emphasises the role not only of the characteristic variety, but also the so-called rank one variety. For a quasilinear system (1.1), the latter is quite straightforward to define: the system imposes a linear constraint on the derivative du_{z} at each z ∈ ℳ, depending only on p = u(z), hence defining a p-dependent linear subspace ℰ of m × n matrices (p ∈ 𝒰 ); the rank one variety is the projective variety R^{ℰ} in P(ℰ) of rank one matrices in ℰ up to scale. Thus it is a p-dependent projective variety lying in the Segre variety of all m × n matrices of rank one, up to scale. The latter is canonically isomorphic to the product P(ℝ^{n})×P(ℝ^{m}) of the projective spaces of 𝔱 ≅ ℝ^{n} and ℝ^{m}, because any rank one matrix factorizes as a product θ ⊗ S of a column vector and a row vector, each determined uniquely up to scale.

Thus the rank one variety has projections onto both P(ℝ^{n}) and P(ℝ^{m}); the former is the characteristic variety X^{ℰ} of the system. I am unaware of a standard name for the latter projection C^{ℰ}, so I refer to it as the cocharacteristic variety of the system. Note that, geometrically speaking, the ℝ^{m} appearing here is the tangent space of the codomain 𝒰 at a given point p ∈ 𝒰 . In this more invariant language, the p-dependent linear subspace ℰ defining the equation is really a vector subbundle of the vector bundle 𝔱 ⊗ T 𝒰 with fibre 𝔱 ⊗ T_{p}𝒰 at p ∈ 𝒰 , and the cocharacteristic variety C^{ℰ} is a subbundle of P(T 𝒰 ), i.e., it defines a family of cones in the tangent spaces of 𝒰 , equipping 𝒰 with a potentially interesting geometric structure.

A key observation about the method of hydrodynamic reductions is that the dispersion relation arises because the characteristic momenta are projections of points on the rank one variety. Indeed if u = U ∘ R and R satisfies (1.2) then, by the chain rule,

du = R^{∗}dU ∘ dR = ∑_{a=1}^{N} dR^{a} ⊗ ∂_{a}U(R) = ∑_{a=1}^{N} f_{a}(R)θ_{a}(R) ⊗ ∂_{a}U(R)

where θ_{a}(R) = ∑_{j=1}^{n} κ_{aj}(R)dt_{j}, and (assuming κ_{a1} = 1 as before) f_{a}(R) = ∂_{t_{1}}R^{a}. For this to yield solutions of (1.1) for given U and any such R, θ_{a}⊗∂_{a}U must must be a section of ℰ for each a ∈ {1, … N}, and this is clearly also sufficient. Since θ_{a} ⊗∂_{a}U is a rank one matrix, its span [θ_{a} ⊗ ∂_{a}U] must lie in the rank one variety R^{ℰ}; thus [θ_{a}] is characteristic and [∂_{a}U] is cocharacteristic. The latter condition means that U maps coordinate lines in 𝒱 ⊆ ℝ^{N} onto curves in 𝒰 whose tangent lines belong to the cocharacteristic variety C^{ℰ} at each point. This is a special kind of net (see e.g. [1]) with respect to the geometric structure C^{ℰ} defines on 𝒰 .

In the classes of examples studied in [3, 6, 8], such a net turns out to be sufficient to recover the hydrodynamic reduction, as the coordinate derivatives ∂_{a}U implicitly contain information about the characteristic momenta. This can be understood using the correspondence between characteristic and cocharacteristic varieties given by the rank one variety. For well-behaved systems, this correspondence is a bijection, i.e., for each [θ] in the characteristic variety, there is a unique [S] in the cocharacteristic variety such that [θ⊗S] is in the rank one variety, and vice versa. This is the case in Example 1.1: for given U, the cocharacteristic variety consists of all [u, v] in P(ℝ^{2}) and the characteristic variety is the Veronese embedding of P(ℝ^{2}) as a conic in P(ℝ^{3}). In general such a bijective rank one correspondence means that the characteristic and cocharacteristic varieties are simply different projective embeddings of the same underlying abstract variety.

Unfortunately this is not sufficient in general to describe hydrodynamic reductions purely in terms of the cocharacteristic variety, because it is necessary also to encode the compatibility condition of (1.2), and this depends on the embedding of X^{ℰ}. Motivated again by [3, 6, 8], I describe a class of compliant quasilinear systems for which the characteristic embedding can be recovered from the cocharacteristic one, and a class of cocharacteristic nets for which the following result holds.

Theorem 1.1. Let ℰ ⩽ 𝔱^{∗}⊗T 𝒰 be a compliant quasilinear system and N ⩽ dim 𝒰 . Then up to natural equivalences, there is a bijection between generic N-component hydrodynamic reductions of ℰ and generic N-dimensional cocharacteristic nets in 𝒰 .

It follows that hydrodynamic integrability for compliant systems is equivalent to the existence sufficiently many cocharacteristic nets in 𝒰 , a condition which is amenable to analysis by differential geometric and/or representation theoretic techniques (see e.g. [3, 13, 19, 20]).

Note that Example 1.1 is not compliant. However, even when a quasilinear system is not compliant, there may be a reformulation of the system (for example by prolongation or differential covering) which is, and this can be done for the dKP equation (see Remark 2.4).

The structure of the paper is as follows. In Section 2, I explain in more detail the affine geometry of first order quasilinear systems and the characteristic correspondence. Section 3 reviews the algebraic geometry of projective embeddings in order to define the notion of a compliant quasilinear system. In Section 4, I turn to the differential geometry of nets, and explain what is a cocharacteristic net. This turns out to include conjugate nets as a special case. The proof of Theorem 1.1 is then given in Section 5.

Throughout the paper, all manifolds are assumed connected, and can be real or complex, and all maps between manifolds are smooth over the base field (hence holomorphic in the complex case). For a vector bundle ℰ over a manifold ℳ, the space of (smooth) k-forms on ℳ with values in ℰ will be denoted Ω_{ℳ}^{k} (ℰ), or Ω_{ℳ}^{k} (V) if ℰ = ℳ × V is a trivial bundle; when k = 0 this is just the space of (smooth) sections of ℰ or functions ℳ → V. The derivative of a map F: ℳ → 𝒩 between manifolds is denoted dF ∈ Ω_{ℳ}^{1} (F^{∗}T 𝒩 ), but in the case that 𝒩 is a vector space V, F^{∗}T V ≅ ℳ × V is trivial, so dF ∈ Ω_{ℳ}^{1} (V).

2. Quasilinear systems and their characteristic geometry

2.1. Affine geometry of quasilinear systems. First order quasilinear systems (1.1) are translation invariant, and so the natural context for their study is affine geometry. An open subset ℳ of an affine space is an example of a flat affine manifold. More precisely, if the affine space is modelled on an n-dimensional vector space 𝔱, then there is an exact 1-form dt ∈ Ω_{ℳ}^{1} (𝔱) which defines a bundle isomorphism T ℳ ≅ ℳ × 𝔱. Under this isomorphism, the constant 𝔱-valued functions are identified with the infinitesimal generators of the local free and transitive translation action on ℳ. The affine coordinate t: ℳ → 𝔱 is defined up to an additive constant in 𝔱, which is usually fixed by specifying an origin z ∈ ℳ with t(z) = 0. I will not do this herein, nor will I assume ℳ is an open subset of an affine space: it suffices to have an exact 1-form dt, the tautological 1-form, inducing T ℳ ≅ ℳ × 𝔱.

