Infinitesimal Classification of First Order Systems of Two Partial Differential Equations

in Two Variables

S. V. Duzhin ^{∗}

June 11, 2010

1 Introduction

We give a classification of symbols of partial differential systems that consist of two equations imposed on two functions in two independent variables, with respect to the natural action of the group of linear changes of dependent and independent variables.

The classification turns out to be finite, consisting of 5 types in the real case and of 4 types in the complex case. Apart from the elliptic, parabolic and hyperbolic types [1] constituting the well-known classification of second order equations for a function u(x, y), in the present case we also have two non-equivalent ‘degenerate’ types.

Bifurcation diagrams for the singular orbits in the space of symbols imply certain properties of the singularities of linear and quasilinear systems of the considered type. I am indebted to S. V. Chmutov who explained me the meaning of versal deformations and bifurcation diagrams in the general context of an arbitrary Lie group action on a manifold.

In the last section of the paper we find, for every type of the symbol, the topological type of the corresponding integral Grassmanian — an object important in the study of singularities of solutions of partial differential equations [5].

∗This work was partially supported by the following grants: RFBR 98-01-00555a, RFBR 00-15-96084, NWO-RFBR 047-008-005.

All theorems stated in this text are quite straightforward, so we confine ourselves just to short comments, omitting the complete proofs.

2 Classification of Symbols

A system of k-th order partial differential equations imposed on m functions of n independent variables determines a submanifold E in the jet space J_{k}(X, Y) where dim X = n, dim Y = m. The symbol of the system at a point θ ∈ E is defined as the intersection of the tangent space to E and the tangent space to the fiber F_{θ} of the natural projection J_{k} → J_{k−1} containing θ. If a ∈ X and b ∈ Y are the source and the target of the jet θ, respectively, then the tangent space T_{θ}(F_{θ}) is naturally isomorphic to S^{k}V^{∗} ⊗ W where V = T_{a}(X), W = T_{b}(Y) and S^{k} denotes the k-th symmetric power (see [2]). In this sense, a symbol can be thought of as a linear subspace of the space S^{k}(R^{n})^{∗} ⊗ R^{m}.

Denote by G_{θ} be the group of those local diffeomorphisms of the pair (X × Y, (a, b)) which, after being lifted to the jet space J_{k}(X, Y), do not move the point θ.

Theorem 1 Let θ, θ′ ∈ π_{k,0}^{−1}(a, b) be two points lying in the same fiber of the natural projection π_{k,0} : J_{k}(X, Y) → J_{0}(X, Y) = X × Y. Then:

1. There exists a local diffeomorphism ϕ in a neighbourhood of (a, b) such that ϕ^{(k)}(θ) = θ′.

2. The groups G_{θ} and G_{θ′} are conjugate via ϕ.

3. The action of G_{θ} in T_{θ}(F_{θ}) agrees with the action of G_{θ′} in T_{θ′}(F_{θ′}) under the identification T_{θ}(F_{θ}) ≅ T_{θ′}(F_{θ′}) ≅ S^{k}V^{∗} ⊗ W.

4. In the case X = R^{n}, Y = R^{m} and θ = (0, 0, …, 0) in the standard jet coordinates the group G_{θ} consists of those local diffeomorphisms (x, y) ↦ (x̃, ỹ) which satisfy \frac{∂ỹ}{∂x}|_{(0,0)} = 0.

5. The action of this group in T_{θ}(F_{θ}) depends only on the linear part of the diffeomorphism and decomposes as G → GL(V) × GL(W) → GL(T_{θ}(F_{θ})), where the last arrow is nothing but the natural left-right action of the direct product GL(R^{n}) × GL(R^{m}) in S^{k}(R^{n})^{∗} ⊗ R^{m}.

Thus, in a purely algebraic setting, the problem of symbol classification can be stated as follows. Fix a quadruple of natural numbers n, m, k, r, and let V and W be linear spaces of dimensions n and m over a certain field. Describe the orbits of the group GL(V) × GL(W) in the Grassmanian of rdimensional planes in S^{k}V^{∗} ⊗ W with respect to the action that comes from the natural left-right action of this group in the underlying space S^{k}V^{∗} ⊗ W. The problem makes sense over an arbitrary field; we will be only concerned with the reals and the complexes.

In the special case m = 1, k = 2, r = 1 (corresponding to one second order equation) we obtain the problem of classification of (real or complex) quadratic forms in n variables. Its well-known solution gives rise to the usual notion of type (hyperbolic, parabolic, elliptic) for the second order equations.

