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The geometry of differential constraints for a class of evolution PDEs

The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in order to obtain explicit differential constraints for PDEs belonging to this family. Several examples, with applications in non-linear stochastic filtering theory, stochastic perturbation of soliton equations and non-isospectral integrable systems, are discussed in detail to verify the effectiveness of the method.


Introduction
The method of differential constraints is a well known and general method for determining particular explicit solutions to a partial differential equation (PDE) reducing the PDE to a system of ordinary differential equations (ODEs) through a suitable ansatz on the form of the solution. In particular, given a system of evolution equations of the form ∂ t (u k ) = F k (x, t, u, u σ ), where (x i , u k ) ∈ M × N and u k σ are the derivatives of u k with respect to x j the number of times defined by the multi-index σ, we can look for solutions of the form u(x, t) = K(x, w 1 (z), ..., w L (z)), (1) where K : M × R L → N and z : M × R → R are smooth functions. Replacing this ansatz in the initial evolution equation, we may obtain a system of ODEs for the functions w i with respect to the variable z. We remark that, for general functions K and z, the system of ODEs for w i is overdetermined and has no solutions. When the system for the functions w i admits solutions, the ansatz (1) is said compatible with the equation and K is called a differential constraint for ∂ t (u k ) = F k (x, t, u, u σ ). This method appears with different names in several papers and books (see, e.g., [8,17,24,27,29,31,33]) and is equivalent to append to the original equation a suitable overdetermined system of PDEs of the form I = {I 1 (x, t, u, u σ ) = 0, ..., I K (x, t, u, u σ ) = 0}. Indeed, if I admits a finite dimensional solution, this can be described by a function of the form (1) and the requirement that I and ∂ t (u k ) = F k (x, t, u, u σ ) have common solutions can be interpreted as the compatibility condition for the ansatz. In that situation the system I is called differential constraint as well and, in order to distinguish between the two approaches, some Authors refer to direct differential constraints for the formulation with the function K and to indirect differential constraints when the system I is considered (see, e.g. [31,33]).
In general, checking that the ansatz (1) or the overdetermined system I are compatible with the evolution equations ∂ t (u k ) = F k (x, t, u, u σ ) is a difficult task. Even in the simplest case M = R, in order to verify that I is a differential constraint, we have to solve a system of strongly non-linear PDEs for the unknown functions I k . For this reason many Authors look for differential constraints imposing some restrictions on the form of the functions K and z, or, equivalently, on the form of the constraints I k or, finally, on the form of the equation ∂ t (u k ) = F k (x, t, u, u σ ).
In this paper we are interested in evolution equations with a particular form of the functions F k and we make suitable assumptions on the form of the differential constraints. In particular we deal with the problem of finding differential constraints for evolution PDEs of the form where the functions F k i do not depend on t. Furthermore we chose the new variable z(x, t) = t and we look for differential constraints that are independent of any possible choice of the smooth functions c i (t). The choice of equations of the form (2) is not a matter of computational convenience, but is triggered by many theoretical and applied problems arising in different branches of mathematics. First of all, evolution equations of the form (2) appear in the theory of stochastic processes and in particular in the study of finite dimensional solutions to stochastic partial differential equations (SPDEs). This topic is discussed in many applications of SPDEs: see, for examples, [9,10,23] for applications to non-linear filtering problems, [14,15,16] for applications to financial problems and [22,37,39] for applications to stochastic soliton equations. In this framework the problem of finding finite dimensional solutions can be reduced to the problem of finding differential constraints for evolution equations of the form (2) for any choice of c k (t). Indeed we can (formally) see the functions c k (t) as the derivatives 1 of some stochastic process which can assume any possible values. Moreover, equations of the form (2) have applications to integrable system theory, and our results turn out to be useful for dealing with non-isospectral deformations of integrable systems (see [4,7,19]). In fact the local non-isospectral deformation of KdV hierarchy can be reduced to the usual isospectral KdV hierarchy by means of a time dependent transformation (see e.g. [20] and the examples of Subsections 6.2 1 The usual stochastic process considered in SPDE theory is Brownian motion that does not admit derivative almost surely. For this reason we say that the deterministic problem proposed in this paper is formally equivalent to the stochastic one. A detailed discussion of the possible application of differential constraints in SPDE theory will be provided in a following paper exploiting the geometric techniques developed here. and 6.5). Finally, evolution equations of the form (2) can be considered in infinite dimensional control theory (see [3,28]). In this framework our method can be interpreted as an application of usual methods of geometric control theory to the explicit computation of the reachable sets of some particular point (see [1] for the finite dimensional case and [25] for the infinite dimensional one). It is worth to remark that, although the problem of finding differential constraints for equations of the form (2) has been faced many times, to the best of our knowledge this is the first time that the abstract form (2) of the problem has been recognized (since the previous Authors consider particular forms of the functions F i k ) and that the problem has been tackled by using the geometrical framework of differential constraints for PDEs. In particular, the use of the differential constraints method permits to consider, from a theoretical point of view, a very general form for the functions F k i and allows us to obtain a useful algorithm for the explicit computation of the solutions to equations (2) (see Subsection 6.1).
On the other hand, the problem of finding differential constraints for PDEs of the form (2) is an interesting challenge in itself. In order to address this issue we provide a geometrical framework for the description of differential constraints method which allows us to simplify the formulation of the problem. In particular, we associate with any autonomous evolution equation of the form ∂ t (u k ) = F k i (x, u, u σ ) an evolution vector field V Fi on the space of infinite jets J ∞ (M, N ) of the functions from M into N . In this setting the time independent overdetermined system I, or equivalently the time independent function K, can be described as a particular finite dimensional submanifold K of J ∞ (M, N ) and we prove that I is a differential constraint for the system (2) for any c i (t) if and only if V Fi ∈ T K ∀i = 1, . . . s. This geometrical reformulation gives new insight into the problem of finding differential constraints. First of all it provides a general and powerful method for dealing with many different evolution equations which have been previously faced with different techniques and in different frameworks. The second important result following by our general approach is the derivation of some necessary conditions for the existence of differential constraints for systems of the form (2). These conditions form an infinite dimensional analogous of necessary condition of the well-known Frobenius theorem. Indeed the existence of a differential constraint for (2) ensures that the functions F i , restricted on some subset K of J ∞ (M, N ), form a module with respect to the Lie brackets [F i , F j ] induced by the vector fields Lie brackets [V Fi , V Fj ]. This imposes severe conditions on F i : in particular, if K = J ∞ (M, N ), F i must form a Lie algebra on J ∞ (M, N ). Moreover, in order to obtain a sufficient condition for the existence of differential constraints for systems of the form (2), we introduce the notion of characteristic flow for a general evolution vector field. In particular this definition, generalizing to higher order the standard notion of characteristic of a first order scalar evolution equation, is an infinity dimensional analogous of the flow of a vector field in finite dimensional framework. Thereafter we divide the functions F i in two sets H i and G j so that the functions G j form a Lie algebra and their evolution vector fields V Gj admit generalized characteristic flow. Under these assumptions we prove that, if the vector fields V Hi admit a differential constraint H, then the complete set of V Fi admits a differential constraint K that can be explicitly computed starting from H and using the characteristic flows of G i . In addition we provide a generalization of this theorem to the case of H i , G i forming a finite dimensional Lie algebra on a real analytic submanifold H of J ∞ (M, N ) which is also a differential constraint for V Hi . The proofs of both these results are constructive so that we can compute explicitly the differential constraints in many interesting examples. In particular some of the examples have been chosen in order to show the flexibility and the effectiveness of our geometrical approach with respect to the standard differential constraints method (see Subsections 6.2, 6.4 and also [12] for other examples on the same topics). Other examples instead have been proposed for their relevance in applied mathematical problems such as non-linear stochastic filtering theory (see Subsection 6.3), stochastic perturbation of integrable equations (see Subsections 6.4 and 6.5) and non-isospectral deformation of integrable systems (see Subsection 6.2 and 6.5).
The paper is organized as follows: after recalling some basic facts on the geometry of J ∞ (M, N ) in Section 2, in Section 3 we provide a geometric characterization of differential constraints for systems of evolution PDEs of the form (2). Hence, in Section 4, we discuss the problem of characteristics in J ∞ (M, N ) and in Section 5 we apply previous results to the explicit construction of differential constraints (or reduction functions) for evolution PDEs of the form (2). Finally, in Section 6, we apply the results of the previous sections to several explicit examples.

