Korteweg – de Vries equation: solitons and undular bores

The Korteweg – de Vries (KdV) equation is a fundamental mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is known to possess a number of families of exact analytic solutions. Two of them: solitons and nonlinear periodic travelling waves – are of particular interest from the viewpoint of fluid dynamics applications as they occur as typical asymptotic outcomes in a broad class of initial/boundary-value problems. Two different major approaches have been developed in the last four decades to deal with the problems involving solitons and nonlinear periodic waves: inverse scattering transform and the Whitham method of slow modulations. We review these methods and show relations between them. Emphasis is made on solving the KdV equation with large-scale initial data. In this case, the long-time evolution of an initial perturbation leads to formation of an expanding undular bore, a modulated travelling wave connecting two different non-oscillating flows. Another problem considered is the propagation of a soliton through a variable environment in the framework of the variable-coefficient KdV equation. If the background environment varies slowly, the solitary wave deforms adiabatically and an extended small-amplitude trailing shelf is generated. On a long-time scale, the trailing shelf evolves, via an intermediate stage of an undular bore, into a secondary soliton train.