Definition 2.1. Let u: ℳ → 𝒰 be a map between manifolds, with domain ℳ a flat affine n-manifold as above, and derivative du ∈ Ω_{ℳ}^{1} (u^{∗}T 𝒰 ). Then the affine derivative of u is the unique ψ ∈ Ω_{ℳ}^{0} (𝔱^{∗} ⊗ u^{∗}T 𝒰 ) with du = ⟨ψ, dt⟩, where angle brackets denote contraction of 𝔱 with 𝔱^{∗}.

In particular if f is a scalar-valued function on ℳ, then its affine derivative is in Ω_{ℳ}^{0} (𝔱^{∗}); if this function is constant, f is said to be affine, and 𝔥 will denote the space of affine functions on ℳ.

Remarks 2.2. In view of the definition, it is natural to denote the affine derivative by du/dt; indeed if ℳ is 1-dimensional, this is the ordinary derivative of a function of 1-variable, expressed as a ratio of 1-forms. More generally, in local affine coordinates,

dt = (dt_{1}, … dt_{n}) and \frac{du}{dt} = (∂_{t_{1}}u, … ∂_{t_{n}}u).

(the partial derivatives of u).

The affine derivative restricts to a linear map from 𝔥 to 𝔱^{∗}, with kernel the constant functions; if this map surjects (as it does locally) then there is a natural affine coordinate t: ℳ → 𝔥^{∗} defined by ⟨t(z), f⟩ = f(z). This identifies ℳ locally with the affine hyperplane ⟨t, 1⟩ = 1 in 𝔥^{∗}; in particular, ⟨dt, c⟩ = 0 for any constant function c, so dt takes values in 𝔱, and agrees with the tautological 1-form, justifying the notation.

Although the above invariant language is not essential, it provides a convenient and computationally concise way to formalize the theory of quasilinear systems.

Definition 2.3. A quasilinear system (QLS), on maps from a flat affine space ℳ with tautological 1-form dt ∈ Ω_{ℳ}^{1} (𝔱) to an m-manifold 𝒰 , is vector subbundle ℰ ⩽ 𝔱^{∗} ⊗ T 𝒰 over 𝒰 ; a solution of the QLS ℰ is a map u: ℳ → 𝒰 with du/dt ∈ Ω_{ℳ}^{0} (u^{∗}ℰ).

If ℰ_{p} ⩽ 𝔱^{∗} ⊗ T_{p}𝒰 has codimension k for each p ∈ 𝒰 , then it is locally the kernel of k independent linear forms on 𝔱^{∗} ⊗ T_{p}𝒰 depending smoothly on p, and so the QLS imposes k linear constraints on the derivative of u. Identifying 𝔱 with ℝ^{n} or ℂ^{n} (via affine coordinates t_{i}) and T 𝒰 locally with 𝒰 × ℝ^{m} or 𝒰 × ℂ^{m}, the t_{i} components of these linear forms combine, for each j ∈ {1, … n}, into k × m matrix valued functions A_{j}(p) of p ∈ 𝒰 , yielding the coordinate expression (1.1) for the QLS.

Example 2.1. Consider the first order formulation of the generalized dKP equation (1.3) for u = (u, v): ℳ = ℝ^{3} → 𝒰 = ℝ^{2}. Thus the fibre of ℰ at (u, v) ∈ 𝒰 = ℝ^{2} is the set of (u_{t}, u_{x}, u_{y}) ⊗ (1, 0) + (v_{t}, v_{x}, v_{y}) ⊗ (0, 1) satisfying (1.4), which may be solved for (v_{t}, v_{y}) to give

ℰ_{(u,v)} = {(u_{t}, u_{x}, u_{y}) ⊗ (1, 0) + (u_{y}, v_{x}, u_{x} + τ(u)u_{t}) ⊗ (0, 1)}.

Example 2.2. A QLS in n = 2 dimensions is often called a hydrodynamic system; it takes the form B(u)∂_{t}u = C(u)∂_{x}u for affine coordinates (x, t) and matrix valued functions B, C. Multiplying by dx∧dt, this system can be rewritten as (B(u)dx+C(u)dt)∧du = 0.

This has an analogue in any dimension n. For later use, the unknown map will be denoted R: ℳ → 𝒱 , where the codomain 𝒱 is an N-manifold. The hydrodynamic system is then determined by a section A ∈ Ω_{𝒱}^{0} (𝔱^{∗} ⊗ End(T 𝒱 )), and R solves this system if

(2.1)  R^{∗}A ∧ \frac{dR}{dt} = 0 in Ω_{ℳ}^{0} (∧^{2}𝔱^{∗} ⊗ R^{∗}T 𝒱 ),

where the wedge product is on 𝔱^{∗} and the action of End(T 𝒱 ) on T 𝒱 is used to multiply coefficients. Thus ℰ is the kernel of the bundle map 𝔱^{∗} ⊗ T 𝒱 → ∧^{2}𝔱^{∗} ⊗ T 𝒱 sending ψ to A ∧ ψ. The QLS can also be formulated as an exterior differential system (EDS) on ℳ ×𝒱 using the projections π_{ℳ} and π_{𝒱} to ℳ and 𝒱 : for any 1-form α on 𝒱 , π_{𝒱}^{∗} α(A) ∈ Ω_{ℳ ×𝒱}^{1} (𝔱^{∗}) and π_{ℳ}^{∗} dt ∈ Ω_{ℳ ×𝒱}^{1} (𝔱), hence their contracted wedge product ⟨π_{𝒱}^{∗} α(A)∧π_{ℳ}^{∗} dt⟩ is a 2-form on ℳ × 𝒱 . Then R solves (2.1) if and only if these 2-forms pullback to zero by (id_{ℳ} , R): ℳ → ℳ × 𝒱 , i.e., the graph of R is an integral manifold of the EDS (differential ideal) generated by these 2-forms.

Evidently if R = Ψ ∘ R̃ for some diffeomorphism Ψ of 𝒱 , then dR = R̃^{∗}dΨ ∘ dR̃, and so the system transforms to ⟨R̃^{∗}Ã, dt⟩ ∧ dR̃ = 0 where Ã = (dΨ)^{−1} ∘ Ψ^{∗}A ∘ dΨ.

The fundamental case for hydrodynamic integrability is when A can be simultaneously diagonalized by such a diffeomorphism Ψ, with the eigenvectors tangent to coordinate lines on 𝒱 . Thus there are coordinates r_{a} : a ∈ A = {1, … N} on 𝒱 called Riemann invariants, and functions κ_{a}: 𝒱 → 𝔱^{∗} called characteristic momenta such that A = ∑_{a∈A} κ_{a}dr_{a}⊗∂_{r_{a}}. These coordinates may be assumed (for local questions) to realise 𝒱 as an open subset of ℝ^{N}. Setting θ_{a} = ⟨κ_{a}, dt⟩ for a ∈ A, the above EDS is generated by the 2-forms θ_{a} ∧ dr_{a} : a ∈ A on ℳ × 𝒱 , and the QLS ℰ is spanned by κ_{a} ⊗ ∂_{r_{a}} : a ∈ A.

Note that there is some residual gauge freedom in the system: each κ_{a} may be scaled by a function f_{a} on 𝒱 (scaling equivalence), and each r_{a} may be replaced by r′_{a} = ρ_{a}(r_{a}) for functions ρ_{a} of one variable (reparametrization equivalence); these do not change ℰ, or the differential ideal generated by θ_{a} ∧ dr_{a} : a ∈ A.

I now turn to the examples that this paper seeks to unify. These examples have in common that they all come from a PDE system, for a function w: ℳ → 𝒲 with values in an affine space 𝒲 , which is not a priori a QLS, but which may be reformulated as a QLS on the derivative of w. There are three cases: general grassmannian equations [6, 7], in which the derivative of w satisfies an algebraic constraint; hessian or Hirota type equations [8, 19, 20], in which w is the derivative of a scalar function φ whose second derivative satisfies an algebraic constraint, and invariant wave equations [3], in which the equation on w is linear in the second derivative with coefficients depending only on the first derivative (thus the equation is invariant under translation of w). These three classes of examples will be referred to as types G, H and I respectively.

