The complex Toda chains and the simple Lie algebras: II. Explicit solutions and asymptotic behaviour

We propose a compact and explicit expression for the solutions of the complex Toda chains related to the classical series of simple Lie algebras g. The solutions are parametrized by a minimal set of scattering data for the corresponding Lax matrix. They are expressed as sums over the weight systems of the fundamental representations of g and are explicitly covariant under the corresponding Weyl group action. In deriving these results we start from the Moser formula for the A_r series and obtain the results for the other classical series of Lie algebras by imposing appropriate involutions on the scattering data. Thus we also show how Moser's solution goes into that of Olshanetsky and Perelomov. The results for the large-time asymptotics of the A_r-CTC solutions are extended to the other classical series B_r-D_r. We exhibit also some 'irregular' solutions for the D_2n+1 algebras whose asymptotic regimes at t → ±∞ are qualitatively different. Interesting examples of bounded and periodic solutions are presented and the relations between the solutions for the algebras D₄, B₃ and G₂ are analysed.