Preliminaries
In this section we collect some basic facts about (infinite) jet bundles in order to provide the necessary geometric tools for our aims.
Given the trivial fiber bundle M × N → M , where M, N are open sets of R m and R n respectively, we denote by x i the cartesian coordinate system of M and by x i , u j the cartesian coordinate system of M × N . The coordinates x i , u j induce a global coordinate system x i , u j , u j σ on the k-order jet bundle J k (M, N ), where σ = (σ 1 , ..., σ m ) ∈ N m 0 , |σ| ≤ k is a multi-index denoting the number σ l of derivatives of u j with respect x l . It is well known that J k (M, N ) admits a natural structure of finite-dimensional smooth vector bundle on M and, considering the natural projections π k,h : J k (M, N ) → J h (M, N ), it is possible to define the inverse limit J ∞ (M, N ) of the sequence Unfortunately the space J ∞ (M, N ) is not a finite-dimensional manifold, being the inverse limit of a sequence of spaces of increasing dimension. From a topological point of view J ∞ (M, N ) is a Fréchet manifold modeled on R ∞ (see [35]) and any open set U of J ∞ (M, N ) contains a set of the form U ′ = π −1 k (V ) for some k ∈ N and some open set V ⊂ J k (M, N ). This means that, for any σ with |σ| > k, the coordinates u i σ vary in the whole R. Furthermore, the differential structure of J k (M, N ) induces a natural differential structure in J ∞ (M, N ) (see [26,6] for a complete description). Hereafter we denote by F k the algebra of real-valued smooth functions defined on J k (M, N ) and we deduce the differential structure of J ∞ (M, N ) from the geometric smooth algebra F defined as the direct limit of the sequence Let G ⊂ F be a finitely generated subalgebra of F , which means that there are a finite number of functions g 1 , ..., g l ∈ G such that any g ∈ G is of the form g = G(g 1 , ..., g l ) for a unique smooth function G. It is possible to associate with G in a unique way a finite dimensional manifold M G (see [30]). For this reason in the following we identify the subalgebra G with the manifold M G such that G = C ∞ (M G ). The inclusion i : G → F induces a unique projectionπ : The algebra F is a graded algebra and a vector field X on J ∞ (M, N ) is a derivation on the space F which respects the order. It is well known that the Cartan distribution C on J ∞ (M, N ) generated by the vector fields defines an integrable connection on J ∞ (M, N ). Hence, for any vector field X on J ∞ (M, N ), we can write We remark that if X = X v + X h is a symmetry of C then also X h and X v are symmetries of C.
Definition 2.1 A vertical vector field X that is a symmetry of C is called evolution vector field. If X is an evolution vector field there exists a unique smooth function where F = (f 1 , ..., f n ) and D σ = (D 1 ) σ1 ...(D m ) σm . We call F the generator of the evolution vector field X and we write X = V F .
If V F and V G are two evolution vector fields, then [V F , V G ] is also an evolution vector field and there exists an unique H ∈ F n = F × ... × F such that [V F , V G ] = V H . Therefore the commutator between evolution vector fields induces a commutator in F n and we define [ We conclude this section recalling that a subset E of J ∞ (M, N ) is a submanifold of J ∞ (M, N ) if for any p ∈ E there exists a neighborhood U p of p such that π h (E ∩ U p ) is a submanifold of J h (M, N ) for h > H p . If, for any p ∈ E, all the submanifolds π h (E ∩U p ) with h > H p have the same dimension L, we say that E is an L-dimensional submanifold of J ∞ (M, N ). In particular, given an L-dimensional manifold B and a smooth immersion K : B → J ∞ (M, N ), for any point y ∈ B there exists a neighborhood V of p such that K(V ) is a finite dimensional submanifold of J ∞ (M, N ).
Any canonical submanifold E can be locally described as the set of zeros of a finite number of smooth independent functions f 1 , ..., f L and of all their differential consequences D σ (f i ).