Examples 2.3. Let 𝒲 be an affine space modelled on a vector space V with dim V = q. Then the derivative of a smooth map w: ℳ → 𝒲 may be viewed as a V -valued 1-form dw ∈ Ω_{ℳ}^{1} (V) (strictly speaking this is w^{∗}ds∘dw where ds: T 𝒲 → V is the tautological 1-form of 𝒲 )—hence it has an affine derivative u = dw/dt: ℳ → 𝔱^{∗} ⊗ V

Alternatively, for each z ∈ ℳ, the graph of the linear map u(z) ∈ 𝔱^{∗}⊗V ≅ Hom(𝔱.V) is in Gr_{n}(𝔱 ⊕V), the grassmannian of n-dimensional subspaces 𝔱 ⊕V, so u may be viewed as a map Γ_{u}: ℳ → Gr_{n}(𝔱 ⊕ V); z ↦ (dt_{z}, dw_{z})(T_{z}ℳ). This viewpoint reveals PGL(𝔱 ⊕ V) as a natural symmetry group [6, 7], with 𝔱^{∗} ⊗ V being identified as the open subset of n-dimensional subspaces of 𝔱 ⊕ V on which the projection to 𝔱 is an isomorphism.

Now an arbitrary u: ℳ → 𝔱^{∗} ⊗ V arises locally in this way from w: ℳ → 𝒲 if and only if du/dt: ℳ → 𝔱^{∗} ⊗ 𝔱^{∗} ⊗ V takes values in S^{2}𝔱^{∗} ⊗ V. With respect to affine coordinates t = (t_{1}, … t_{n}) on ℳ, u has components u_{i}: ℳ → V, and this is simply the statement that for all i, j ∈ {1, … n}, ∂_{t_{i}}u_{j} = ∂_{t_{j}}u_{i}, i.e., the integrability condition to write u_{i} = ∂_{t_{i}}w: ℳ → V for each i ∈ {1, … n}.

There are now two ways to impose equations on u. First Γ_{u} can be required to take values in an m-dimensional submanifold of Gr_{n}(𝔱 ⊕ V). Let 𝒰 be the corresponding submanifold of the open subset 𝔱^{∗} ⊗ V. The integrability condition on u may be conveniently described using the map (id_{ℳ} , u): ℳ → ℳ ×𝒰 as follows: let p ∈ Ω_{ℳ ×𝒰}^{0} (𝔱^{∗}⊗V) be the tautological coordinate given by projection onto 𝒰 ⊆ 𝔱^{∗}⊗V and let ⟨dp∧dt⟩ ∈ Ω_{ℳ ×𝒰}^{2} (V) be the wedge product of dp with π_{ℳ}^{∗} dt ∈ Ω_{ℳ ×𝒰}^{1} (𝔱) where their values are contracted using the natural map 𝔱^{∗} ⊗ V × 𝔱 → V. Then the symmetry of du/dt means equivalently that (id_{ℳ} , u)^{∗}⟨dp ∧ dt⟩ = 0. This motivates a second class of quasilinear constraints on u that (id_{ℳ} , u)^{∗}⟨dp ∧ Φ⟩ = 0 for further differential forms Φ ∈ Ω_{ℳ ×𝒰}^{k} (𝔱) of the form Φ = ∑_{I,j} F_{I,j}(p)dt_{I} ⊗ ∂_{t_{j}}, where I = {i_{1}, … i_{k}} is a multi-index with 1 ⩽ i_{1} < i_{2} < ⋯ < i_{k} ⩽ n and dt_{I} = dt_{i_{1}} ∧ dt_{i_{2}} ∧ ⋯ ∧ dt_{i_{k}}. Thus the QLS is the EDS generated by components of ⟨dp∧dt⟩ and the additional differential forms ⟨dp∧Φ⟩.

The three classes G,H,I of QLS are special cases of these EDS. (G) This is the case of a general submanifold 𝒰 , with no additional differential forms Φ. Since 𝒰 has dimension m, it has codimension m′ = qn−m and so may be described locally by equations F_{k}(p) = 0 for k ∈ {1, … m′}. If u: ℳ → 𝒰 with (id_{ℳ} , u)^{∗}⟨dp ∧ dt⟩ = 0 then u comes from a smooth map w: ℳ → 𝒲 which solves the system

F_{k}(dw/dt) = 0 for all k ∈ {1, … m′}.

For any p ∈ 𝒰 , the fibre ℰ_{p} of the QLS is the intersection of 𝔱^{∗} ⊗ T_{p}𝒰 with S^{2}𝔱^{∗} ⊗ V. (H) This is an important special case of (G) where 𝒲 = V = 𝔱^{∗} and the submanifold is a hypersurface in the lagrangian grassmannian of maximal isotropic subspaces in 𝔱 ⊕ V = 𝔱 ⊕ 𝔱^{∗} with respect to the natural symplectic form given by ω(X_{1} + ξ_{1}, X_{2} + ξ_{2}) = ⟨ξ_{1}, X_{2}⟩ − ⟨ξ_{2}, X_{1}⟩. The lagrangian grassmannian has S^{2}𝔱^{∗} as an open subset which, given affine coordinates t = (t_{1}, … t_{n}), has coordinates p_{ij} = p_{ji} (i, j ∈ {1, … n}). Hence the hypersurface 𝒰 in S^{2}𝔱^{∗} is given locally by F (…, p_{ij}, …) = 0 for some function F of \frac{1}{2}n(n + 1) variables. Any u: ℳ → 𝒰 with (id_{ℳ} , u)^{∗}⟨dp ∧ dt⟩ = 0 now comes from a scalar function φ on ℳ (via w = dφ/dt) which solves the equation

F (…, ∂_{t_{i},t_{j}}^{2}φ, …) = 0.

In this case, for any p ∈ 𝒰 , T_{p}𝒰 ⩽ S^{2}𝔱^{∗} is the kernel of dF_{p}, so ℰ_{p} is the intersection of 𝔱^{∗} ⊗ T_{p}𝒰 with S^{3}𝔱^{∗}. (I) Let q = 1 (so that V is one-dimensional), with 𝒰 open in 𝔱^{∗} ⊗ V ⊆ Gr_{n}(𝔱 ⊕ V), and suppose Q(p) ∈ Ω_{ℳ ×𝒰}^{0} (S^{2}𝔱) is a quadratic form on 𝔱^{∗} with coefficients depending only on p. Thus in affine coordinates t = (t_{1}, … t_{n}) with p = (p_{1}, … p_{n}), Q = ∑_{i,j} F_{ij}(p_{1}, … p_{n})∂_{t_{i}} ⊗ ∂_{t_{j}} with F_{ij} = F_{ji}, and defines Φ ∈ Ω_{ℳ ×𝒰}^{n−1} (𝔱) by the partial contraction Φ = ∑_{i,j} F_{ij}(p_{1}, … p_{n})∂_{t_{i}} ⌟ (dt_{1} ∧ ⋯ ∧ dt_{n}) ⊗ ∂_{t_{j}}. Now if (id_{ℳ} , u) pulls back both ⟨dp ∧ dt⟩ and ⟨dp ∧ Φ⟩ to zero, then u comes from a scalar function w on ℳ which solves the equation

∑_{i,j} F_{ij}(∂_{t_{1}}w, … ∂_{t_{n}}w)∂_{t_{i},t_{j}}^{2}w = 0.

At each p ∈ 𝒰 , ℰ_{p} is the kernel of Q(p) ⊗ id_{V} in 𝔱^{∗} ⊗ T_{p}𝒰 ∩ S^{2}𝔱^{∗} ⊗ V.

