Pfaffian systems that do not satisfy their integrability conditions

Question

Remark: In this question I am first and foremost interested in a local problem and local solutions therefore I assume all functions are defined on open sets of real coordinate spaces and I will not bother with explictly considering domains. To simplify notation some maps will actually be partial maps defined on an open subset. Summation convention on repeated indices is understood. All functions are taken to be $C^\infty$.

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Background: A Pfaffian system is a set $\theta^\alpha=\theta^\alpha_i dx^i$ of pointwise linearly independent $1$-forms ($i=1,...,m$, $\alpha=1,...,n$, $n\le m$, i.e. we have $n$ such $1$-forms in $m$ variables).

A Pfaffian equation $$ \theta^\alpha\approx 0 $$ is a partial differential equation for a submanifold $\phi:\mathbb R^{m-n}\rightarrow\mathbb R^{m}$ which is a solution iff $$ 0=\phi^\ast\theta^\alpha=\theta^\alpha_i(\phi(u))\frac{\partial\phi^i}{\partial u^a}du^a. $$

If $\Lambda^\alpha_{\ \beta}$ is an invertible matrix whose elements are functions of the $x^i$, the system $$ \bar\theta^\alpha =\Lambda^\alpha_{\ \beta}\theta^\beta$$ is equivalent to the system $\theta^\beta$.

Per the standard terminology, the Pfaffian system is integrable if for each $x_0$ there is a solution (integral submanifold) whose image contains the point $x_0$. The Pfaffian system is completely integrable if there is an equivalent system whose generators are exact, i.e. there exists an invertible matrix $\Lambda^\alpha_{\ \beta}$ such that $$ \Lambda^\alpha_{\ \beta}\theta^\beta=dF^\alpha. $$The system is closed if there are $1$-forms $\xi^\alpha_\beta$ such that $$ d\theta^\alpha=\xi^\alpha_\beta\wedge\theta^\beta, $$ or equivalently if $$ d\theta^\alpha\wedge\theta^1\wedge...\wedge\theta^n=0. $$

The Frobenius integrability theorem essentially states that a Pfaffian system being closed is a sufficient condition for complete integrability. (The other direction of implications completely integrable $\Rightarrow$ integrable $\Rightarrow$ closed is easily seen.)

Assuming now $m$ "base" variables $x^i$ and $n$ "fibre" variables $y^\alpha$ and $n+m$ "total variables, a differential equation $$ \frac{\partial\phi^\alpha}{\partial x^i}(x)+\Pi^\alpha_i(x,\phi(x))=0 $$ which is often called a total differential equation has its existence conditions determined by the Frobenius theorem. The Frobenius condition in this case is $$ R^\alpha_{ij}=\frac{\partial\Pi^\alpha_j}{\partial x^i}-\frac{\partial\Pi^\alpha_i}{\partial x^j}+\frac{\partial\Pi^\alpha_i}{\partial y^\beta}\Pi^\beta_j-\frac{\partial\Pi^\alpha_j}{\partial y^\beta}\Pi^\beta_i=0. $$ If this equation is satisfies then for any initial condition $(x_0,y_0)$ the total differential equation has a (essentially unique) solution with $\phi^\alpha(x_0)=y_0^\alpha$. It also follows (modulo some analytical muckery regarding the smooth dependence on initial conditions) that then there exists a function $$ \Phi^\alpha(x,y_0) $$ of $m+n$ variables such that for fixed $y_0$ the function (as a function of the remaining varibles) is the unique solution corresponding to the initial values $(x_0,y_0)$, that is the general solution of the total differential equation is parametrized by $n$ constants/parameters.

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Questions:

I am essentially looking for a modern (more explanation later on what I mean modern) treatment of Pfaffian systems which also take into account what happens when the integrability conditions are not satisfied identically.

For the case of a Pfaffian system $\theta^\alpha\approx 0$ and from a geometric point of view I expect that integral submanifolds of maximal ($m-n$) dimension do not exist, but lower dimensional integral manifolds might.

For total differential equations of the form $\phi^\alpha_{,i}(x)+\Pi^\alpha_{i}(x,\phi(x))=0$ I expect that a general point $(x_0,y_0)\in \mathbb R^{m+n}$ has no solution passing through it. However solutions that satisfy the algebraic constraint $$ R^\alpha_{ij}(x,\phi(x))=0 $$can still exist. I am interested in eg. how can we describe the "general solution" of the differential equation. Maybe instead of $n$ parameters, the general solution depends on $< n$ constants?

