The complex Toda chains and the simple Lie algebras - Solutions and large time asymptotics

The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends related to the classical series of simple Lie algebras are classified. It is shown that the CTC models have much richer variety of asymptotic regimes than the real Toda chain (RTC). Besides asymptotically free propagation (the only possible regime for the RTC), CTC allows bound-state regimes, various intermediate regimes when one (or several) group(s) of particles form bound state(s), singular and degenerate solutions. These results can be used, for example, in describing the N-soliton train interactions of the nonlinear Schrödinger equation. Explicit expressions for the solutions in terms of minimal sets of scattering data are proposed for all classical series B_r-D_r.