In this note we will solve, and derive the consequences of, the symbol classification problem in the case n = 2, m = 2, k = 1, r = 2, which, simple as it is, has never been studied in the literature, to the best of our knowledge. The algebraic problem to study is that of classifying two-dimensional linear spaces of linear operators in the plane with respect to the independent linear changes both in the source and in the target.

A complete list of all quadruples (n, m, k, r) for which the classification of symbols is finite, will be given in a separate paper [4].

3 Planes of linear operators in the plane

The direct product GL(R^{2}) × GL(R^{2}) acts in End(R^{2}) according to the rule

(X, Y) · A = XAY.

We are to find the orbits of the induced action on the set of 2-planes in End(R^{2}).

In the four-dimensional space End(R^{2}) we single out a three-dimensional cone K that consists of all degenerate operators. The trace left by this cone on a two-plane L in End(R^{2}) and considered up to linear transformations of the plane, is an important invariant of L which is sufficient to distinguish all the orbits but two. The latter are made up of planes that are entirely contained in K and differ in that, for one of them, all the operators of the plane possess a common 1-dimensional kernel, while for the other, they have one and the same 1-dimensional image.

Theorem 2 Two-dimensional planes of linear operators in R^{2} under the linear changes of variables in the source and in the target split into five orbits. Therefore, the symbol of any system of partial differential equations with n = m = r = 2 belongs to one of the five possible types.

The orbits are listed in this table:

Type  dim(O) L ∩ K  Normal form of L  System of d.e.
Hyperb.  4  two lines ⟨( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )⟩ \{ \begin{matrix}u_{y} = 0 \\ v_{x} = 0 \end{matrix}
Ellipt.  4  one point ⟨( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )⟩ \{ \begin{matrix}v_{x} = −u_{y} \\ v_{y} = u_{x} \end{matrix}
Parab.  3  one line ⟨( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )⟩ \{ \begin{matrix}v_{x} = 0 \\ v_{y} = u_{x} \end{matrix}
Deg. I  1  all plane ⟨( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )⟩ \{ \begin{matrix}u_{y} = 0 \\ v_{y} = 0 \end{matrix}
Deg. II  1  all plane ⟨( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )⟩ \{ \begin{matrix}v_{x} = 0 \\ v_{y} = 0 \end{matrix}

These five orbits can be conveniently vizualized in terms of elementary projective geometry. The Grassmanian of two-planes in the four-dimensional space End(R^{2}) is isomorphic to the variety of projective lines in the threedimensional projective space P (End R^{2})) ≅ RP^{3}. The cone of degenerate operators shows up as a non-degenerate quadric Q ⊂ RP^{3}. The action of the group of left and right linear changes carries over into the action of the group of projective transformations of RP^{3} that preserve the quadric Q. A line l belongs to the ellptic, parabolic or hyperbolic orbit if it intersects Q in 0, 1 or 2 points, respectively. The two families of straightline generators of the quadric correspond to the two degenerate types.

4 Versal Deformations

Let G be a Lie group smoothly acting on a manifold M. A versal deformation [3] of a point p ∈ M is a local mapping f : (Λ, 0) → (M, p) of a certain parameter space Λ ≅ R^{s} whose image is transverse to the orbit G(p) and has dimension equal to the codimension of G(p).

The partition of the space of parameters Λ into subsets corresponding to different orbits is called bifurcation diagram.

If the orbit of the point p has dimension equal to the dimension of M, then its bifurcation diagram is trivial and it has no versal deformations with s > 0. The type of such point p is said to be stable.

Theorem 3 The versal deformations of the normal forms listed in Theorem 2 can be chosen in the following way:

1. The hyperbolic type is stable.

2. The elliptic type is stable.

3. For the parabolic type {v_{x} = λu_{y}, v_{y} = u_{x}}.

4. For the first degenerate type {u_{y} = λ_{1}v_{x}, v_{y} = λ_{2}u_{x} + λ_{3}v_{x}}.

5. For the second degenerate type {v_{x} = λ_{1}u_{y}, v_{y} = λ_{2}u_{x} + λ_{3}u_{y}}.

The bifurcation diagram for a parabolic point is one-dimensional and consists of an elliptic region on one side of the point considered and of a hyperbolic region on the other side.

Bifurcation diagrams for both degenerate types are similar; they can be represented as a neighbourhood of zero in a three-space split by the cone λ_{3}^{2} + 4λ_{1}λ_{2} = 0 into an elliptic region (inside the cone) and a hyperbolic region (outside the cone); the points of the cone itself belong to the parabolic type.

