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

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

2005-05-01

arxiv.org/abs/nlin/0505003v2

Phyl. Trans. of Royal Society A (2008), v.366(1867), March 28, 923-945

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Major changes. 23 pages; LaTeX

Scientific paper

In the example of the Schr\"odinger/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 discussed also.

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