On the mKdV equations related to the Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$

Abstract

We outline the derivation of the mKdV equations related to the Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$. First we formulate their Lax representations and provide details how they can be obtained from generic Lax operators related to the algebra $sl(6)$ by applying proper Mikhailov type reduction groups ${\mathbb{Z}}_h$. Here $h$ is the Coxeter number of the relevant Kac-Moody algebra. Next we adapt Shabat's method for constructing the fundamental analytic solutions of the Lax operators $L$. Thus we are able to reduce the direct and inverse spectral problems for $L$ to Riemann-Hilbert problems (RHP) on the union of $2h$ rays $l_{\nu }$. They start from the origin of the complex $\lambda$-plane and close equal angles $\pi /h$. To each $l_{\nu }$ we associate a subalgebra ${\mathfrak{g}}_{\nu }$ which is a direct sum of $sl(2)$-subalgebras. Thus to each regular solution of the RHP we can associate scattering data of $L$ consisting of scattering matrices $T_{\nu }\in {\mathcal{G}}_{\nu }$ and their Gauss decompositions. The main result of the paper is to extract from $T_0$ and $T_1$ related to the rays $l_0$ and $l_1$ the minimal sets of scattering data ${\mathcal{T}}_k$, $k=1,2$. We prove that each of the minimal sets ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ allows one to reconstruct both the scattering matrices $T_{\nu }$, $\nu =0,1,\dots 2h$ and the corresponding potentials of the Lax operators $L$.

arXiv comment21 pages, 2 Figures 3 Tables