Differential constraints and PDEs reduction
In this section we propose a geometric reformulation of differential constraints method for a family of evolution PDEs. In particular we introduce the notion of reduction function and we discuss its relation with differential constraints seen as finite dimensional submanifolds of J ∞ (M, N ).
3.1 Differential constraints: from the reduction function to the submanifold Let us consider a system of evolution PDEs of the form where F i ∈ F n and k = 1, . . . , n.
Definition 3.1 Given a system of evolution PDEs of the form (4) and an L-dimensional manifold B, let K : M × B → N be a smooth function. We say that K is a reduction function for (4) if there are smooth functions f j such that is a solution to the system (4) for any c 1 (t), ..., c s (t) if and only if b j (t) are solutions to the system of ODEs In this framework it is natural to associate with K a function R K : and in coordinates we have . Furthermore the following theorem holds. Proof. If K is a reduction function for the system (4), we have Choosing c i = δ 1,i and applying ∂ σ x to both sides of the previous equation we get It is easy to show that the functions g j i do not depend on x ∈ M being R K * (∂ x i ) = D i so the thesis follows choosing Since the submanifold K = R K (M × B) is a finite dimensional canonical submanifold, K can be locally described as the set of zeros of a finite number of smooth independent functions f 1 , ..., f L and of all their differential consequences D σ (f i ). Therefore a necessary and sufficient condition for not an immersion (and so K is not a submanifold). Even so, for the sake of simplicity, in the following we always consider submanifolds K of J ∞ (M, N ). In particular, if K is a real analytic function with respect the x i variables, a necessary and sufficient condition for K to be a submanifold is that ∂ b i (K(·, b)) are linearly independent as functions from M into N . In the smooth case it can happen that ∂ b i (K(·, b)) are linearly independent but R K is not an immersion. However, this situations can be considered as exceptional: indeed the set of K such that R K is an immersion is an open everywhere dense subset of C ∞ (M, N ) with respect the Whitney topology (see [18]).
Remark 3.4 An interesting consequence of Theorem 3.2 is that, if s = 1, any solution U (x, t) : M × R → N to the system (4) for c 1 = 1 is a reduction function. Indeed in this case we have that

Differential constraints: from the submanifold to the reduction function
In this section we discuss the problem of computing the reduction function K starting from the knowledge of a suitable canonical submanifold K. The resulting algorithm will be used in the examples of Section 6.
In order to prove that with any differential constraint K for (4) it is possible to associate a reduction function K, we need to recall the following technical result.
T H = C). Denoting by π the canonical projection π : Proof. A proof of this result can be found in [6], Chapter 4, Proposition 2.3.
which is a differential constraint for equation (4). Then, for any point and K is a reduction function for the system (4).
Proof. The manifold K with respect the projection π is a finite dimensional fibred manifold with base M . Moreover, since D i ∈ K, the Cartan distribution C is a finite dimensional flat connection of (K, π, M ) and, for any ) and a local trivialization R : Therefore Theorem 3.6 ensures that there exists a smooth function K : V × B → N such that R K = R and Theorem 3.2 guarantees that K is also a reduction function for the system (4).
Remark 3.8 There are two obstructions for a global version of Theorem 3.7. The first one is that, if M is not simply connected, C may admit only a local trivialization and not a global one and the second is that, if C is a non-linear connection on K, it may not admit a global trivialization, since non-linear ODEs can blow-up. Obviously, if M is simply connected and C| K has at most linear grow for some coordinate system on K, Theorem 3.7 admits a global version.
Given an L-dimensional canonical submanifold K which is a differential constraint for (4), Theorem 3.7 provides an explicit construction procedure for the function K and for the system of ODEs for the parameters b i . Indeed let (x i , y j ) be an adapted coordinate system for the fibred manifold (K, π, M ), which means that x i is the standard coordinate system on M and y j is an adapted coordinate system for the fiber π −1 (x), with j = 1, ..., L − m. The coordinates y i can be chosen among the functions u k , u l σ and, in general, it is possible to find smooth . In the coordinate system (x i , y j ) the vector fields D i have the form and, ∀x 0 ∈ M and y x0 ∈ π −1 (x 0 ), the solution y j =K j (x, y x0 ) to the system provides the local trivialization of the flat connection C = span{D i }. The explicit expression of the function K can be obtained by rewriting u l as functions of (x i , y j ) leading to K l (x, y x0 ) = f l (x,K(x, y x0 )).
Moreover the system of ODEs for the parameters y j x0 can be obtained expressing V Fi in the coordinates (x i , y j )

A necessary condition for existence of differential constraints
The main goal of the general theory of differential constraints is to find a reduction function for a system of the form (4) for s = 1. As proven in [31] (see also Remark 3.4) this problem admits an infinite number of solutions. On the other hand, the problem of existence of reduction functions (or differential constraints) for s > 1 is completely different: actually, in the general case, there are no reduction functions at all.
In this section we address the problem of existence of a reduction function for a system of the form (4) starting from the following remark.
Remark 3.9 If a system of evolution PDEs of the form (4) admits a differential constraint K, the set S = span{V F1 , ..., V Fs }, is a finite dimensional module on K.
The following Proposition provides a useful characterization for the vector fields V Fi .
Proof. The proof is given for the case N = M = R and H = J ∞ (M, N ); the general case is a simple generalization of this one. Since S is s-dimensional, for any point p ∈ J ∞ (M, N ) there exist a neighborhood U of p and an integer h ∈ N 0 such that the matrix A = (D h+j−1 x (F i ))| i,j=1,...,s is non-singular. Moreover, since the commutator of two evolution vector fields is an evolution vector field, there exist some F i,j ∈ F such that [V Fi , V Fj ] = V Fi,j and, by the definition of evolution vector field, we have Deriving with respect to x the previous relations we obtain and combining (5) and (6) we find Since the matrix A is non-singular we get D x (λ h i,j ) = 0. In Section 5 we will consider two particular cases for the functions F i ∈ F n . In the first case V Fi form a finite dimensional module of constant dimension on all J ∞ (M, N ). In this case Proposition 3.10 ensures that V Fi form a Lie algebra, since the only functions f in J ∞ (M, N ) such that D i (f ) = 0 are the constants. In the second case we suppose that V Fi form a finite dimensional module on a real analytic finite dimensional submanifold H of J ∞ (M, N ).

Characteristic vector fields in J ∞ (M, N )
In this section we define the notion of generalized characteristic flow for an evolution vector field and we discuss the connection with the usual characteristic flow for scalar first order evolution PDEs. These results will play a central role in the explicit construction of differential constraints in Section 5.

Characteristics of scalar first order evolution PDEs
It is well known that, if N = R and F ∈ F \ F 0 , the evolution vector field V F is not the prolongation of a vector field on J 0 (M, N ) and does not admit flow in J ∞ (M, N ), which is why the equation may not admit solutions even for smooth initial data, or may admit infinite solutions for any smooth initial data. For this reason the problem of finding solutions to evolution PDEs is usually solved only in specific situations (for example the linear or semilinear cases) where it is possible to use the powerful techniques of analysis. Anyway, a classical geometric approach to scalar first order evolution PDEs (see, e.g., [11]) shows that something can be done in order to solve equation (7) even when V F does not admit flow in J ∞ (M, N ). Indeed given a first order scalar autonomous PDE it is possible to solve (8) considering the following system of ODEs on J 1 (M, N ) If Φ a is flow of the vector field on J 1 (M, N ) corresponding to the previous system and we define φ i a = (Φ * a (x i )) and η a = Φ * a (u), the solution U (x, t) to the PDE (8) with initial data U (x, 0) = f (x) is given by Moreover it is possible to uniquely extend the flow Φ a to J k (M, N ) as the solution to the following system of ODEs and the vector field corresponding to the flow Φ a on J ∞ (M, N ) is given bȳ We call Φ a the characteristic flow of F andV F its characteristic vector field.

Characteristics in the general setting
In this section we propose an extension of the notion of characteristic vector field and characteristic flow to multidimensional and higher order case. This extension is based on the geometric analysis of J ∞ (M, N ) presented in [26]. We start by recalling the definition of one-parameter group of local diffeomorphisms on J ∞ (M, N ) which reduces to the classical one in the finite dimensional setting.
Definition 4.2 Given an evolution vector field V F , we say that V F (or its generator F ) admits characteristics if there exist suitable smooth functions h 1 , ..., h n ∈ F such that the vector fieldṼ If we restrict to the scalar case (N = R), which is discussed in Subsection 4.1, the following Theorem provides a complete characterization of evolution vector fields admitting characteristics. Proof. The proof that any F ∈ F 1 admits characteristic flow is given in Subsection 4.1. The proof of the converse can be found in [26]. . For example if we consider M = R 2 with coordinates (x, y) and F = u xy , Theorem 4.3 ensures that F = u xy does not admit characteristics on J ∞ (R 2 , R) but, considering the submanifold E ⊂ J ∞ (R 2 , R) generated by the equation u yy = 0, it is easy to prove that V F ∈ T E and that V F admits characteristics on E.
If we do not restrict to the scalar case the situation becomes more complex and, to the best of our knowledge, a complete theory of characteristics in J ∞ (M, N ) for N = R has not been developed. Indeed in this case we can find F ∈ F n 1 such that V F admits characteristics. For example if we consider M = R and N = R 2 (with coordinates x and (u, v) respectively) and F = (v xx , 0) ∈ F 2 , the flow of the vector field V F is given by the following transformation In this paper, in order to deal with the general case, we propose a stronger definition of characteristics that, although imitating in some respects the scalar case, is weak enough to include many cases of interest. Given an open subset U ⊂ J ∞ (M, N ) we denote by the set of smooth functions defined on U , that is the union of the sets of smooth functions defined on π k (U ) ⊂ J k (M, N ). Given a subalgebra G 0 ⊂ F | U , we denote by G k the algebra generated by smooth composition of functions of the form D σ (f ), where f ∈ G 0 and σ is a multi-index with |σ| ≤ k.  N ), a finitely generated subalgebra G 0 of F | U generating F | U and g 1 , ..., g n ∈ F such that the vector fieldV In the scalar case an evolution vector field admits characteristics if and only if admits strong characteristics: indeed in this caseV F (F 1 ) ⊂ F 1 . Moreover, if we consider the evolution vector field V F of the previous example (with generator F = (v xx , 0)), it is easy to check that V F admits strong characteristics. In fact the subalgebra G 0 generated by x, u, v, v x , v xx is such that V F (G 0 ) ⊂ G 0 . Actually we do not know any example of evolution vector field admitting characteristics which are not strong characteristics.
Remark 4.7 In Definition 4.6 we consider a general subalgebra G 0 generating F instead of restricting to the case G 0 = F k for some k ∈ N. This is a crucial point because, in the vector case N = R n (with n > 1), [26]) and the vector fieldsV F satisfyingV F (F k ) ⊂ F k turn out to be tangent to the prolongations of infinitesimal transformations in J 0 (M, N ). A well-known consequence of this fact is that, in the vector case, the only infinitesimal symmetries of a PDE which can be defined using finite jet spaces J k (M, N ) are Lie-point symmetries. On the other hand, if we allow G 0 to be a general subalgebra generating F , we obtain a larger and non-trivial class of evolution vector fields admitting strong characteristics. Proof. The vector fieldV F admits flow on the space of functions G 0 since G 0 is finite dimensional. In order to show thatV F admits flow on all F | U and so (since U depends on a generic point) on F we prove by induction thatV where h i j ∈ G 1 , beingV F (G 0 ) ⊂ G 0 and x i ∈ G 0 . We recall that G k is generated by functions of the form D i (g) with g ∈ G k−1 . Sō sinceV F (g) ∈ G k−1 and h i j ∈ G 1 . HenceV F admits flow on G k and the flow on G k is compatible with the flow on G k−1 being G k−1 ⊂ G k . The problem of the previous construction is that in general the domain U k of the flow in G k depends on k. This means that, if we denote with P h,k the natural projection of G h on G k with h > k, it might happen that P −1 h,k (U k ) = U h . But this is not actually the case. Indeed since G 0 generates F | U , then F 0 | U ⊂ G 0 and so F k | U ⊂ G k . In particular u i σ ∈ G k if |σ| ≤ k. But by Remark 7.2 and Corollary 7.3 (see Appendix) Φ a (u i σ ) is polynomial in u j σ ′ for |σ ′ | sufficiently large. This means that u j σ ′ can vary in all R and so the domain of definition of Φ a in U ⊂ J ∞ (M, N ) is not empty and is of the form U ′ = π −1 ∞,k (U k ) for k sufficiently large. Since U ′ is an open subset of J ∞ (M, N ) this concludes the proof. Definition 4.9 Let y 1 , ..., y l , ... ∈ F | U be a sequence of functions defined in an open set U . We say that Y = {y i }| i∈N is a local adapted coordinate system with respect to a subalgebra G 0 generating F | U , if there exists a sequence k 1 , ..., k l , ... ∈ N, with k i < k i+1 , such that y 1 , ..., y ki is a coordinate system for G i .

Remark 4.10
The flow of a vector field with strong characteristics solves a triangular infinite dimensional system of ODEs. Indeed if we consider an adapted coordinate system with respect to a subalgebra G 0 we haveV F (y i ) = f (y 1 , .., y k1 ) for i = 1, ..., k 1 , V F (y i ) = f (y 1 , .., y k2 ) for i = k 1 + 1, ..., k 2 and so on. So we can start by solving the system for i = 1, ..., k 1 and then solve the system for i = k 1 + 1, ..., k 2 , since the system is of triangular type.
The main trouble when working with a family of evolution vector fields admitting characteristic flows is that the sum or the Lie brackets of two of them usually do not admit characteristic flow. In order to overcome this problem we give the following Definition. If F 1 , ..., F s , correspond to evolution vector fields with strong characteristics admitting a common filtration, then also cF i + dF j (where c, d ∈ R) and [F i , F j ] correspond to vector fields with strong characteristics. Furthermore cF i + dF j and [F i , F j ] admit the same common filtration of F 1 , ..., F k .

