Solution of the Semi-Infinite Toda Lattice for Unbounded Sequences

The semi-infinite Toda lattice is the system of differential equations d_n(t)/dt = _n(t)(b_n+1(t) – b_n(t)), db_n(t)/dt = 2(_n²(t) – _n–1²(t)), n = 1, 2, ..., t > 0. The solution of this system (if it exists) is a pair of real sequences _n(t), b_n(t) which satisfy the conditions _n(0) = _n,, b_n(0) = b_n, where _n > 0 and b_n are given sequences of real numbers. It is well known that the system has a unique solution provided that both sequences _n and b_n are bounded. When at least one of the known sequences _n and b_n is unbounded, many difficulties arise and, to the best of our knowledge, there are few results concerning the solution of the system. In this letter we find a class of unbounded sequences _n and b_n such that the system has a unique solution. The results are illustrated with a typical example where the sequences _i(t), b_i(t), i = 1, 2, ... can be exactly determined. The connection of the Toda lattice with the semi-infinite differential-difference equation d²/dt² log h_n = h_n+1 + h_n–1 – 2h_n, n = 1, 2, ... is also discussed and the above results are translated to analogous results for the last equation.