Remark 2.4. Up to differential coverings, the dKP equation can be reformulated as a QLS in each of these ways [3, 6, 8]. For example, the equation

(2.2)  w_{xt} + w_{t}w_{tt} = w_{yy}

has type I and so defines a QLS on the first derivatives f = w_{t}, g = w_{x}, and h = w_{y}:

f_{x} = g_{t}, f_{y} = h_{t}, g_{y} = h_{x} and f_{x} − ff_{t} = h_{y}.

On the other hand, by differentiating (2.2) with respect to t, it follows that u = w_{t} satisfies the dKP equation (u_{x} + uu_{t})_{t} = u_{yy}. Finally, writing (2.2) as (w_{x} + \frac{1}{2}w_{t}^{2})_{t} = w_{yy} yields a first order form

w_{x} + \frac{1}{2}w_{t}^{2} = v_{y}, v_{t} = w_{y}

of type G, and a further potential φ with φ_{y} = v and φ_{t} = w reduces this system to the equation φ_{xt} + \frac{1}{2}φ_{tt}^{2} = φ_{yy} of type H.

2.2. The characteristic correspondence. The projective bundle P(𝔱^{∗} ⊗ T 𝒰 ) → 𝒰 has a subbundle R whose fibre at p ∈ 𝒰 is

R_{p} := {[ξ ⊗ Z] ∈ P(𝔱^{∗} ⊗ T_{p}𝒰 ) | ξ ∈ 𝔱^{∗}, Z ∈ T_{p}𝒰 }.

This is the image of the Segre embedding of P(𝔱^{∗}) × P(T_{p}𝒰 ), consisting of the projectivizations of rank one tensors.

Definition 2.5. Let ℰ ⩽ 𝔱^{∗} ⊗ T 𝒰 be a QLS.

• The rank one variety of ℰ is R^{ℰ} := R ∩ P(ℰ).

• The characteristic and cocharacteristic varieties of ℰ are the projections X^{ℰ} and C^{ℰ} of R^{ℰ} to 𝒰 × P(𝔱^{∗}) and P(T 𝒰 ) respectively.

• The characteristic correspondence of ℰ is the diagram

\begin{aligned}& R^{ℰ} \\ 𝒰 × P(𝔱^{∗}) ⊇ X^{ℰ} & ^{π_{X}\underline{}} & C^{ℰ} ⊆ P(T 𝒰 ) & ^{π_{C}^{\overline{}}} \end{aligned}

where π_{X} and π_{C} are the natural projection maps.

For convenience, R^{ℰ}, X^{ℰ} and C^{ℰ} will be assumed to be fibre bundles over 𝒰 whose fibres are projective varieties, so that the characteristic correspondence is a double fibration.

Remark 2.6. For (complex, or real hyperbolic) determined systems, the characteristic variety X^{ℰ} is a hypersurface, hence of dimension n−2. Thus P(ℰ_{p}) does not meet generic fibres of R_{p} over P(𝔱^{∗}), and the fibres it does meet, it generically meets in dimension zero, hence in a single point because these fibres are projective subspaces in the Segre embedding. For m ⩾ n, the same is true for the fibres over P(T_{p}𝒰 ), and hence the characteristic correspondence maps are isomorphisms, and C_{p}^{ℰ} has codimension (m − 1) − (n − 2) = m − n + 1 in P(T_{p}𝒰 ).

Example 2.1 bis. For the generalized dKP equation (1.3), the fibre of the rank one variety R^{ℰ} at (u, v) consists of the nonzero elements with (u_{t}, u_{x}, u_{y}) and (u_{y}, v_{x}, u_{x} + τ(u)u_{t}) linearly dependent, which forces

u_{t}v_{x} = u_{y}u_{x}, u_{y}v_{x} = (u_{x} + τ(u)u_{t})u_{x}, and u_{y}^{2} = u_{t}(u_{x} + τ(u)u_{t}).

The last equation is solved by

u_{t} = cλ_{0}^{2}, u_{x} = c(λ_{1}^{2} − τ(u)λ_{0}^{2}), and u_{y} = cλ_{0}λ_{1}

for constants c, λ_{0}, λ_{1}. Hence cλ_{0}^{2}v_{x} = c^{2}λ_{0}λ_{1}(λ_{1}^{2} − uλ_{0}^{2}) and cλ_{0}λ_{1}v_{x} = c^{2}λ_{1}^{2}(λ_{1}^{2} − τ(u)λ_{0}^{2}), so either cλ_{0} = cλ_{1} = 0 or λ_{0}v_{x} = cλ_{1}(λ_{1}^{2} − uλ_{0}^{2}). Now if λ_{0} = 0, cλ_{1} = 0, otherwise c is divisible by λ_{0}; thus, without loss, c = λ_{0}, v_{x} = λ_{1}^{2}(λ_{1}^{2} − τ(u)λ_{0}^{2}) and

R_{(u,v)}^{ℰ} = {(λ_{0}^{2}, λ_{1}^{2} − τ(u)λ_{0}^{2}, λ_{0}λ_{1}) ⊗ (λ_{0}, λ_{1}) : λ_{0}, λ_{1} ∈ ℝ},

the cocharacteristic variety is C^{ℰ} = P^{1}, and the characteristic variety X^{ℰ} is a u-dependent conic in P^{2}.

Example 2.2 bis. For a (diagonalizable) hydrodynamic system, with Riemann invariants and characteristic momenta r_{a}, κ_{a} : a ∈ A, the description of ℰ immediately yields that [κ_{a} ⊗ ∂_{r_{a}}] are in the rank one variety. It follows generically (if the κ_{a} are pairwise independent) that

X^{ℰ} = {[κ_{a}] : a ∈ A}, C^{ℰ} = {[∂_{r_{a}}] : a ∈ A} and R^{ℰ} = {[κ_{a} ⊗ ∂_{r_{a}}] : a ∈ A}.

Examples 2.3 bis. For a QLS ℰ of type G, H or I, it is straightforward to describe the characteristic correspondence using the description of ℰ given previously. (G) At each p ∈ 𝒰 ,

X_{p}^{ℰ} = {[ξ] ∈ P(𝔱^{∗}) | ξ ⊗ v ∈ T_{p}𝒰 for some nonzero v ∈ V }
C_{p}^{ℰ} = {[ξ ⊗ v] ∈ P(T_{p}𝒰 )}
R_{p}^{ℰ} = {[ξ ⊗ ξ ⊗ v] ∈ P(𝔱^{∗} ⊗ T_{p}𝒰 )}.

Now P(𝔱^{∗}) × P(V) has dimension n + q − 2, so P(T_{p}𝒰 ) generically meets it in a variety of dimension n + q − m′ − 2, so the system is determined for m′ = q, in which case for any [ξ] ∈ X^{ℰ} there is a unique [v] ∈ P(V) such that [ξ ⊗ v] ∈ C^{ℰ}. The references [6, 7] concern the cases q = m′ = 2, with n ∈ {3, 4}. (H) In this case dF_{p} induces a quadratic form Q_{p} on 𝔱^{∗} for each p ∈ 𝒰 , and

X_{p}^{ℰ} = {[ξ] ∈ P(𝔱^{∗}) | Q_{p}(ξ) = 0}
C_{p}^{ℰ} = {[ξ ⊗ ξ] ∈ P(S^{2}𝔱^{∗}) | Q_{p}(ξ) = 0}
R_{p}^{ℰ} = {[ξ ⊗ ξ ⊗ ξ] ∈ P(S^{3}𝔱^{∗}) | Q_{p}(ξ) = 0}.

