SYMMETRY AND REDUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS: PEAKON AND THE TODA CHAIN SYSTEMS

We are analyzing several types of dynamical systems which are both integrable and important for physical applications. The first type are the so-called peakon systems that appear in the singular solutions of the Camassa-Holm equation describing special types of water waves. The second type are Toda chain systems, that describe molecule interactions. Their complexifications model soliton interactions in the adiabatic approximation. We analyze the algebraic aspects of the Toda chains and describe their real Hamiltonian forms.