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

Abstract

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 $\mathbb{C}$ but in the Banach algebra ${\cal L}$ of bounded operators and

2. to derive solutions of the original lattice ($\mathbb{C}$-solutions) by applying a functional $\tau$ to the ${\cal L}$-solutions constructed in 1.

The main advantage of this process is that the derived solution still contains an element of $\cal L$ 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.