Solitons & Nonlinear Wave Equations
Solitons are stable nonlinear travelling waves that retain their shape and speed
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:
- Infinitely many symmetries and conservation laws, starting with x-translations and mass conservation.
- Exactly solvable by a nonlinear analog of Fourier transforms.
- Determinant formula for generating multi-soliton solutions.
Nonlinear superposition principle, matrix and operator
commutator formulations, Hamiltonian structures and Poisson brackets, zero
curvature connections in differential geometry, and much more.