Two-dimensional Toda field equations related to the exceptional algebra $\mathfrak{g}_2 $ : Spectral properties of the lax operators

Abstract

We analyze spectral properties of the Lax operator corresponding to the two-dimensional Toda field equations related to the algebra \(\mathfrak{g}_2 \). We construct two minimal sets of scattering data \(\mathcal{T}_s \), s = 1, 2, understanding the map between the potential and each of the sets \(\mathcal{T}_s \) as a generalized Fourier transformation. We construct explicit recursion operators with special factorization properties.