TITLE: Which nonlinear PDEs are of interest to algebraic geometers and why?
QUESTION [50 upvotes]: 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.

REPLY [4 votes]: I give two reason to show that nonlinear PDE  is of interest to algebraic geometers.
First: Monge-Ampere equation is nonlinear PDE and  is of interest to algebraic geometers, Here is one of examples
In fact if $X$ be a projective variety and canonical Ring $$\bigoplus_{m\geq 0}H^0(X,K_X^{m})$$
is finitely generated and Kodaira dimension is positive then 
$$\pi:X\to X_{can}=proj(\bigoplus_{m\geq 0}H^0(X,K_X^{m}))$$
gives a canonical metric which is twisted by Weil-Petersson metric on $X$ via
$$Ric(g_{can})=-g_{can}+g_{WP}$$
Which $g_{WP}$ corresponds to moduli space of Calabi-Yau fibers.
In fact we can write twisted Kahler Einstein metric in the form of Monge-Ampere equation by using Yau-Vafa semi Ricci flat metric.

Second: is the concept of surgery. We believe that surgery in PDE gives
  surgery in algebraic geometry and vise versa. Surgery in algebraic
  geometry here means flips and flops. 
Surgery in PDE<===>Surgery in Geometry<===> Surgery in Algebraic
  geometry

In fact in the proof of Poincare conjecture Perelman used Surgery in PDE<===>Surgery in Geometry , he took PDE surgery by using Ricci flow 
and for proof of finding canonical metrics for varieties which do not have definite first chern class (long standing conjecture) we need to use 
Surgery in PDE<===> Surgery in Algebraic Geometry
albiet Surgery in Algebraic Geometry may not exists in general.