(I) Again there is a quadratic form Q_{p} (up to scale) on 𝔱^{∗} for each p ∈ 𝒰 , and

X_{p}^{ℰ} = {[ξ] ∈ P(𝔱^{∗}) | Q_{p}(ξ) = 0}
C_{p}^{ℰ} = {[ξ ⊗ v] ∈ P(𝔱^{∗} ⊗ V) | Q_{p}(ξ) = 0}
R_{p}^{ℰ} = {[ξ ⊗ ξ ⊗ v] ∈ P(S^{2}𝔱^{∗} ⊗ V) | Q_{p}(ξ) = 0}.

Types H and I are thus determined equations wherever Q_{p} ≠ 0.

3. Algebraic geometry: compliant quasilinear systems

3.1. Projective embeddings. Motivated by the previous section, I focus now on the case that the fibres over each p ∈ 𝒰 of the characteristic, cocharacteristic and rank one varieties are projective embeddings of the same abstract variety. This is a standard situation in elementary algebraic geometry, where projective embeddings are described using line bundles (see e.g. [15]).

Let Ξ ⊆ P(V) be a projective variety and let L → Ξ be the restriction to Ξ of the dual tautological line bundle O_{V} (1) → P(V), whose fibre at ℓ ∈ P(V) is the dual vector space to ℓ ⩽ V: O_{V} (1)_{ℓ} = ℓ^{∗}. Since the space of regular global sections H^{0}(P(V), O_{V} (1)) is V^{∗}, restriction to Ξ defines a canonical linear map V^{∗} → H^{0}(Ξ, L). Furthermore, this linear map is injective unless Ξ is contained in a hyperplane in P(V).

Definition 3.1. A linear system on an abstract variety Ξ is a line bundle L → Ξ together with a linear subspace W of H^{0}(Ξ, L), and its base locus is B = {x ∈ Ξ | ∀ w ∈ W, w(x) = 0}. The linear system is said to be complete if W = H^{0}(Ξ, L) and basepoint-free if B = ∅.

The reason for these definitions is that if (L, W) is a basepoint-free linear system on Ξ, then there is a map Ξ → P(W^{∗}) sending x ∈ Ξ to the element of P(W^{∗}) corresponding to the hyperplane {w ∈ W : w(x) = 0} (the annihilator of this hyperplane). If this map is an embedding, the linear system is said to be very ample. In particular, if the complete linear system yields an embedding, the line bundle L is said to be very ample.

Lemma 3.2. Let L_{1} and L_{2} be line bundles over Ξ. Then there is a canonical linear map

H^{0}(L_{1}) ⊗ H^{0}(L_{2}) → H^{0}(L_{1} ⊗ L_{2}); ℓ_{1} ⊗ ℓ_{2} ↦ ℓ

defined by pointwise multiplication: ℓ(x) = ℓ_{1}(x) ⊗ ℓ_{2}(x).

Proof. The map sending (ℓ_{1}, ℓ_{2}) ∈ H^{0}(L_{1}) × H^{0}(L_{2}) to ℓ_{1}(x) ⊗ ℓ_{2}(x) is regular in x, so induces a section of L_{1} ⊗ L_{2}; this is bilinear in (ℓ_{1}, ℓ_{2}), and so induces a map on the tensor product. □

Similarly, using the canonical isomorphism (L_{1}^{∗} ⊗ L_{2}) ⊗ L_{1} ≅ L_{2}, there is a canonical pointwise multiplication map

H^{0}(Ξ, L_{1}^{∗} ⊗ L_{2}) ⊗ H^{0}(Ξ, L_{1}) → H^{0}(Ξ, L_{2})

with transpose

(3.1)  Φ: H^{0}(Ξ, L_{2})^{∗} → H^{0}(Ξ, L_{1}^{∗} ⊗ L_{2})^{∗} ⊗ H^{0}(Ξ, L_{1})^{∗}.

Proposition 3.3. Suppose L_{1} and L_{2} are very ample line bundles on Ξ. For j ∈ {1, 2}, let W_{j} = H^{0}(Ξ, L_{j}), ϕ_{j}: Ξ → P(W_{j}^{∗}) the corresponding projective embeddings, V = H^{0}(Ξ, L_{1}^{∗} ⊗ L_{2}), and Φ: W_{2}^{∗} → W_{1}^{∗} ⊗ V^{∗} be given by (3.1). Then for any x ∈ Ξ, Φ maps ϕ_{2}(x) into ϕ_{1}(x) ⊗ L(x) for some one dimensional subspace L(x) of V^{∗}.

Proof. Let ℓ_{1} ∈ W_{1} and ℓ ∈ V; then for any x ∈ Ξ and α ∈ ϕ_{2}(x), ⟨Φ(α), ℓ_{1} ⊗ ℓ⟩ = ⟨α, ℓ_{1}(x)ℓ(x)⟩, which vanishes whenever ℓ_{1}(x) = 0, i.e., ℓ_{1} ∈ ker ϕ_{1}(x). Let e_{1}, … e_{k−1} be a basis of ker ϕ_{1}(x) and extend by e_{k} to a basis of W_{2}. Write Φ(α) = ∑_{j=1}^{k} ε_{j} ⊗ α_{j} where ε_{1}, … ε_{k} is the dual basis and α_{j} ∈ V^{∗}; it then follows by evaluating on e_{j} ⊗ v for j ∈ {1, … k − 1} and any v ∈ V that Φ(α) = ε_{k} ⊗ α_{k}. But ε_{k} vanishes on ker ϕ_{1}(x) and so Φ(α) ∈ ϕ_{1}(x) ⊗ ⟨α_{k}⟩. □

In general, there is no reason to suppose that Φ here injects; indeed V could even be zero. The injectivity can be viewed as a relative ampleness condition.

Definition 3.4. Let L_{1} and L_{2} be line bundles on Ξ. If the canonical map Φ in (3.1) is injective (equivalently Φ^{∗} surjects), L_{2} is said to be more ample than L_{1}.

Example 3.1. The degree d line bundles O_{V} (d) over a projective space P(V) are very ample for d ⩾ 1 and then O_{V} (d_{1}) is more ample than O_{V} (d_{2}) iff d_{1} ⩾ d_{2}.

These ideas apply fibrewise to the bundles of projective varieties X^{ℰ} and C^{ℰ}: the corresponding fibrewise dual tautological line bundles pull back to line bundles L_{X} → X^{ℰ} and L_{C} → C^{ℰ}.

For a line bundle L over a bundle of projective varieties Y → 𝒰 , let H^{0}(L) → 𝒰 denote the bundle of fibrewise regular sections. Then there are canonical maps

(3.2)  𝒰 × 𝔱 → H^{0}(L_{X}) and T^{∗}𝒰 → H^{0}(L_{C})

given by restricting fibrewise sections of the dual tautological line bundles over U × P(𝔱^{∗}) and P(T 𝒰 ) to X^{ℰ} and C^{ℰ} respectively.

If X^{ℰ} and C^{ℰ} are not contained (fibrewise) in any hyperplane, these maps are injective, hence fibrewise linear systems, and surjectivity means that these linear systems are complete. In this situation, the projective embeddings can be recovered from the line bundles, and the relative ampleness of Definition 3.4 can be used (fibrewise) to compare them.

3.2. Compliant quasilinear systems. The following definition covers a wide class of QLS, yet makes the algebraic geometry of the characteristic and cocharacteristic projective embeddings as straightforward as possible, allowing an easy comparison between them.

