FROM THE NON-ABELIAN TO THE SCALAR TWO-DIMENSIONAL TODA LATTICE

We extend a solution method used for the one-dimensional Toda lattice in [1], [2] to the two-dimensional Toda lattice. The idea is 1. to study the lattice not with values in C but in the Banach algebra L of bounded 2. operators and 3. to derive solutions of the original lattice (C-solutions) by applying a functional (tau) to the 4. L-solutions constructed in 1. The main advantage of this process is that the derived solution still contains an element of L as parameter that may be chosen arbitrarily. Therefore, plugging in different types of operators, we can systematically construct a huge variety of solutions. In the second part we focus on applications. We start by rederiving line-solitons and briefly discuss discrete resonance phenomena. Moreover, we are able to find conditions under which it is possible to superpose even countably many line-solitons.