^{1,2,a)}, F. Müller-Hoissen

^{1,a)}and N. Stoilov

^{1,2,a)}

### 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 Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, 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.

O.C. would like to thank the Mathematical Institute of the University of Göttingen for hospitality in November 2013 — April 2014, when part of this work has been carried out. Special thanks go to Dorothea Bahns. Since October 2014, O.C. has been supported via an Alexander von Humboldt fellowship for postdoctoral researchers. N.S. has been supported by the Marie Curie Actions Intra-European fellowship HYDRON (FP7-PEOPLE-2012-IEF, Project No. 332136). The authors are also grateful to Aristophanes Dimakis, for sharing some of his insights, and to Eugene Ferapontov and Maxim Pavlov for some very motivating discussions.

I. INTRODUCTION II. THE LINEARIZATION METHOD III. BINARY DARBOUX TRANSFORMATIONS IN BIDIFFERENTIAL CALCULUS A. Darboux transformations for “Riemann equations” IV. MATRIX RIEMANN EQUATIONS AND INTEGRABLE DISCRETIZATIONS A. Riemann equation B. Semi-discrete Riemann equation 1. Cole-Hopf transformation 2. Darboux transformations C. Discrete Riemann equation 1. Cole-Hopf transformation 2. Darboux transformations D. Hierarchies 1. Riemann hierarchy 2. Semi-discrete Riemann hierarchy 3. Discrete Riemann hierarchy V. SOME INTEGRABLE EQUATIONS ASSOCIATED WITH RIEMANN EQUATIONS OR THEIR INTEGRABLE DISCRETIZATIONS A. Self-dual Yang-Mills equation 1. The sdYM Riemann system 2. Breaking multi-soliton-type solutions of the sdYM equation 3. A non-autonomous chiral model in three dimensions B. A matrix version of the two-dimensional Toda lattice 1. Cole-Hopf transformation for the “Riemann system” 2. Darboux Transformations for the “Riemann system” C. A matrix version of Hirota’s bilinear difference equation 1. First version 2. Second version VI. (2+1)-dimensional matrix nonlinear Schrödinger system A. “Riemann system” associated with the (2+1)-dimensional matrix NLS system 1. “Riemann system” associated with the scalar (2+1)-dimensional NLS equation 2. The linearization method B. Multi-soliton solutions of the (2+1)-dimensional NLS equation, parametrized by solutions of Riemann equations VII. MATRIX BURGERS AND KP EQUATIONS A. Burgers equation 1. Cole-Hopf transformation 2. Darboux transformations B. The second equation of the Burgers hierarchy C. KP VIII. MATRIX DAVEY-STEWARTSON SYSTEM A. Another “Riemann equation” 1. Cole-Hopf transformation in the scalar case B. “Riemann system” associated with the matrix Davey-Stewartson system IX. CONCLUDING AND FURTHER REMARKS

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