Nonlinear evolution equations related to Kac-Moody algebras $A_r^{(1)}$: spectral aspects

Abstract

We analyze three types of integrable nonlinear evolution equations (NLEE) related to the Kac-Moody algebras $A_r^{(1)}$. These are $\mathbb{Z}_h$-reduced derivative NLS equations (DNLS), multicomponent mKdV equations and 2-dimensional Toda field theories (2dTFT). We outline the basic tools of this analysis: i) the gradings of the simple Lie algebras using their Coxeter automorphisms; ii) the construction of the relevant Lax representations; and iii) the spectral properties of the Lax operators and their reduction to Riemann-Hilbert problems. We also formulate the minimal set of scattering data which allow one to recover the asymptotics of the fundamental analytic solutions to $L$ and its potential.

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