Building differential constraints
In this section we consider a system of PDEs of the form (4) such that some of the evolution vector fields V Fi admit strong characteristics and a common filtration. In this setting we show how it is possible to construct a differential constraint for the system (4) starting from the knowledge of a suitable submanifold of J ∞ (M, N ). The construction, which is completely explicit, take the cue from of the moving frame method (see [13]).  N) such that H ∩ U is the annihilator of f i , we say that a distributions ∆ = span{V G1 , . . . , V G h } is transversal to H in U if there exist r 1 , . . . , r h such that the matrix (V Gi (f rj ))| i,j=1,...,h has maximal rank in U . In the following the sequence f i will be chosen so that r j = j and f i is a local coordinate system adapted with respect to the filtration G k for k sufficiently large.
Lemma 5.2 Let G 1 , ..., G h be a subalgebra of F n admitting strong characteristics and a common filtration. Let Φ i a i be the characteristic flow of G i and H be a canonical finite dimensional submanifold of J ∞ (M, N ) such that the distribution T H ⊕ span{V G1 , ..., V G h } has constant rank and the distribution ∆ = span{V G1 , . . . , V G h } is transversal to H. Then there exists a suitable neighborhood of the origin is a finite dimensional submanifold of J ∞ (M, N ).
Proof. In the following, for the sake of clarity, we write such that H is the annihilator of f i , for any point p ∈ H there exists a neighborhood U such that the matrix (V Gi (f j ))| i,j=1,...,h has maximal rank in U . Therefore, considering the submanifoldH defined as the annihilator of the functions f i ∈ F | U (i = 1, . . . , h), the equations can be solved with respect to α. This means that, possibly restricting the open set U , there exist a smooth function A(p) = (A 1 (p), ..., A h (p)) defined on U such that 1 (p))...) ∈H. In the following we prove that K is the annihilator of the functions and, since K j are independent and adapted with respect to the filtration G k for k sufficiently large, K is a submanifold of J ∞ (M, N ).
We start by proving that if p 0 ∈ K∩U , then K j (p 0 ) = 0 (for j > h). Indeed, if p 0 ∈ K∩ U , the point p 0 can be reached starting from p ∈ H by means of composition of suitable flows Φ i a i . On the other hand, for any p 0 ∈ K ∩ U , there exists A(p 0 ) = (A 1 , . . . , Since H ⊂H and the transversality condition ensures that equation (10) have a unique solution, ...) = 0 for any j and in particular for j > h. In order to prove the other inclusion we have to ensure that p 0 can be reached starting from a point p ∈ H by means of the flows 1 (p 0 ))...) = p, the definition of A(p 0 ) ensures that f i (p) = 0 for i = 1, . . . h whereas by hypothesis we have Hence f i (p) = 0 ∀i ∈ N and p ∈ H.

Lemma 5.3
In the hypotheses and with the notations of Lemma 5.2,V Gj ∈ T K and D i ∈ T K.
Proof. We recall that a vector field V ∈ T K if and only if V (K j ) = 0, where K j are given by (11). Since for any j (with j > h) there exists a suitable k such that f 1 , . . . f j ∈ G k , it is possible to chose as coordinates in K ∩ U ∩ G k the functions f i (i = 1, . . . h) and some functions y 1 , . . . , y r (with r = dim(G k ) − h) such that V Gi (y l ) = 0. In particular, for any j > h, there exists a smooth function L j such that y 1 (p), ...y r (p)).
In order to prove that D i ∈ T K, we consider By definition, being D i ∈ T H, we have that D α i ∈ T K and, by Theorem 7.1 (see Appendix), there exist smooth functions C i j (α, p) such that Moreover, since Φ i a i are diffeomorphisms, span{D α 1 , ..., D α m } and span{D 1 , ..., D m } have the same dimension. Hence the matrix C i j is invertible for any α, ensuring that D i ∈ T K.

Remark 5.4
The functions K j defined by (11) are a set of independent invariants for the vector fieldsV Gi . Furthermore, since K is finite dimensional, it is possible to add a finite number of functions z i such that (z i , K j ) form an adapted coordinate system with respect the filtration G k for k sufficiently large.
Theorem 5.5 In the hypotheses and with the notations of Lemma 5.2, let V F be an evolution vector field such that V F ∈ T H, dim(span{V F , V G1 , ..., V G h }) = h + 1 and Proof. Given the (m + h + 1)-dimensional distribution If we prove that Φ i a * (∆) = ∆, we have ∆| K ⊂ T K and, in particular, V F ∈ T K. Considering the coordinate system z i , K j of Remark 5.4 we can suppose, possibly relabeling some invariant z i with K j for some j, that we have exactly h coordinates z i . Eliminating some element of the form ∂ K j , the sequence V F ,V Gj , D k , ∂ K l form a basis of T J ∞ (M, N ) and for any vector field X ∈ T J ∞ (M, N ) there exist suitable functions b, c i , d j , e l depending on a and p ∈ U a such that From the definition of X a and using [V Gi , ∆] ⊂ ∆ and [V Gi , ∂ K l ] ∈ ∆, we obtain that the functions e l must solve the equations Moreover, since X 0 = X ∈ ∆, e l (0, p) = 0 and, from the previous equation, we get e l (a, p) = 0 for any a, which ensures X a ∈ ∆ for any a.
Remark 5.6 Theorem 5.5 still holds if we consider r functions F i ∈ F such that dim(span{V F1 , ..., V Fr , V G1 , ..., V G h }) = r + h, V Fi ∈ T H for any i = 1, ..., r and for some constants λ l i,j , µ k i,j ∈ R.
Theorem 5.7 In the hypotheses and with the notations of Lemma 5.2, if F, G i are real analytic, H is defined by real analytic functions and, denoting by L = F, G 1 , . . . , G h the Lie algebra generated by F and G i , we have Proof. We note that the functions K i defined by (11) are real analytic if the vector fieldsV Gi and the submanifold H are real analytic. The vector field V F is in T K if for any p 0 ∈ K and any K i we have We know that if p 0 ∈ K there exists an α = (a 1 , ..., a h ) ∈ R h and p 1 ∈ H such that Moreover, being K i invariants of Φ j a j we have Since the previous expression is real analytic it is sufficient to prove that any derivative of any order with respect a i evaluated in (a 1 , ..., a h ) = 0 is zero. It is easy to verify that ∂ k1 where we use the notationV ...) ∈ T K| H and so for any p 1 ∈ H and any K ī Remark 5.8 IfV Gi , V F are real analytic and H is defined by real analytic equations, Theorem 5.5 implies Theorem 5.7. On the other hand Theorem 5.5 turns out to be very useful when we consider smooth (not analytic) invariant manifolds H.

