Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies

TL;DR

This paper introduces a generalized Riemann-Hilbert-Birkhoff decomposition that broadens integrable hierarchy models by including higher powers and non-zero backgrounds, unifying various known models and enabling new integrable systems.

Contribution

It develops a new formalism for integrable hierarchies incorporating adjustable parameters for grading, background solutions, and gauge choices, extending the scope of existing models.

Findings

Constructed new integrable models with higher grading and non-zero backgrounds.

Unified framework encompassing known hierarchies like mKdV and AKNS.

Demonstrated broad applicability of the extended formalism.

Abstract

We propose a generalized Riemann-Hilbert-Birkhoff decomposition that expands the standard integrable hierarchy formalism in two fundamental ways: it allows for integer powers of Lax matrix components in the flow equations to be increased as compared to conventional models, and it incorporates constant non-zero vacuum (background) solutions. Two additional parameters control these features. The first one defines the grade of a semisimple element that underpins the algebraic construction of the hierarchy, where a grade-one semi-simple element recovers known hierarchies such as mKdV and AKNS. The second parameter distinguishes between zero and non-zero constant background (vacuum) configurations. Additionally, we introduce a third parameter associated with an ambiguity in the definition of the grade-zero component of the dressing matrices. While not affecting the decomposition itself, this parameter classifies different gauge realizations of the integrable equations (like for example, Kaup-Newell, Gerdjikov-Ivanov, Chen-Lee-Liu models). For various values of these parameters, we construct and analyze corresponding integrable models in a unified universal manner demonstrating the broad applicability and generative power of the extended formalism.