# Riemann-Hilbert problem, integrability and reductions

## TL;DR

This paper explores integrable models with dihedral reduction groups, demonstrating their solutions via Riemann-Hilbert problems and introducing new nonlinear evolution equations with dihedral symmetries.

## Contribution

It establishes the connection between integrable models with dihedral reduction groups and Riemann-Hilbert problems, and presents two new nonlinear evolution equations with such symmetries.

## Key findings

- Solutions are equivalent to solving Riemann-Hilbert problems with symmetry-dependent contours.
- Two new nonlinear evolution equations with dihedral symmetries are introduced.
- The Lax representation facilitates the analysis of these integrable models.

## Abstract

The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R \simeq \mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $\mathbb{D}_h$ symmetries are presented.