Geometric aspects of Miura transformations

TL;DR

This paper explores the geometric properties of Miura transformations, showing how they connect different integrable systems and geometric structures, and constructing generalized transformations in algebraic and geometric contexts.

Contribution

It introduces generalized Miura transformations within geometric and algebraic frameworks, linking integrable curve flows and moving frames across different geometries.

Findings

Miura transformations relate integrable curve flows in various geometries.

Generalized transformations are constructed in algebraic and geometric settings.

Transformations induce transitions between different moving frames.

Abstract

The Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi-Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV-type hierarchies to the KdV-type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Other geometric formulations are also investigated.