This talk presents a study on a famous example of a nonlinear wave equation, the Korteweg-de Vries equation (KdV). First we will consider the linearized KdV equation and describe the long time asymptotics of the solutions by using the method of stationary phase. Then we will study the nonlinear case. Finally, we will solve the Cauchy problem for the KdV equation with rapidly decaying initial data by the inverse scattering method, which represents a nonlinear analog of the Fourier transform to solve linear partial differential equations.
If time permits the Lax approach for the KdV equation will be presented.