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I give some further context here. I know that this problem is quite classical and its solution was even known in the 19th century. But for some reason I have extreme difficulties in finding a good and concise reference that treats this and contains the proofs.

I have searched in differential geometry books, but those usually only contain a geometric discussion of Frobenius' theorem but do not consider solutions of PDEs when the integrability conditions are not satisfied.

I am not particularly familiar with the PDE literature but looked in a few books, which only contained treatments of Laplacian/Poissonian/wave equations, boundary conditions, etc. not Pfaffians. As far as I understand, Pfaffians belong more to differential geometry than PDE theory especially that the structure of Pfaffians can be understood via ODEs.

I have found extensive discussions in very old books like those of Schouten/Eisenhart/Forsyth, however I have mainly three issues with these books,

1. They use very archaic terminology and notations that makes reading them extremely difficult.
2. It seems to me that pre-Bourbaki mathematicians had a very different idea what constitutes a proof than the post-Bourbaki mathematicians.
3. They frequently assume analyticity and have often seen power series proofs in these books. I don't like that. As far as I am aware exterior differential systems more complicated than Pfaffians can only be treated in the analytic category. But the Frobenius theorem only requires $C^\infty$ (in fact $C^{\text{sufficiently high}}$) functions and thus the integrability of Pfaffian systems is treatable in the $C^\infty$ category. This is something I absolutely insist on, I have no interest in any treatment that uses analyticity.

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Sooooo, I'd really like to get my hands on a reasonably modern and rigorous reference on integrability conditions of Pfaffian PDEs which consider the case when the integrability conditions are not satisfied and only use $C^\infty$ arguments at most.

I also note that I am not necessarily interested in constructive solution techniques that can aid with explicitly finding the solutions/integral manifolds. I am only interested in "theoretical" characterization of integral submanifolds and/or solutions of the Pfaffian system.

Answer 1

Since everything is local and $C^\infty$, it is not hard to derive sufficient conditions for there to exist solutions. Analyticity is not actually needed, but some assumption of regularity is necessary.

Here is one such result:

Let the Pfaffian system $\mathcal{I}$ be generated by $\theta^1,\ldots,\theta^s$ and choose $1$-forms $\omega^1,\ldots,\omega^m$ to complete to a basis of $1$-forms. (If desired, one can choose functions $x^1,\ldots, x^m$ such that $\theta^1\wedge\cdots\theta^s\wedge\mathrm{d}x^1\wedge\cdots\mathrm{d}x^m\not=0$ and let $\omega^i=\mathrm{d}x^i$, but this is not necessary.)

Now compute the functions $R^\sigma_{ij}=-R^\sigma_{ji}$ such that $$ \mathrm{d}\theta^\sigma \equiv \tfrac12 R^\sigma_{ij}\,\omega^i\wedge\omega^j\ \mod \theta^1,\ldots,\theta^s. $$ Next, compute the functions $R^\sigma_{ijk}$ such that $$ \mathrm{d}R^\sigma_{ij} \equiv R^\sigma_{ijk}\,\omega^k\ \mod \theta^1,\ldots,\theta^s, $$ and, inductively, define $R^\sigma_{i_1i_2\cdots i_{k+1}}$ for $k\ge3$ by $$ \mathrm{d}R^\sigma_{i_1i_2\cdots i_k} \equiv R^\sigma_{i_1i_2\cdots i_ki_{k+1}}\,\omega^{i_{k+1}}\ \mod \theta^1,\ldots,\theta^s. $$ Let $\mathcal{R^k}$ for $k\ge 2$ denote the $C^\infty$ ideal spanned by the functions $R^\sigma_{i_1i_2\cdots i_j}$ for $j\le k$. Obviously, this is an increasing sequence of ideals, and any $m$-dimensional integral manifold of $\mathcal{I}$ will lie in the zero locus of $\mathcal{R^k}$ for all $k\ge 2$.

If, at any level of $k$, $\mathcal{R}^k$ becomes equal to the ring of all $C^\infty$ functions on the domain, then there is no integral manifold of the Pfaffian system on which $\omega^1\wedge\omega^2\wedge\cdots \wedge\omega^m$ is nonvanishing.

If, on the other hand, $\mathcal{R}^{k+1}=\mathcal{R}^k$ for some $k\ge 2$ and the zero locus $Z_k$ of $\mathcal{R}^k$ consists of $\mathcal{R}^k$-ordinary zeroes, then the smooth manifold $Z_k$ is smoothly foliated by $m$-dimensional integral manifolds of $\mathcal{I}$.

