Some Wadati-Konno-Ichikawa type integrable systems and their constructions

TL;DR

This paper derives a Wadati-Konno-Ichikawa type integrable hierarchy from a matrix spectral problem, explores its bi-Hamiltonian structure, and connects it to the generalized short pulse equation, providing new algebraic constructions.

Contribution

It introduces a new integrable hierarchy based on sl(2, R) and links it to the generalized short pulse equation, with algebraic and Hamiltonian structures analyzed.

Findings

Hierarchy has bi-Hamiltonian structure

Higher grading affine algebraic construction proposed

Generalized short pulse equation arises from negative WKI flow

Abstract

A standard-form Wadati-Konno-Ichikawa(WKI) type integrable hierarchy is derived from a corresponding matrix spectral problem associated with the Lie algebra sl(2, R). Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the trace identity. Then, the higher grading affine algebraic construction of some special cases is proposed. We also show that eneralized short pulse equation arises naturally from the negative WKI flow.