Remark 5.9
It is important to note that Theorems 5.5 and 5.7 hold also if H is a manifold with boundary. In this case if V G1 , ..., V G h ∈ T (∂H) we obtain that K is also a local manifold with boundary.

The general algorithm
For the convenience of the reader we start this subsection describing the general algorithm we use in the examples. Given a PDE of the form where G k ∈ F n admit characteristic flows, the first step is to compute the characteristic flows of G k integrating the characteristic vector fieldsV G k and obtaining the local diffeomorphisms Φ i a . Then, considering the differential constraint H for F given by the equations f i (x, u, u σ ) = 0 and their differential consequences D σ (f i ) = 0, we choose h independent functions g i between the functions f i , D σ (f i ) such that the matrix (V G k (g i )) is non-singular and we solve the equations where α = (a 1 , ..., a h ), obtaining a i = A i (x, u, u σ ). Hence, the new differential constraint K is obtained by replacing α with (A 1 (x, u, u σ ), ..., A h (x, u, u σ )) in the expression Φ * (A 1 (x,u,uσ),...,A h (x,u,uσ)) (h i ) = 0, where h i are all the other functions f j , D σ (f k ). We remark that, in order to integrate the system of ODEs representing the evolution equation on K and the connection C| K representing the reduction function K(x, b), we have to compute the coordinate expressions for the vector fields V F , V G k , D i restricted to K. For this purpose we choose a coordinate system given by x i , some coordinate system y i on H and the functions a i = A i (x, u, u σ ).
Using coordinates (x i , y j , a k ) the vector fields V F , V G k , D i have a rather simple expression. Obviously D i (x j ) = δ j i , but we have also D i (a k )| K = 0. Furthermore, if the hypotheses of Theorem 5.5 hold, V G k (a k ), V F (a k ) depend only on the coordinates a k . Indeed on H, or equivalently on the submanifold a k = 0 in K, we have ∂ a k = −V G k (the minus sign owing to the fact that we use the pull-back in (13)). This means that for α = (a 1 , ..., a h ) we have ∂ a k = Φ * α (−V G k ) and it is easy to prove that ∂ a k = jC j k (α)V Gj and sō where C is the inverse matrix ofC. SinceV G k = V G k − lh l D l (for some functionsh l ∈ F ) and D i (a k ) = 0, the expression V G k (a l ) depends only on a 1 , ..., a h . The situation is completely similar for V F . Remark 6.1 It is important to note that, in the hypotheses of Theorem 5.5, V G k (a l ) and V F (a l ) do not depend on the choice of H and on the functions g i but only on the order we choose to apply the pull-back in equation (13).

A non-linear transport equation with dissipation
Let us consider the following equation Since, for c 1 (t) = 1, c 2 (t) = 0, equation (14) is a non-linear transport equation ∂ t (u) = uu x (see for example [38]), if c 2 (t) ≤ 0 and u ≥ 0 the term u can be considered as a dissipation factor. Furthermore being equation (14) for c 1 (t) = 1, c 2 (t) = 0 the first equation of an Hamiltonian hierarchy (see, e.g., [32]), the complete equation (14) can be considered as a non-isospectral perturbation.
We can see (14) as an evolution PDE of the form (12) with In this case, since V F , V G1 , V G2 are not linearly independent, Theorem 5.5 can not be applied. However Lemma 5.2 and 5.3 are enough to provide a differential constraint K for (14), since V F ∈ T K being V F = 0. The characteristic vector fields for G 1 and In order to construct a differential constraint for (14) we consider the differential constraint H for V F defined by f 1 := u − x 2 = 0 and all its differential consequences x and conversely we have Using the previous expressions for a, b in Φ * α (f 3 ) = 0 we find that K is defined by the vanishing of the function 2u 2 u xx − xu 3 x + uu 2 x and all its differential consequences. Choosing the coordinate system (x, a, b) on K we have Hence the function A(x, t) and B(x, t) have to satisfy Therefore, using the expression of u in terms of x, a, b we obtain Indeed Remark 6.1 provides a more general result. In fact, if we consider A(t), B(t) given by equations (15) and (16), and we chose any manifold H of the form u = H(x) (with H(x) a non-linear function) we have that any function U (x, t) solution to is a solution to equation (14). Hence in this case the use of the implicit form for the constraint turns out to be more effective than the direct use of the reduction function K.

