A comprehensive reference, with many worked-out examples, is Contact Geometry and Nonlinear Differential Equations (2007). Here is a review.

Methods from contact and symplectic geometry can be used to solve
highly non-trivial nonlinear partial and ordinary differential
equations without resorting to approximate numerical methods or
algebraic computing software. This book explains how it's done. It
combines the clarity and accessibility of an advanced textbook with
the completeness of an encyclopedia. The basic ideas that Lie and
Cartan developed at the end of the nineteenth century to transform
solving a differential equation into a problem in geometry or algebra
are here reworked in a novel and modern way. Differential equations
are considered as a part of contact and symplectic geometry, so that
all the machinery of Hodge-deRham calculus can be applied. In this way
a wide class of equations can be tackled, including quasi-linear
equations and Monge-Ampere equations (which play an important role in
modern theoretical physics and meteorology).

The concrete application that might be what you are looking for is to the Knizhnik–Zamolodchikov equation (which models the propagation of sound beams in a non-linear medium).