Motivation
I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a number of important achievements in the study of PDEs, suffice it to mention the construction of finite-gap solutions to integrable PDEs (see e.g. this book) and the geometric approach to PDEs developed by A.M. Vinogradov et al. which revolves around the concept of diffiety (the word itself was merged from "differential" and "variety") and which the authors themselves consider, at least to some extent, as a "translation" of ideas from algebraic geometry and commutative algebra to the realm of PDEs, see e.g. these two books.
On the other hand, it appears (as far as my googling skills allow me to tell) that the other way around, i.e., in the applications of nonlinear PDEs in algebraic geometry, the interaction is at least somewhat less intense. I was able to come up with basically just two things: the Novikov conjecture (proved by Shiota)
on the relation of the KP equation to the Schottky problem and the applications of the Monge-Ampere equations in Kahler geometry.
Question
Which are the other applications of the nonlinear PDEs in (broadly understood) algebraic geometry? In other words, which nonlinear PDEs are of interest to algebraic geometers and why?
EDIT: It should be obvious, but to play it safe I would like to spell this out loud and clear: please feel free to share not only the already known cases where PDEs have helped the algebraic geometers but also the more open-problem-type cases where, say, there is a PDE that could be of use in algebraic geometry but some crucial bits of information about this PDE (for instance, about the existence of solution(s) with the desired properties) are still missing.
Best Answer
Strictly speaking, this is not meant as an answer to the question---it's more like a suggestion that you might find it interesting to also ask a related, but different, question.
I would say the most interesting feature of many interactions between nonlinear (especially integrable) PDE and algebraic geometry, from the point of view of an algebraic geometer, is that they show us new structures that nobody knew existed classically.
One instance of this is the story that you're referring to relating the conformal field theory of free fermions, KP, and the geometry of moduli of bundles, mediated (geometrically) by the Sato Grassmannian. This is what leads to Shiota's characterization of the Schottky locus. That's well before my time, but I wouldn't have guessed that most classical algebraic geometers thought of this as a very big advance on the Schottky problem at the time: if you'd like equations for the Schottky locus, or even some nice algebro-geometric description of it, you probably aren't satisfied with what Shiota tells you (I'm a huge fan of this story, so my remark isn't meant to convey my own opinion of this work, rather to guess at what some others might have thought). Algebraic geometers were interested in moduli of curves, bundles, etc. long before this story appeared, but a whole new world emerged from it.
In a different direction, the discovery of integrability in topological string theory (as formulated, for example, in Witten's conjecture, later Kontsevich's theorem) shows us striking new structure, again governed by integrable PDE (KdV, n-KdV) in the intersection theory on moduli spaces of curves (and later 2D Toda in the Gromov-Witten theory of curves). Again, I would argue (as someone who, admittedly, is incredibly far from expert in the subject) that the most interesting part of the story is the amazingly rich structure (involving integrable PDE, matrix models, and intersection theory on moduli spaces) that was revealed by these discoveries.
EDIT: What I wrote initially takes a slightly misleading tone about applications back to classical AG: for example, Krichever's stunning work on the trisecant conjecture, which grows directly out of integrable PDE, is surely something of classical interest!