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What does integrability of finite-gap or soliton potentials mean?

Yu. V. Brezhnev
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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 $Ψ$-function, trace formula and the Jacobi problem provide a complete solution. The $Θ$-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 discussed also.

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Yu. V. Brezhnev. 2006-04-17. What does integrability of finite-gap or soliton potentials mean?. https://arxiv.org/abs/nlin/0505003

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