Definition 3.5. A QLS ℰ ⩽ 𝔱^{∗} ⊗ T 𝒰 is compliant if all of the following conditions hold:

(1) the characteristic correspondence maps π_{X} and π_{C} are isomorphisms, with ζ^{ℰ} = π_{X} ∘ π_{C}^{−1} : C^{ℰ} → X^{ℰ} denoting the induced isomorphism;

(2) the canonical maps 𝒰 ×𝔱 → H^{0}(L_{X}) and T^{∗}𝒰 → H^{0}(L_{C}) of (3.2) are isomorphisms;

(3) V^{ℰ} := H^{0}(L_{C} ⊗ (ζ^{ℰ})^{∗}L_{X}^{∗})^{∗} → 𝒰 is a vector bundle over C^{ℰ}, and L_{C} is more ample than (ζ^{ℰ})^{∗}L_{X}, so that the canonical vector bundle map

Φ^{ℰ} : T 𝒰 → 𝔱^{∗} ⊗ V^{ℰ} ,

defined fibrewise as in (3.1), using the isomorphisms in (2), is injective.

(4) if rank(V^{ℰ}) ⩾ 2, no 2-dimensional submanifold Σ of 𝒰 has Φ^{ℰ} (T Σ) ⊆ 𝔱^{∗} ⊗ V^{ℰ} everywhere decomposable (i.e., with all elements of rank one).

Compliancy may seem rather restrictive due to the number of conditions involved. However, by Remark 2.6, condition (1) is expected for determined QLS. A significant part of condition (2) is that the characteristic variety X^{ℰ} does not lie in a projective hyperplane bundle, but this could be taken as part of what it means for a QLS to be dispersionless (I have not found a rigorous definition of “dispersionless” for arbitrary QLS in the literature).

The key condition here is (3), giving a tensor product decomposition of T 𝒰 . This fails in Example 1.1 because the characteristic embedding is more ample than the cocharacteristic one, rather than vice-versa. However all of the QLS in Examples 2.3 are generically compliant. Condition (4) implies that the generic tangent space to any 2-dimensional submanifold Σ of 𝒰 contains indecomposable elements (via the injection Φ^{ℰ}). This is a technical condition needed to ensure that the decomposition in (3) is enough to recover the compatiblity of the hydrodynamic system from the cocharacteristic variety.

The following is an immediate consequence of Proposition 3.3 (fibrewise), and allows the recovery of the characteristic embedding from the cocharacteristic one.

Proposition 3.6. Let ℰ be a compliant QLS. Then for any [X] ∈ C^{ℰ}, there exist nonzero κ ∈ ζ([X]) and v ∈ V^{ℰ} such that Φ^{ℰ} (X) = κ ⊗ v.

4. Differential geometry: cocharacteristic nets

4.1. Nets and conjugate nets.

Definition 4.1. Let 𝒩 be an N-manifold and A = {1, … N}.

• A pre-net on 𝒩 is a direct sum decomposition T 𝒩 = ⨁_{a∈A} D_{a} into rank one distributions D_{a} ⩽ T 𝒩 for a ∈ A; thus each D_{a} is tangent to a foliation of 𝒩 with one dimensional leaves.

• A pre-net D_{a} : a ∈ A on 𝒩 is integrable if for every subset B ⊆ A, D_{B} := ⨁_{b∈B} D_{b} is an integrable distribution (i.e., tangent to a foliation with #B dimensional leaves); an integrable pre-net is called a net.

Recall that by the Frobenius Theorem (see e.g. [16]), a constant rank distribution is tangent to a foliation if and only if its sheaf of smooth sections is closed under Lie bracket.

Proposition 4.2. For a pre-net D_{a} : a ∈ A on 𝒩 , the following are equivalent.

(1) D_{a} : a ∈ A is integrable, i.e., a net.

(2) For every B ⊆ A with #B = 3, D_{B} is an integrable distribution, and if N = 3, the same is true when #B = 2.

(3) For every B ⊆ A with #B = 2, D_{B} is an integrable distribution.

(4) For each a ∈ A, D_{A\{a}} is an integrable distribution.

(5) Near any p ∈ 𝒩 , there are local coordinates r_{a} : a ∈ A such that D_{a} = span{∂_{r_{a}}}.

Proof. It is trivial that (1) ⇒ (2).

(2) ⇒ (3). This is vacuous for N ⩽ 2 and built into the statement for N = 3. For N ⩾ 4 and #B = 2, there exist distinct b, c ∈ A \ B, so that [D_{B}, D_{B}] ⩽ D_{B∪{b}} ∩ D_{B∪{c}} = D_{B}.

(3) ⇒ (4). For any b, c ∈ A \ {a} and any local sections X_{b} and X_{c} of D_{b} and D_{c} respectively, [X_{b}, X_{c}] is a section of D_{b} ⊕ D_{c} ⩽ D_{A\{a}}, and hence D_{A\{a}} is integrable.

(4) ⇒ (5). The r_{a} are pullbacks of local coordinates on the one dimensional local leaf spaces of D_{A\{a}}.

(5) ⇒ (1). For B ⊆ A, D_{B} is the joint kernel of dr_{a} : a ∈ A \ B, hence integrable. □

Example 4.1. Consider the hydrodynamic system R^{∗}A ∧ (dR/dt) = 0 where A is a section of 𝔱^{∗} ⊗ End(T 𝒱 ) over 𝒱 , as in Example 2.2. If A is everywhere simultaneously diagonalizable with distinct eigenfunctions, then the rank one eigendistributions define a pre-net on 𝒱 . The condition that these eigendistributions are tangent to coordinate lines (leading to the QLS ℰ spanned by κ_{a} ⊗ ∂_{r_{a}} : a ∈ A) is precisely the condition that this pre-net is a net.

4.2. Conjugate and cocharacteristic nets. For application to hydrodynamic integrability, a special class of nets will be needed. Suppose that D_{a} : a ∈ A is a pre-net on 𝒩 , and that T 𝒩 ⩽ 𝔱^{∗} ⊗ V for a vector space 𝔱^{∗} and a line bundle V → 𝒩 ; then each D_{a} defines a line subbundle M_{a} of 𝒩 × 𝔱^{∗} with D_{a} = M_{a} ⊗ V.

One may then require, for all a, b ∈ A and for any sections X_{b} of D_{b} and σ_{a} of M_{a}, that d_{X_{b}}σ_{a} is a section of M_{a} ⊕ M_{b} ⩽ 𝒩 × 𝔱^{∗}. If this holds then for all a, b ∈ A, [X_{a}, X_{b}] is a section of D_{a} ⊕ D_{b} and D_{a} : a ∈ A is therefore a net, which will be called a conjugate net.

Example 4.2. The terminology here comes from the classical situation (see e.g. [1]) that 𝒩 is an affine space with translations 𝔱^{∗} (or indeed a flat affine manifold). Then the tangent bundle of 𝒩 is isomorphic to 𝒩 × 𝔱^{∗} and a net D_{a} : a ∈ A is conjugate if for all a, b ∈ A, the coordinate lines along the any surface tangent to D_{a} ⊕ D_{b} are conjugate, i.e., orthogonal with respect to the second fundamental form of the surface.

Definition 4.3. Let ℰ ⩽ 𝔱^{∗} ⊗ T 𝒰 be a compliant QLS with T 𝒰 ⩽ 𝔱^{∗} ⊗ V^{ℰ} and cocharacteristic variety C^{ℰ} ⩽ P(T 𝒰 ). An N-dimensional cocharacteristic net in 𝒰 is a parametrized submanifold U: 𝒱 → 𝒩 ⊆ 𝒰 , with 𝒱 open in ℝ^{N}, such that:

(1) the net on 𝒩 spanned by ∂_{a}U : a ∈ A = {1, … N} satisfies [∂_{a}U] ∈ C^{ℰ}; and

(2) if V^{ℰ} has rank one, the net is conjugate.

