D-modules and Algebraic Solutions of PDEs

I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When I was doing a perusal on "A primer of algebraic D-modules by S. C. Coutinho" the justification on the importance of D-modules; they provide an algebraic tool towards the solution of differential equations. This is the story I always hear! Do someone have a reference or more information about D-modules and algebraically solution of PDEs ?.

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It's probably worth fixing the typo in the title. Also, I think there is probably a D-modules tag, which would be appropriate. –  Peter Samuelson Oct 9 '11 at 19:22
May I say that the tag analysis-of-pde is inapropriate, since the purpose of D-modules is to bring an algebraic rather analytic point of view ? –  Denis Serre Oct 9 '11 at 19:37
There's the classic book by Borel et al. –  Mariano Suárez-Alvarez Oct 9 '11 at 21:15
Denis -- D-modules exist on complex analytic manifolds, not just on algebraic ones. Besides, the question explicitly asks for applications of D-modules to PDE's, so I think the PDE tag is appropriate here. –  algori Oct 10 '11 at 4:23
To be honest, the relevant chapter in Coutinho's book is pretty good. Borel et al is fairly heavy artillery, and more oriented towards technical stuff than providing intuition. –  Ketil Tveiten Oct 10 '11 at 8:05

Perhaps you are looking for something deeper, but right there at the beginning of Hotta, Takeuchi, and Tanisaki's book on D-mods in the introduction is the connection to Linear PDEs.

I quote:

Therefore, systems of linear partial differential equations can be identified with the D-modules having some finite presentations like (0.0.3), and the purpose of the theory of linear PDEs is to study the solution space HomD(M, O). Since the space HomD(M, O) does not depend on the concrete descriptions (0.0.2) and (0.0.3) of M (it depends only on the D-linear isomorphism class of M), we can study these analytical problems through left D-modules admitting finite presentations. In the language of categories, the theory of linear PDEs is nothing but the investigation of the contravariant functor HomD(•, O) from the category M(D) of D-modules admitting finite presentations to the category M(C) of C-modules.

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Yes, that is certainly very nice. But apart from the Bernstein application mentioned in Paul Siegel's answer I haven't seen any other result about PDE's proved by the theory of D-modules. –  Vít Tuček Nov 7 '12 at 19:46

My guess is that this is a reference to Bernshtein's proof using the theory of D-modules that every constant coefficent partial differential operator $D$ has a fundamental solution, i.e. there is a distributional solution to the PDE $Du = \delta$ where $\delta$ is the Dirac distribution. This was an important theorem in analysis due to Malgrange and Ehrenpreis in the 1950's, and I think it came as a bit of a surprise that it can be done purely algebraically - all of the analytic input is encapsulated by a few basic facts about distributions. Everyone knows that the fundamental theorem of algebra is proved in analysis class; maybe this means the fundamental theorem of analysis will one day be proved in algebra class!

Here is a link to Bernshtein's paper: http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/bernstein-mod-dif-FAN.pdf. This was written in the 70's, and I think since then the argument has been cleaned up and made even more algebraic.

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Nice paper indeed –  Ongaro Nyang' Oct 15 '11 at 5:11
If I am not mistaken, today there is a short proof of this result, since one can write down explicit Green functions for the general case. –  Vít Tuček Nov 7 '12 at 19:45