Title: Introduction to Solitons and the Inverse Scattering Transform

Solitons are localized solutions of nonlinear evolution equations such
as the Korteweg-deVires (KdV) Equation, $\partial_t u + 6 u \partial_x
u + \partial_x3 u = 0$, and the Nonlinear Schr\"odinger (NLS)
Equation, $i\partial_t \psi +\partial_x2 \psi + 2 |\psi|^2 \psi =
0$. The KdV Equation is the limit of long water waves in a shallow
narrow channel, while the NLS is the limit of a sufficiently short,
intense optical pulse traveling down an optical fiber in a
communication network. These equations are key physical equations for
a wide variety of physical phenomena and are what are called
"Integrable Equations". This lecture will describe solitons, some of
the mathematics of solitons, and some of their applications.