Whitham Deformations of the Korteweg-de Vries Equation

Abstract

In this thesis, a new independent approach to obtain a dispersionless version of the KdV hierarchy is presented and used to describe a class of solutions that are accessible by the generalized hodograph method.

The Korteweg-de Vries (KdV) equation is a dispersive and non-linear partial differential equation (PDE) in one spatial dimension and time. It originates from the observation and description of solitary shallow-water waves and later became famous in the context of the Fermi–Pasta–Ulam–Tsingou problem. An aspect that has drawn a lot of attention is that the KdV equation possesses infinitely many conserved quantities and corresponding symmetries – giving rise to the structure of an integrable hierarchy.

The focus of the thesis is on a dispersionless version of the integrable KdV hierarchy which is obtained by applying adiabatic theory from classical mechanics. The resulting KdV Whitham hierarchy is again integrable, but in a more general way. While the dispersive equation admits stable solitary waves as solutions, on the dispersionless side breaking waves occur. The theoretical description of the dispersionless hierarchy yields algebraic-geometric and differential-geometric structures which are defined on the space of the conserved quantities of the dispersive hierarchy. In particular, Euler–Poisson–Darboux equations, known from classical differential geometry, can be used to characterize solutions of the KdV Whitham hierarchy.