Solution of the Semi-Infinite Toda Lattice for Unbounded Sequences |
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Authors: | Ifantis E K Vlachou K N |
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Institution: | (1) Department of Mathematics, University of Patras, 26500 Patras, Greece |
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Abstract: | 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(
n
2(t) –
n–1
2(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/dt
2 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. |
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Keywords: | semi-infinite Toda lattice Jacobi matrices continued fractions semi-infinite differential-difference Darboux equation |
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