Thus, the diagram of orbit adjacency looks like this:

Degenerate I

Degenerate II Parabolic Elliptic

Hyperbolic

[figure]

[figure]

[figure]

[figure]

In the complex case all these results remain valid with one exception: the elliptic and hyperbolic orbits merge into one.

5 Degeneration of Symbols for Linear and Quasilinear Systems

So far, we have been concerned with the symbol of the system E ⊂ J_{1}(R^{2}, R^{2}) at one fixed point θ ∈ E. Let us now see what can be said about the dependence of the symbol σ(E, θ) on this point θ.

For a general non-linear system the set {σ(E, θ)} constitutes a six-para-meter family, because the manifold E is six-dimensional.

For a quasilinear system

\{ \begin{matrix}a_{1}(x, y, u, v)u_{x} + b_{1}(x, y, u, v)u_{y} + c_{1}(x, y, u, v)v_{x} + d_{1}(x, y, u, v)v_{y} = 0 \\ a_{2}(x, y, u, v)u_{x} + b_{2}(x, y, u, v)u_{y} + c_{2}(x, y, u, v)v_{x} + d_{2}(x, y, u, v)v_{y} = 0 \end{matrix}

(1)

this family is constant along the fibers of the projection J_{1} → J_{0} and thus in fact depends only on four parameters.

For a linear system

\{ \begin{matrix}a_{1}(x, y)u_{x} + b_{1}(x, y)u_{y} + c_{1}(x, y)v_{x} + d_{1}(x, y)v_{y} = 0 \\ a_{2}(x, y)u_{x} + b_{2}(x, y)u_{y} + c_{2}(x, y)v_{x} + d_{2}(x, y)v_{y} = 0 \end{matrix}  (2)

this family only depends on two parameters.

The results obtained in the previous section lead to the following conclusions about the behaviour of the regions in the parameter spaces where the symbol of the system belongs to one or another type.

Theorem 4 For a linear system (2), the degenerations of the symbol of types I and II can be eliminated by a small movement. The parabolic points of a typical system form a curve in the plane (x, y), on one side of which the symbol of the system is elliptic while on the other side it is hyperbolic. This picture is stable with respect to small deformations.

For a quasilinear systems (1) all degenerations of the symbol are stable. In a neighbourhood of every point (x, y, u, v) that belongs to either of the two degenerate types I or II, there is an entire curve made up of points of the same type; in a transversal to this curve one can always find parabolic, elliptic and hyperbolic points which, up to a diffeomorphism, fill up a quadratic cone, its inner part and its outer part.

6 Integral Grassmanians

According to [5], with every point θ of a submanifold E ⊂ J_{k}(X, Y), one can associate an algebraic variety IG(E, θ) which consists of all n-planes tangent to E and integral for the metasymplectic structure naturally defined on the jet bundle. This variety is called the integral Grassmanian of E at θ; its topology yields important information about the singularities that may appear in the solutions of the given system.

It is readily verified that the algebraic (and hence the topological) type of the integral Grassmanian IG(E, θ) is completely determined by the type of the symbol σ(E, θ) in the sense of the previous classification and thus Theorem 2 allows to describe all possible types of integral Grassmanians for the first order systems 2 × 2.

Theorem 5 The integral Grassmanians for the first order systems with two dependent and two independent variables may belong to one of the following five topological types defined by the type of the symbol:

Symbol type  Integral Grassmanian

Elliptic  Sphere S^{2}
Hyperbolic  Torus S^{1} × S^{1}

Parabolic Torus with a meridian collapsed to a point

Degenerate I

[figure]

Two intersecting planes in RP^{3}

Degenerate II

[figure]

Three-manifold

S^{2} × [0, 1]/(x, 0) ∼ (−x, 1)

References

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2. H. Goldshmidt. Integrability criteria for systems of nonlinear partial differential equations. J. Differential Geometry 1 (1967), p. 269–307.

3. V. I. Arnold, A. N. Varchenko, S. Gussein-Zade. Singularities of differentiable mappings (in Russian). Vol. 1. Nauka, Moscow, 1982.

4. S. V. Chmutov, S. V. Duzhin. Simple and unimodular cases in the classification of linear spaces of vector-valued homogeneous forms. In preparation).

5. V. V. Lychagin. Topological aspects of geometrical theory of differential equations. Lecture Notes in Math. 1214, Springer, 1985.