SOME INTEGRABLE FINITE-DIMENSIONAL SYSTEMS AND THEIR CONTINUOUS COUNTERPARTS

The discrete spectral problem of Ablowitz-Ladik is considered in the c ase in which the potential has a finite support of length L. The spect ral transform is explicitly computed and a recurrence relation on the length L for computing it in L algebraic step is given. This spectral transform can be used to generate, via the scattering method, a finite -dimensional version of the dynamical systems associated to the Ablowi tz-Ladik spectral problem. A special case in which the potential can b e constrained to evolve in time on a semiline is proposed. The truncat ed soliton, i.e. the potential obtained by putting to zero the one sol iton outside an interval of length L, is examined in detail. The suffi cient and necessary condition for having a soliton contained in the tr uncated soliton solution is derived. Finally, the continuous counterpa rt of these finite-dimensional systems is considered. The spectral tra nsform is derived via a Riccati equation and the special case of the t runcated soliton is studied.