Description:
"Solitons" (waves that behave like particles) were initially discovered in efforts to model physical phenomena and have since become valuable in various fields of science and engineering. This mini-course will not focus on those applications but will instead explore the fascinating mathematical structure of "soliton equations," which allow these special nonlinear partial differential equations to be solved explicitly. With the help of the mathematical software package Mathematica, students will discover how soliton equations can be understood as algebro-geometric objects "disguised" as differential equations.

Topics covered in the mini-course:

• Soliton equations: KdV, KP, ncKP, nonlinear Schrödinger, Toda lattice, sine-Gordon, the KP Hierarchy

• Traveling waves and elliptic curves

• Lax pairs and isospectrality

• Tau-functions and Hirota bilinear form

• Ordinary differential operators and pseudo-differential operators

• Grassmannian manifolds as the phase space of KP solutions