The Korteweg–de Vries Equation

The Korteweg–de Vries (KdV) equation, which is a nonlinear partial differential equation of the third order, is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. This chapter aims to study the KdV equation and the inverse scattering method (based on Schrodinger and Gelfand–Levitan equations) used to solve it. The KdV equation preserves mass, momentum, energy and many other quantities. The nonlinear KdV equation is transformed into the linear Gelfand–Levitan equation. The spectrum of the Schrodinger operator is invariant by the Hamiltonian flow defined by the KdV equation. The isospectral sets related to invariant manifolds defined by putting these invariants equal to generic constants are compact, connected and infinite-dimensional tori.