'Riemann Equations' in Bidifferential Calculus

TL;DR

This paper explores a unified framework using bidifferential calculus to generate solutions for various integrable equations, including discrete and continuous Riemann equations, and relates them to well-known models like Yang-Mills and NLS.

Contribution

It introduces a universal solution-generating method within bidifferential calculus that links Riemann equations to multiple integrable systems and soliton solutions.

Findings

Unified approach to integrable equations via bidifferential calculus

Solution-generating methods including binary Darboux transformations

Explicit multi-soliton solutions related to Riemann equations

Abstract

We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of `Riemann equations' are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional NLS, KP and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated `Riemann equations'. For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the `Riemann equations' associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes `breaking' multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.