98 citations · 686 across the 33 of their papers we have counts for
34 papers
On the Structure of Set-Theoretic Polygon Equations
Folkert Müller-Hoissen
Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared…
A vectorial binary Darboux transformation of the first member of the negative part of the AKNS hierarchy
Folkert Müller-Hoissen
Using bidifferential calculus, we derive a vectorial binary Darboux transformation for the first member of the "negative" part of the AKNS hierarchy. A reduction leads to the first…
Differential Calculi on Associative Algebras and Integrable Systems
Aristophanes Dimakis, Folkert Müller-Hoissen
After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a "symmetry" of a generalized zero curvature equation and derive…
Non-isospectral extension of the Volterra lattice hierarchy, and Hankel determinants
Xiao-Min Chen, Xing-Biao Hu, Folkert Müller-Hoissen
For the first two equations of the Volterra lattice hierarchy and the first two equations of its non-autonomous (non-isospectral) extension, we present Riccati systems for function…
Matrix KP: tropical limit and Yang-Baxter maps
Aristophanes Dimakis, Folkert Müller-Hoissen
We study soliton solutions of matrix Kadomtsev-Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attache…
NLS breathers, rogue waves, and solutions of the Lyapunov equation for Jordan blocks
Oleksandr Chvartatskyi, Folkert Müller-Hoissen
The infinite families of Peregrine, Akhmediev and Kuznetsov-Ma breather solutions of the focusing Nonlinear Schroedinger (NLS) equation are obtained via a matrix version of the Dar…
Generalized Volterra lattices: binary Darboux transformations and self-consistent sources
Folkert Müller-Hoissen, Oleksandr Chvartatskyi, Kouichi Toda
We study two families of (matrix versions of) generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differentia…
Self-Consistent Sources for Integrable Equations via Deformations of Binary Darboux Transformations
Oleksandr Chvartatskyi, Aristophanes Dimakis, Folkert Müller-Hoissen
We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation…
Simplex and Polygon Equations
Aristophanes Dimakis, Folkert Müller-Hoissen
It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order." We describe simplex equations (inc…
KdV soliton interactions: a tropical view
Aristophanes Dimakis, Folkert Mueller-Hoissen
Via a "tropical limit" (Maslov dequantization), Korteweg-deVries (KdV) solitons correspond to piecewise linear graphs in two-dimensional space-time. We explore this limit.
KP solitons, higher Bruhat and Tamari orders
Aristophanes Dimakis, Folkert Mueller-Hoissen
In a tropical approximation, any tree-shaped line soliton solution, a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili (KP-II) equation, determines a…
The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework
Aristophanes Dimakis, Nils Kanning, Folkert Müller-Hoissen
The non-autonomous chiral model equation for an matrix function on a two-dimensional space appears in particular in general relativity, where for a certain reduc…
KP line solitons and Tamari lattices
Aristophanes Dimakis, Folkert Mueller-Hoissen
The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "criti…
Solutions of matrix NLS systems and their discretisations: A unified treatment
Aristophanes Dimakis, Folkert Muller-Hoissen
Using a bidifferential graded algebra approach to integrable partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and ful…
Riemannian Geometry of Bicovariant Group Lattices
Aristophanes Dimakis, Folkert Muller-Hoissen
Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can…
Differential Geometry of Group Lattices
Aristophanes Dimakis, Folkert Muller-Hoissen
In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differ…
On generalized Lotka-Volterra lattices
Aristophanes Dimakis, Folkert Muller-Hoissen
Generalized matrix Lotka-Volterra lattice equations are obtained in a systematic way from a "master equation" possessing a bicomplex formulation.
Bicomplexes and Backlund transformations
Aristophanes Dimakis, Folkert Muller-Hoissen
A bicomplex is a simple mathematical structure, in particular associated with completely integrable models. The conditions defining a bicomplex are a special form of a parameter-de…
Bicomplex formulation and Moyal deformation of (2+1)-dimensional Fordy-Kulish systems
Aristophanes Dimakis, Folkert Muller-Hoissen
Using bicomplex formalism we construct generalizations of Fordy-Kulish systems of matrix nonlinear Schroedinger equations on two-dimensional space-time in two respects. Firstly, we…
Bicomplexes and Integrable Models
Aristophanes Dimakis, Folkert Muller-Hoissen
We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings a…
Bi-differential calculus and the KdV equation
Aristophanes Dimakis, Folkert Muller-Hoissen
A gauged bi-differential calculus over an associative (and not necessarily commutative) algebra A is an N-graded left A-module with two covariant derivatives acting on it which, as…
Pseudo-Riemannian Metrics in Models Based on Noncommutative Geometry
Aristophanes Dimakis, Folkert Muller-Hoissen
Several examples and models based on noncommutative differential calculi on commutative algebras indicate that a metric should be regarded as an element of the left-linear tensor p…
Bi-differential calculi and integrable models
Aristophanes Dimakis, Folkert Muller-Hoissen
The existence of an infinite set of conserved currents in completely integrable classical models, including chiral and Toda models as well as the KP and self-dual Yang-Mills equati…
Noncommutative geometry and a class of completely integrable models
A. Dimakis, F. Muller-Hoissen
We introduce a Hodge operator in a framework of noncommutative geometry. The complete integrability of 2-dimensional classical harmonic maps into groups (sigma-models or principal…