Solution of the generalized periodic discrete Toda equation. II: Theta function solution. (English) Zbl 1192.37093
The paper deals with the so called hungry periodic discrete (hpd) Toda equations described by
\[
I_n^{t+M} = I_n^t + V_n^t-V_{n-1}^{t+1}\;,\;V_n^{t+1} = {{I_{n+1}^tV_n^t} \over{I_n^{t+M}}}
\]
with the periodic boundary conditions
\[
I_n^t = I^t_{n+N}\;,\;V_n^t = V^t_{n+N}
\]
and \(N\), \(M\) positive integers, \(t\) is the time variable and \(n\) is the position. The following assumption is made
\[
\prod_{n=1}^N I_n^{t+M} = \prod_{n=1}^N I_n^t \neq \prod_{n=1}^N V_n^{t+1} = \prod_{n=1}^N V_n^t
\]
to ensure the existence of a unique solution. This solution is constructed using the auxiliary \(\tau\)-(tau)-function.
Reviewer: Vladimir Răsvan (Craiova)
MSC:
37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
14H42 | Theta functions and curves; Schottky problem |
14H45 | Special algebraic curves and curves of low genus |
39A23 | Periodic solutions of difference equations |