The KdV Zero-Dispersion KdV Limit
and Densities of Dirichlet Spectra
Abstract
This talk introduces the problem of the zero-dispersion limit for
the Korteweg-de Vries (KdV) equation as a special representative of
problems such as the semiclassical limit of nonlinear fields (quantum
or classical) and the continuum limit of lattice dynamics. A history
of the problem will be given and samples of limiting phenomenology
will be shown. Theory shows that the conserved densities and fluxes
of the whole KdV hierarchy have limits that are characterized in terms
of the solution of a maximization problem. The maximizer is shown to
be a limiting density of half-line Dirichlet spectra of the associated
\Schrodinger operator. This enables one both to strengthen the limits
asserted for the conserved densities and fluxes, and to establish the
limit of the associated Weyl functions.
