Though originating in the study of linear partial differential equations, microlocal analysis has become an invaluable tool in the study of nonlinear pde. Of particular importance has been the application of paraproducts/paradifferential operators and related tools developed by Bony, Coifman, Meyer and others. The theory of paradifferential operators fits nicely within the framework of microlocal analysis pioneered by Calderón, Zygmund, Hörmander, Kohn, Nirenberg, etc. However, there is another rather distinct school of (or approach to) microlocal analysis.
This second school, developed by Sato, Kashiwara, Kawai and others, makes liberal use of tools from algebra as well as the theory of sheaves (hence algebraic microlocal analysis). Additionally, analytic functions (as opposed to $C^\infty$ functions) play a much more prominent role in algebraic microlocal analysis. As far as I'm aware (and this isn't terribly far as concerns algebraic microlocal analysis), one can obtain very similar theories of linear pde using either microlocal analysis or algebraic microlocal analysis (though, of course, some differences surely exist). If I'm wrong about this, I'd certainly be interested to hear more. However, I'm not familiar with any work applying algebraic microlocal analysis to the study of nonlinear pde. This leads to my question(s). I have heard that there is at least some work applying algebraic microlocal analysis to nonlinear pde. Could anyone point me towards some references? Additionally, I'd love to hear any thoughts or intuition anyone has about the ability to apply (or the limitations of the applicability of) algebraic microlocal analysis to the study of nonlinear pde.
Edit: So, I was able to locate a couple references that are somewhat relevant to this question. In a set of lecture notes [1] by Pierre Schapira, he indicates that chiral algebras can be used to construct a theory of nonlinear pde in the spirit of $\mathcal{D}$-module theory. He gives [2] and [3] as references, which he says sketch the aforementioned theory.
[1] P. Schapira. An Introduction to $\mathcal{D}$-Modules. Lecture Notes, https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf, 2017.
[2] A. Beilinson and V. Drinfeld. Chiral Algebras. Amer. Math. Soc. Coll. Publ., 51, Providence, RI, 2004.
[3] M. Kapranov and Y. Manin. Modules and Morita Theorem for Operads. Amer. J. Math., 123:811-838, 2001.
