2Partial Differential Equations
II Integrable Systems
2 Partial Differential Equations
For the remainder of the course, we are going to look at PDE’s. We can view
these as infinite-dimensional analogues of ODE’s. So what do we expect for
integrable PDE’s? Recall that If an 2
n
-dimensional ODE is integrable, then we
n
first integrals. Since PDE’s are infinite-dimensional, and half of infinity is still
infinity, we would expect to have infinitely many first integrals. Similar to the
case of integrable ODE’s, we would also expect that there will be some magic
transformation that allows us to write down the solution with ease, even if the
initial problem looks very complicated.
These are all true, but our journey will be less straightforward. To begin with,
we will not define what integrability means, because it is a rather complicated
issue. We will go through one method of “integrating up” a PDE in detail,
known as the inverse scattering transform, and we will apply it to a particular
equation. Unfortunately, the way we apply the inverse scattering transform to a
PDE is not obvious, and here we will have to do it through “divine inspiration”.
Before we get to the inverse scattering transform, we first look at a few
examples of PDEs.