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I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a potential function (which was addressed in a separate question). In that question, Robert Bryant noted that the characteristic variety consists of three points. Concretely, I'm wondering if this means that there are at most three solutions to this system of equations (perhaps modulo some affine transformations).

More generally, does the number of points in the characteristic variety bound the number of real analytic solutions for a given system of PDEs?

My understanding is that the characteristic variety gives something like the formal power series solutions to the system, so there shouldn't be more real analytic solutions than that. However, my knowledge of $D$-modules is really lacking, so I was wondering if anyone could either correct me on this or else point me to a good reference to learn more. I've been trying to read through Bryant's Exterior Differential Systems, and I apologize if this question is obvious for those who understand the theory.

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    $\begingroup$ You might also want to look at the book Cartan for Beginners by Ivey and Landsberg. In general, the family of local solutions is infinite dimensional and one counts the number of solutions by parameterizing the space of solutions by how many functions (and how many inputs each function takes) uniquely determine a local solution. Also, it is impossible to do this count effectively for anything but a system of PDEs that is involutive (or "in involution"). $\endgroup$ – Deane Yang Apr 5 '19 at 14:49
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    $\begingroup$ Also, there is a finite dimensional family of local solutions only if the system can be solved using the Frobenius theorem. In that case, the characteristic variety is empty. $\endgroup$ – Deane Yang Apr 5 '19 at 14:53
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    $\begingroup$ @AliTaghavi, in general there is no connection without more information. Again, keep in mind that the codimension will also be infinite, so one has to count using functions, rather than numbers. $\endgroup$ – Deane Yang Apr 5 '19 at 16:17
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    $\begingroup$ @AliTaghavi, what's being discussed here is not index theory. Index theory is about global solutions to a linear elliptic PDE, where the kernel and cokernel are finite dimensional. The discussion here is about local solutions to nonlinear PDEs that are not necessarily elliptic. $\endgroup$ – Deane Yang Apr 6 '19 at 19:32
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    $\begingroup$ @AliTaghavi, as for counting using functions, here is the basic example: If you have a real analytic system of PDEs of the form $$ \partial_tu + A^i\partial_i u = f, $$ where $1 \le i \le n$, and $u = (u^1, \dots, u^m)$, then, by the Cauchy-Kovalevski theorem, given any real analytic $\mathbb{R}^m$-valued function $u_0(x^1, \dots, x^n)$, there exists a unique local solution $u$ to the system of PDEs such that $u(0,x) = u_0(x)$. Therefore, you can say that local solutions depend on $m$ functions of $n$ variables, in the sense that each $u_0$ uniquely determines a solution $u$. $\endgroup$ – Deane Yang Apr 6 '19 at 19:39
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The wave equation in the plane is $\partial^2_x-\partial^2_y=(\partial_x+\partial_y)(\partial_x-\partial_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.

Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently smooth solution, by a theorem of Ofer Gabber. For the wave equation, this is the foliation by the two directions spanned by the vector fields $\partial_x+\partial_y, \partial_x-\partial_y$.

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    $\begingroup$ Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution? $\endgroup$ – Gabe K Apr 5 '19 at 13:57
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    $\begingroup$ The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity. $\endgroup$ – Ben McKay Apr 5 '19 at 14:55
  • $\begingroup$ @BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE? $\endgroup$ – Ali Taghavi Apr 5 '19 at 15:21
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    $\begingroup$ @AliTaghavi: A precise statement is difficult to make, as it requires a long discussion. There is a precise statement on p. 195 of Bryant et. al., Exterior Differential Systems, theorem 3.20, but it uses terminology that would require extensive discussion to define here. They also give the reference to Gabber's paper: O. Gabber, The Integrability of the characteristic variety, Amer. J. Math., 103, 1981, .p 445-468. $\endgroup$ – Ben McKay Apr 6 '19 at 18:30
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    $\begingroup$ @AliTaghavi, please keep in mind here that both the dimension of the space of solutions and the codimension of the image are infinite dimensional. And the adjoint PDE plays no role here. You're confusing the situation here with the global theory of elliptic PDEs. $\endgroup$ – Deane Yang Apr 6 '19 at 19:29

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