(Note: Given a set of smooth functions $\mathcal{R}$ on a domain, a zero of $\mathcal{R}$, i.e., a point $p$ at which all of the functions in $\mathcal{R}$ vanish, is said to be an ordinary zero of $\mathcal{R}$ if there exist functions $r_1,\ldots,r_q\in \mathcal{R}$ with independent differentials at $p$ such that the zero locus of $\mathcal{R}$ is defined by $r_1=\cdots=r_q=0$ on some open neighborhood of $p$.)

The proof of the above result is simple: It's a local statment, so let $p\in Z_k$ be an ordinary zero of $\mathcal{R}^k$, let $r_1,\ldots,r_q\in \mathcal{R}^k$ be functions with linearly independent differentials at $p$, and let $U$ be an open neighborhood of $p$ on which the $\mathrm{d}r_\ell$ are linearly independent and $Z_k\cap U$ is the locus in $U$ on which all of the $r_\ell$ vanish.

Then, by the implicit function theorem, $Z_k\cap U$ is an embedded submanifold of $U$ of codimension $q$ and, by construction, $\mathcal{R}^k(U)= \langle r_1,\ldots,r_q\rangle$. Consequently, since $\mathcal{R}^{k+1}(U)=\mathcal{R}^k(U)$, it follows that $$ \mathrm{d}r_\ell = G_{\ell\sigma}\theta^\sigma + F_{\ell i}\,\omega^i $$ on $U$ where $F_{\ell i}\in \langle r_1,\ldots,r_q\rangle$.

Since the $F_{\ell i}$ vanish on $Z_k\cap U$, it follows that the matrix $(G_{\ell\sigma})$ must have rank $q$ on $Z_k\cap U$. Consequently, when these relations are pulled back to $Z_k\cap U$, they become $q$ linearly indepdent relations among the $\theta^\sigma$, i.e., $0 = G_{\ell\sigma}\theta^\sigma$.

Meanwhile, since $R^\sigma_{ij}\in \mathcal{R}^k(U)= \langle r_1,\ldots,r_q\rangle$, it follows that the pullback of the Pfaffian system $\mathcal{I}$ to $Z_k\cap U$ (which has rank $s-q$) satisfies the Frobenius integrability condition. Now apply the Frobenius Theorem to this system on $Z_k\cap U$.

References: I don't know exactly where this is written in the 'modern' literature. I think we didn't include it in our book Exterior Differential Systems (B—, Chern, Gardner, Goldschmidt, and Griffiths). I know that I learned it from a paper of Élie Cartan, Les problèmes d'équivalence (Séminaire de Math., exposé D, 11 janvier 1937) [Republished in Partie II of Cartan's Œeuvres Complètes]. [Of course, it's written in Cartan's style, so you won't see anything mention of an open set $U$, ideals of smooth functions, or a formal definition of 'ordinary zero', but the concept is implicit in his argument. I just translated his proof into our 'modern' idiom.]

Now, the above sufficient condition is by no means necessary. Here is an example to show why: On $xyz$-space consider the $1$-form $$ \theta = \mathrm{d}z + z^m x\,\mathrm{d}y, $$ for some integer $m>1$. Then $\mathrm{d}\theta \equiv z^m\,\mathrm{d}x\wedge\mathrm{d}y\mod\theta$ and $\mathrm{d}(z^m)\equiv - mz^{2m-1} x\,\mathrm{d}y\mod\theta$, so $\mathcal{R}^k = \langle z^m\rangle$ for all $k\ge 2$, so the locus $z=0$ (which is the unique $2$-dimensional integral manifold of $\theta$) does not consist of ordinary zeros of $\mathcal{R}^k$ for any $k\ge 2$.

One could, instead, replace $\mathcal{R}^2$ by $\widehat{\mathcal{R}^2}$, the set of all $C^\infty$ functions on the domain that vanish on the zeroes of $\mathcal{R}^2$, and similarly, at each stage, replace $\mathcal{R}^k$ by $\widehat{\mathcal{R}^k}$, the `$C^\infty$ radical' of $\mathcal{R}^k$. There would be a similar theorem to the above for the ordinary zeroes of the resulting limit ideal. However, in most cases, such a computationally intensive procedure is not needed.

The upshot is that one doesn't need to assume anlyticity, per se, but, since arbitrary $C^\infty$ ideals can have almost arbitrarily bad zero loci, and since one can construct $C^\infty$ Pfaffian ideals whose $\mathcal{R}^k$ can be just about any $C^\infty$ ideal, unless one makes some kind of constant rank assumptions, almost anything can happen. What is true is that there are enough constant rank assumptions to cover most cases of interest in the smooth category.