I found a solid background in PDE, together with some physics, to be a useful entry point to Olver's nice book. There's the 'Lectures on Partial Differential Equations' by V.I.Arnold which is fun to read alongside, if not before. Any solid book on mathematical methods in classical mechanics and quantum mechanics should prove useful as well. Finally, I agree with Deane- the most efficient path is to start reading the book, and learn the material you need as you proceed.
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!
Best Answer
A recent book more or less of the kind you want is
V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter, Berlin, 2011.
Probabilistic methods for nonlinear partial differential equations are developed in a series of papers by Yana Belopolskaya, for example
Y. Belopolskaya and W. Woyczynski, Generalized solutions of nonlinear parabolic equations and diffusion processes. Acta Appl. Math. 96, No. 1-3, 55-69 (2007).