Solution of the Cauchy problem for a Toda chain with non-stabilizing initial data (Q2768853)

The author considers the problem of finding exact solutions to infinite Toda lattice described by the system NEWLINE\[NEWLINE\frac{d^2x_n}{dt^2}=e^{x_{n+1}-x_n}-e^{x_n-x_{n-1}}, n\in \mathbb Z.NEWLINE\]NEWLINE Any sequences \(\{v_n\}, \{w_n\}\) associated with initial conditions \(x_n(0)=v_n, \dot x_n(0)=w_n\) give rise to a Jacobi matrix \(J(\{a_n\},\{b_n\})\) with elements \(a_n=w_n, b_{n-1}=\exp\frac{v_n-v_{n-1}}{2}\). Basing on ideas of inverse spectral method the exact solutions are obtained by a special choice of the spectral data for \(J\). Namely, the author requires that the spectrum of an unknown Jacobi matrix contains an isolated interval of absolutely continuous spectrum of multiplicity 2. A reconstruction technique for matrices with such spectral data is developed. The novelty is that the author's approach allows to obtain Jacobi matrices with bounded elements \(a_n, b_n\) that generally are not stabilized as \(n\to \pm \infty \).