How Many Types of Soliton Solutions do We Know?

Abstract

We discuss several ways of how one could classify the various types of soliton solutions related to NLEE that are solvable with the generalized $n \times n$ Zakharov–Shabat system. In doing so we make use of the fundamental analytic solutions, the dressing procedure and other tools characteristic for the inverse scattering method. We propose to relate to each subalgebra $\mathfrak{sl}(p), 2 \leq p \leq n$ of $\mathfrak{sl}(n)$, a type of one-soliton solutions which have $p − 1$ internal degrees of freedom.