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

Authors

  • V.S. Gerdjikov
    Institute of Mathematics and Informatics Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, 1113, Sofia, Bulgaria
    Sankt-Petersburg State University of Aerospace Instrumentation B. Morskaya, 67A, 190000, St-Petersburg, Russia
    Institute for Advanced Physical Studies, 111 Tsarigradsko chaussee, 1784, Sofia, Bulgaria
    Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, Blvd., 1784, Sofia, Bulgaria

DOI:

https://doi.org/10.13108/2021-13-2-115

Keywords:

mKdV equations, Kac–Moody algebras, Lax operators,

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 leave the origin of the complex $\lambda$-plane partitioning it into equal angles $\pi/h$. To each $l_\nu$ we associate a subalgebra $\mathfrak{g}_\nu$ which is a direct sum of $sl(2)$–subalgebras. In this way, 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 states how to find the minimal sets of scattering data $\mathcal{T}_k$, $k=1,2$, from $T_0$ and $T_1$ related to the rays $l_0$ and $l_1$. 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 $L$.

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Published

20.06.2021