The KdV hierarchy is an infinite collection of commuting isospectral
deformations of the one-dimensional Schroedinger operator, and its
spectral theory is related to the initial-value problem for KdV. For
two classes of initial data, the spectral theory is well understood,
and the initial value problem can be considered solved. A potential
rapidly vanishing at infinity can be reconstructed from its spectral
data by using the inverse spectral transform (IVT), and the spectral
data evolves linearly with KdV. An important class of such potentials
are the Bargmann potentials, or soliton solutions of KdV. The spectrum
of a periodic potential consists of an infinite sequence of bands
separated by spectral gaps. For a dense collection of potentials,
there are only finitely many gaps, the eigenfunction is identified as
a section of a line bundle over a corresponding hyperelliptic curve,
and the KdV evolution is linear on the Jacobian of the curve.
It has long been known that finite-gap potentials should be obtainable
as limits of Bargmann potentials, but a precise description of such a
limit was not known. We reformulate the IVT by studying the
singularities of the eigenfunctions of the corresponding Schroedinger
operator, which gives us some additional freedom for describing the
Bargmann potentials. Replacing the isolated singularities with cuts on
the spectral plane, we obtain a new Riemann—Hilbert problem whose
solutions describe potentials of the Schroedinger operator that are
non-vanishing at infinity, but are not periodic, and can be thought of
as a one-dimensional soliton gas. This RH problem can be studied
numerically, and we also study the spectra of the corresponding
Schoedinger operators.
Joint work with Sergey Dyachenko and Vladimir Zakharov.