5. Hydrodynamic integrability

5.1. Integrable hydrodynamic systems. As a warm-up for the proof of the main theorem, and to motivate the definition of hydrodynamic integrability, consider again Example 2.2 in the diagonalizable case, where the QLS ℰ is spanned by κ_{a} ⊗ ∂_{r_{a}} : a ∈ A. Since the system is equivalently the EDS generated by the 2-forms θ_{a} ∧ dr_{a}, where θ_{a} = ⟨κ_{a}, dt⟩, it is compatible if and only if these 2-forms algebraically generate a differential ideal, i.e., for all a ∈ A, dθ_{a} ∧ dr_{a} = 0 mod (θ_{b} ∧ dr_{b})_{b∈A}. This holds if and only if

for all a ∈ A, dκ_{a} ∧ dr_{a} = 0 mod (κ_{b} dr_{b})_{b∈A} (as 𝔱^{∗}-valued 2-forms on 𝒱 ),

i.e., for all a ∈ A, there are (scalar-valued) 1-forms β_{a} = ∑_{b∈A} β_{ab}dr_{b} and functions γ_{ab} on 𝒱 such that

dκ_{a} ∧ dr_{a} = β_{a} ∧ (κ_{a} dr_{a}) + ∑_{b∈A}(κ_{b} dr_{b}) ∧ (γ_{ab} dr_{a})
i.e., ∂_{b}κ_{a} = β_{ab}κ_{a} + γ_{ab}κ_{b} for all b ≠ a.

(5.1)

Fixing β_{a} and γ_{ab}, κ_{a} : a ∈ A may be viewed as a 𝔱^{∗}-valued solution to a linear system on 𝒱 . This linear system in turn has a compatibility condition: differentiating once more,

0 = dβ_{a} ∧ (κ_{a} dr_{a}) − β_{a} ∧ dκ_{a} ∧ dr_{a} + ∑_{b∈A}(γ_{ab} dκ_{b} + κ_{b} dγ_{ab}) ∧ dr_{b} ∧ dr_{a}
= κ_{a} dβ_{a} ∧ dr_{a} + ∑_{b∈A} κ_{b} (γ_{ab}(−β_{a} + β_{b}) + dγ_{ab}) ∧ dr_{b} ∧ dr_{a}
+ ∑_{b,c∈A} κ_{c} γ_{ab}γ_{bc} dr_{c} ∧ dr_{b} ∧ dr_{a}
= κ_{a} dβ_{a} ∧ dr_{a} + ∑_{c∈A} κ_{c} (γ_{ac}(β_{c} − β_{a}) + dγ_{ac} − ∑_{b∈A} γ_{ab}γ_{bc}dr_{b}) ∧ dr_{c} ∧ dr_{a}

for all a ∈ A. This is satisfied for all κ_{a} : a ∈ A if and only if for all a, c ∈ A with c ≠ a

dβ_{a} ∧ dr_{a} = d(β_{a} ∧ dr_{a}) = d(r_{a}dβ_{a}) = 0
∑_{b∈A}(∂_{b}γ_{ac} + γ_{ac}β_{cb} − γ_{ac}β_{ab} − γ_{ab}γ_{bc})dr_{b} ∧ dr_{c} ∧ dr_{a} = 0. (5.2)

If (5.2) holds, and (κ_{a})_{a∈A} is a 𝔱^{∗}-valued solution to (5.1) then θ_{a} ∧ dr_{a} : a ∈ A generate a differential ideal, and ℰ may be called an integrable hydrodynamic system. Such systems were introduced by Tsarev [23] as semi-hamiltonian systems of hydrodynamic type. Such systems are regarded as integrable because of the fundamental observation of Tsarev [23] that they admit generalized hodograph solutions. Interpreted in the affine geometry of Definition 2.1 and Remark 2.2 (using in particular the space of affine functions 𝔥 → 𝔱^{∗} and the tautological affine coordinate t: ℳ → 𝔥^{∗}), his observation is as follows.

Proposition 5.1. Let ℰ be a hydrodynamic system with characteristic momenta κ_{a} : a ∈ A. Then for any lift κ̃_{a} : a ∈ A of κ_{a} : a ∈ A to an 𝔥-valued solution of (5.1), any function R: ℳ → 𝒱 solving the implicit equations

(5.3)  ⟨κ̃_{a}(R), t⟩ = 0

for all a ∈ A, is a solution of ℰ.

Proof. The exterior derivative of (5.3) is ⟨R^{∗}dκ̃_{a}, t⟩ + R^{∗}θ_{a} = 0, since dt takes values in 𝔱. Hence, writing R^{a} = r_{a} ∘ R for the components of R, it follows from (5.1) and (5.3) that

−R^{∗}θ_{a} ∧ dR^{a} = ⟨R^{∗}dκ̃_{a}, t⟩ ∧ dR^{a}
= (⟨κ̃_{a}(R), t⟩β_{a} + ∑_{b∈A} R^{∗}γ_{ab}⟨κ̃_{b}(R), t⟩dR^{b}) ∧ dR^{a} = 0.

for all a ∈ A. □

In the integrable case, the linear system (5.1) is compatible, so there are many such solutions. Tsarev originally presented these systems in particularly convenient gauge: under scaling equivalence κ_{a} ↦ f_{a}κ_{a}, (5.1) is modified by adding f_{a}^{−1}df_{a} to β_{a} and rescaling γ_{ab}. Tsarev used this freedom to ensure that constants are scalar solutions of (5.1). This forces β_{a} = − ∑_{b} γ_{ab}dr_{b} mod dr_{a} and (5.1) becomes

dκ_{a} = ∑_{b∈A} γ_{ab}(κ_{b} − κ_{a})dr_{b} mod dr_{a}, i.e., ∂_{b}κ_{a} = γ_{ab}(κ_{b} − κ_{a}) for b ≠ a.

Since dβ_{a} ∧ dr_{a} = ∑_{b,c}(∂_{c}γ_{ab})dr_{a} ∧ dr_{b} ∧ dr_{c}, the integrability condition (5.2) becomes ∂_{c}γ_{ab} = ∂_{b}γ_{ac} and ∂_{b}γ_{ac} + γ_{ac}γ_{cb} − γ_{ac}γ_{ab} − γ_{ab}γ_{bc} for a, b, c distinct.

5.2. Hydrodynamic reductions. One remaining task before proving the main theorem is to formalize the notion of a hydrodynamic reduction of a QLS ℰ ⩽ 𝔱^{∗} ⊗ T 𝒰 on maps u: ℳ → 𝒰 (for a flat affine manifold ℳ). Recall from the introduction that the data defining an N-component hydrodynamic reduction are a map U: 𝒱 → 𝒰 and maps κ_{a}: 𝒱 → 𝔱^{∗} for a ∈ A = {1, … N} and 𝒱 open in ℝ^{N} with coordinates r_{1}, … r_{N}: 𝒱 → ℝ. These data are required to satisfy two properties:

(1) the hydrodynamic system on maps R: ℳ → 𝒱 defined by dR^{a} ∧ ⟨κ_{a}(R), dt⟩ = 0 (for all a ∈ A) is compatible (where R^{a} = r_{a} ∘ R are the components of R);

(2) if R solves this system, then u = U ∘ R solves ℰ.

Note that the characteristic momenta κ_{a} : a ∈ A are only naturally defined up to scale, and the hydrodynamic system may be rephrased that for all a ∈ A there are scalar valued functions f_{a}(r_{1}, … r_{n}) (depending on the solution) such that dR^{a}/dt = f_{a}(R)κ_{a}(R). As in the introduction, the chain rule for u = U ∘ R implies

\frac{du}{dt} = \frac{R^{∗}dU ∘ dR}{dt} = ∑_{a∈A} \frac{dR^{a}}{dt} ⊗ ∂_{a}U(R) = ∑_{a∈A} f_{a}(R)κ_{a}(R) ⊗ ∂_{a}U(R).