Modified heat equation
Let us consider the following equation which can be seen as an equation of the form (12) with This equation (with c 1 , c 2 constants) has already been studied in [34,36] and coincides with the Zakai equation of the simplest Kalman filter in one dimension if c 1 = 1/2 and c 2 is the derivative of a Brownian motion process (see [2]). The vector fields V G1 , V G2 form an abelian Lie algebra and admit strong characteristics. Furthermore F , G 1 , G 2 ,G 1 = u x ,G 2 = xu x ,G 3 = u form a Lie algebra. Let H be the submanifold of J ∞ (R, R) defined by f 1 := u x = 0 and all its differential consequences f k+1 := D k x (f 1 ) = 0 for any k ∈ N. It is easy to prove that V F , VG i ∈ T H for any i = 1, 2, 3. In this situation F, G 1 , G 2 satisfy the hypotheses of Theorem 5.7 on H and, in order to find the equations of K, we start by computing the characteristic flows of G 1 and G 2 By Theorem 7.1 we obtain the characteristic flows for the derivatives of an order and, using equations Φ * α (u x ) = Φ * α (u xx ) = 0, we get Therefore we can express a, b as functions of u x , u xx as follows and equation Φ α (u xxx ) = 0 defining (together with all its differential consequences) the manifold K is Since the manifold K is four dimensional, we use coordinates (x, u, a, b) on it and, computing the components of the vector fields V Fi , V Gj and D x , we get Hence the equations for U, A, B are The previous system has a unique solution such that U (x 0 , t 0 ) = u 0 , A(x 0 , t 0 ) = a 0 , B(x 0 , t 0 ) = b 0 . Without loss of generality we can suppose that x 0 = 0 and, in order to simplify computation, we also suppose c 1 , c 2 ∈ R. The equations for A, B, U in t derivative form a triangular system (linear in A, U and with Riccati form not dependent on time in B) and can be solved explicitly getting some functions Hence we can explicitly integrate the equations for x and we get The function U (x, t) is the well known Gaussian solution for the modified heat equation.

An integrable two dimensional system
Let us consider the following system In the case c i,j = 0 for i, j ≥ 2 and c i,j = cost for i, j < 2, this system admits an infinite number of higher order symmetries and a recursion operator (see for example [5]). Furthermore denoting by it is easy to prove that V F , V Gi,j form a pro-finite Lie algebra so that this provides a toy-model for the pro-finite Lie algebras used in non-linear filtering problem (see [21]). If we consider the submanifoldH ⊂ J ∞ (R, R 2 ) given by the equation and its differential consequences, obviously D x ∈ TH. Furthermore we have V F , V Gi,j ∈ TH and L 1 = span{V Gi,j } restricted onH is finite dimensional. So puttingL = span{V Gi,j |i, j < k}, if H is a finite dimensional submanifold ofH such thatL has maximal rank on H, D x ∈ T H and V F ∈ T H. The hypotheses of Theorem 5.7 are satisfied: indeed, denoting by L = span{V F , V Gi,j }, we have [L, L]|H = L 1 |H =L|H.
Hereafter, in order to simplify computation, we take c i,j = 0 for i, j > 1, c 0 = 1 and g = v xx − βv. In this case we choose as submanifold H ofH the set of zeros of h = u x − γu. Hence, writing V 1 = V G0,0 , V 2 = V G1,1 , V 3 = V G0,1 and denoting by Φ i the corresponding characteristic flows, we have x (h)) = 0, we obtain that K is defined by On K we use the natural coordinate system ( In this coordinate system we have Fixing (t 0 , x 0 ) and the initial conditions U (t 0 , x 0 ) = u 0 , V (t 0 , x 0 ) = v 0 ..., the functions U, V, V x , A, B, C must satisfy the following overdetermined system of equations In the part of system with the t derivative, the equations for A, B, C do not depend on the other variables and are linear and non-homogeneous with respect to A, B, C. So, considering the matrix where A 0 (t) = A(x 0 , t), etc.. Moreover, since the equations in t for V, V x are linear, we have and, being also the equation for U linear in U and depending on v 0 , v 0 x , ..., we obtain e −γ 2 (s−t0) ((−4βC 0 (s)V 0 (s)V 0 x (s) − 2β 2 B 0 (s) − 2βA 0 (s))V 0 (s) 2 )ds + t t0 e γ 2 (s−t0) (c 0,0 (s)V 0 (s) 2 + c 1,1 (s)V 0 x (s) 2 + c 0,1 (s)V 0 (s)V 0 x (s))ds .

Appendix
In this section we discuss the behavior of the Cartan distribution C under the action of the characteristic flow Φ a associated with an evolution vector field V G . An important consequence of the following Theorem is that Φ * a (u i σ ) is a polynomial function with respect the variable u k σ ′ if |σ ′ | is sufficiently large.
Theorem 7.1 Let V G be an evolution vector field admitting characteristics and let Φ a be the corresponding characteristic flow. Denoting by A the n × n matrix A = (A j i ) := (D i (Φ * a (x j )))| i,j , and by B = (B i j ) the inverse matrix of A, then and, for any f ∈ F , we have Proof. LetṼ G = V G − i h i D i be the characteristic vector field of V G . Since the vector field D a i = Φ * a (D i ) solves the equation In order to prove (19) we show that the vector fieldD a i := j B j i D j solves equation (21) as well. We start by computing ∂ a (A j i ) = ∂ a (D i (Φ * a (x j ))) = D i (Φ * a (Ṽ G (x j ))) = −D i (Φ * a (h j )).
Since B = A −1 the formula for derivative of the inverse matrix gives This means that ∂ a (B i j ) = k,r B k j (D k (Φ * a (h r )))B i r and we get Hence bothD a i and D a i satisfy equation (21) and we haveD a i = D a i .

Remark 7.2
It is important to note that equation (19) holds in all J ∞ (M, N ) while equation (20) holds in F k for k sufficiently large.

Corollary 7.3
Given an evolution vector field V G with corresponding characteristic flow Φ a , the expression Φ * a (u i σ ) is a polynomial function with respect the variable u k σ ′ if |σ ′ | is sufficiently large.
Proof. If we apply Theorem 7.1 to f = u i we get Φ * a (D k (u i )) = j B j k D j (Φ * a (u i )). Since D j (Φ * a (u i )) is a linear function with respect the variable u i σ ′ if |σ ′ | is sufficiently large and B j k ∈ F h for some h ∈ N, applying iteratively Theorem 7.1 we obtain the thesis.