Solitons & Nonlinear Waves

Explain integrable nonlinear evolution equations, solution methods, initial data and wave phenomena.

Begin with KdV Lax compatibility and one-soliton reconstruction. It connects the nonlinear field equation to a differential operator, computes an explicit reflectionless spectral problem, and solves a one-bound-state inverse problem. The KdV learning sequence provides worked practice, and the model record fixes the sign and boundary conventions.

Continue with KdV two-soliton scattering: derive the finite determinant, verify the nonlinear equation and calculate the signed asymptotic position shifts. The collision lesson and numerical laboratory turn those results into calculations you can check.

The remaining map places these examples among initial-value problems, solution methods and wave phenomena planned for the volume. The one- and two-pole calculations do not establish the general theory of arbitrary scattering data or long-time asymptotics.