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Soliton solutions of the Toda hierarchy on quasi-periodic backgrounds revisited. (English) Zbl 1166.37028

The authors after introducing the Toda hierarchy and recalling necessary facts on algebra-geometric quasi-periodic finite-gap solutions briefly review the single and double commutation methods. The phase shift (in the Jacobian variety) caused by inserting one eigenvalue for both methods is computed. The authors also review the direct scattering theory for Jacobi operators with different (quasi)-periodic asymptotics in the same isospectral class. As main result they give a complete description of the effect of the double commutation method on the scattering data. In addition they provide some detailed asymptotic formulas for the Jost functions \(\psi_{\pm}(z,n)\) (which are normalized as \(n\to\pm\infty\)) at the other side, that is, as \(n\to\mp\infty\). Finally they establishe the inverse scattering transform for this setting. The main results here are the time dependence of both the scattering data and the kernel of the Gel’fand-Levitan-Marchenko equation.

MSC:

37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
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