Under the compatibility of the hydrodynamic system (so that it has many independent solutions R), criterion (2) above is therefore equivalent to the property that β_{a} := κ_{a}⊗∂_{a}U is in ℰ for all a ∈ A. But then, where β_{a} is nonzero, [β_{a}] is in the rank one variety R^{ℰ}, i.e., [κ_{a}] ∈ X^{ℰ} is characteristic and [∂_{a}U] ∈ C^{ℰ} is cocharacteristic. Geometrically, the hydrodynamic reduction provides many N-secant solutions u to ℰ, i.e., the projective image of du meets the rank one variety R^{ℰ} in N points.

Definition 5.2. An N-component hydrodynamic reduction of a QLS ℰ ⩽ 𝔱^{∗} ⊗ T 𝒰 with characteristic variety X^{ℰ} is a map

(U, [κ_{1}], … [κ_{N}]): 𝒱 → X^{ℰ} ×_{𝒰} ⋯ ×_{𝒰} X^{ℰ} ,

where 𝒱 is open in ℝ^{N} and the codomain is the N-fold fibre product, such that κ_{a} ⊗ ∂_{a}U is in ℰ for all a ∈ {1, … N} and the hydrodynamic system with characteristic momenta κ_{1}, … κ_{N} is compatible as described in (5.1).

Then [9, 11] ℰ is integrable by hydrodynamic reductions if for all N ⩾ 2 it admits Ncomponent hydrodynamic reductions parameterized by N(n − 2) functions of 1-variable.

For N ⩽ m = dim 𝒰 , a hydrodynamic reduction generically and locally determines an N-dimensional submanifold 𝒩 of 𝒰 (the image of U) together with a net ∂_{a}U : a ∈ A on 𝒩 with [∂_{a}U] ∈ C^{ℰ}. Thus a hydrodynamic reduction defines a net satisfying Definition 4.3 (1). Conversely, if ℰ is a compliant QLS, then for any such net, Proposition 3.6 shows that the embedding of C^{ℰ} into P(𝔱^{∗}⊗V^{ℰ}) gives ∂_{a}U = κ_{a}⊗v_{a} for some local sections v_{a} of U^{∗}V^{ℰ}, where [κ_{a}] are the characteristic momenta corresponding to [∂_{a}U] under the isomorphism ζ^{ℰ} : C^{ℰ} → X^{ℰ}.

Proof of Theorem 1.1. The discussion so far has estabilished a correpondence between hydrodynamic reductions, modulo the compatibility condition, and nets satisfying 4.3 (1). It therefore remains to show that under this correspondence, the compatibility of the hydrodynamic system is equivalent to 4.3 (2), i.e., is automatic if rank V^{ℰ} ⩾ 2 and is equivalent to the net being conjugate otherwise.

The proof of this last step follows the line of argument in [6], cf. also [3] (QLS of type I) and [8] (QLS of type H). First choose a basis ε_{1}, … ε_{n} for 𝔱^{∗} and rescale the characteristic momenta such that κ_{a1} = 1. Under the embedding of C^{ℰ} into P(𝔱^{∗} ⊗ V^{ℰ}), ∂_{b}U = κ_{b} ⊗ v_{b} for some local sections v_{b} of V^{ℰ} over 𝒩 , and hence, in 𝔱^{∗} components, ∂_{b}U_{k} = κ_{bk}v_{b} = κ_{bk}∂_{b}U_{1} for k ∈ {1, … n}. Taking the ∂_{a} derivative of this equation and commuting partial derivatives yields

(∂_{a}κ_{bk})∂_{b}U_{1} − (∂_{b}κ_{ak})∂_{a}U_{1} = (κ_{ak} − κ_{bk})∂_{a}∂_{b}U_{1}.

On dividing by κ_{ak} − κ_{bk}, the right hand side is independent of k and hence

(\frac{∂_{a}κ_{bk}}{κ_{ak} − κ_{bk}} − \frac{∂_{a}κ_{bℓ}}{κ_{aℓ} − κ_{bℓ}} )v_{b} = (\frac{∂_{b}κ_{ak}}{κ_{ak} − κ_{bk}} − \frac{∂_{b}κ_{aℓ}}{κ_{aℓ} − κ_{bℓ}} )v_{a}.

Thus both sides are zero unless v_{a} and v_{b} are linearly dependent, i.e., multiples of some v ∈ V^{ℰ}, say. But then the span of ∂_{a}U = κ_{a} ⊗ v_{a} and ∂_{b}U = κ_{b} ⊗ v_{b} is span{κ_{a}, κ_{b}} ⊗ span{v}, hence decomposable. For rank(V^{ℰ}) ⩾ 2, the set where this holds has empty interior by condition (4) of compliancy, and so the hydrodynamic compatibility criterion is satisfied on the dense complement, hence everywhere by continuity.

It remains to establish the equivalence in the case rank(V^{ℰ}) = 1.

If the compatibility condition ∂_{a}κ_{bk} = γ_{ba}(κ_{ak} − κ_{bk}) for a ≠ b holds, then

∂_{a}∂_{b}U_{k} = (∂_{a}κ_{bk})∂_{b}U_{1} + κ_{bk} ∂_{a}∂_{b}U_{1}
= γ_{ba}(κ_{ak} − κ_{bk})∂_{b}U_{1} + κ_{bk}(γ_{ab}∂_{a}U_{1} + γ_{ba}∂_{b}U_{1})
= γ_{ab}(v_{a}/v_{b})∂_{b}U_{k} + γ_{ba}(v_{b}/v_{a})∂_{a}U_{k}.

Thus ∂_{a}∂_{b}U is in the span of and ∂_{a}U and ∂_{b}U, so the net is conjugate.

Conversely, if the net is conjugate with ∂_{a}∂_{b}U_{k} = α_{ab}∂_{b}U_{k} + β_{ab}∂_{a}U_{k} for a ≠ b, then taking k = 1,

∂_{a}∂_{b}U_{1} = α_{ab}∂_{b}U_{1} + β_{ab}∂_{a}U_{1} = α_{ab}v_{b} + β_{ab}v_{a}.

On the other hand, the ∂_{a} derivative of ∂_{b}U_{k} = κ_{bk}∂_{b}U_{1} yields

κ_{bk} ∂_{a}∂_{b}U_{1} = ∂_{a}∂_{b}U_{k} − (∂_{a}κ_{bk})∂_{b}U_{1} = α_{ab}κ_{bk}v_{b} + β_{ab}κ_{ak}v_{a} − (∂_{a}κ_{bk})v_{b}.

Eliminating ∂_{a}∂_{b}U_{1} between these equations, it follows that

α_{ab}κ_{bk}v_{b} + β_{ab}κ_{bk}v_{a} = α_{ab}κ_{bk}v_{b} + β_{ab}κ_{ak}v_{a} − (∂_{a}κ_{bk})v_{b}

and hence ∂_{a}κ_{bk} = β_{ab}(v_{a}/v_{b})(κ_{ak} − κ_{bk}), which is the compatibility condition. □

Corollary 5.3. A compliant QLS ℰ ⩽ 𝔱^{∗} ⊗ T 𝒰 is integrable by hydrodynamic reductions if and only if 𝒰 admits a family of 3-dimensional cocharacteristic nets parametrized by 3(n − 2) functions of one variable.

In particular, this Corollary applies to generic QLS of types G, H and I, unifying and extending results in [3, 6, 8].

Acknowledgements. I would like to thank the Eduard Čech Institute, grant GA CR P201/12/G028, for financial support, and Robert Bryant, Jenya Ferapontov, Boris Kruglikov and Vladimir Soucěk for invaluable discussions.

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Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.

Email address: D.M.J.Calderbank@bath.ac.uk