title: What does integrability of finite-gap or soliton potentials mean?
abstract: In the example of the Schrödinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field-new explicit Psi-function, trace formula and the Jacobi problem-provide a complete solution. The Theta-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are also discussed.
authors.affiliation: Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA. brezhnev@mail.ru
authors.forename: Yurii V
authors.initials: YV
authors.lastname: Brezhnev