Solitons & Nonlinear Wave Equations


Solitons are stable nonlinear travelling waves that retain their shape and speed in interactions. First discovered empirically in the 1800's from observations of waves made by canal boats, solitons nowadays appear in numerous interesting applications and physical phenomena such as: tsunamis, optical fiber signals, plasmas, atmospheric waves, vortex filaments, superconductivity, and gravitational fields with cylindrical symmetry.

A basic example of a soliton equation is the nonlinear PDE

ut + uux + uxxx = 0

which was first written down by Korteweg & de Vries in 1895 as a model of shallow water waves. Some of its remarkable and beautiful features include: