SOLITON EQUATIONS AND THE ZERO CURVATURE CONDITION IN NONCOMMUTATIVE GEOMETRY

@article{Dimakis1996SOLITONEA,
  title={SOLITON EQUATIONS AND THE ZERO CURVATURE CONDITION IN NONCOMMUTATIVE GEOMETRY},
  author={Aristophanes Dimakis and Folkert Mueller-Hoissen},
  journal={Journal of Physics A},
  year={1996},
  volume={29},
  pages={7279-7286}
}
Familiar nonlinear and in particular soliton equations arise as zero curvature conditions for connections with noncommutative differential calculi. The Burgers equation is formulated in this way and the Cole - Hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework. The KdV, modified KdV equation and the Miura transformation are obtained jointly in a similar setting and a rather straightforward generalization… Expand
Some aspects of noncommutative geometry and physics
An introduction is given to some selected aspects of noncommutative geometry. Simple examples in this context are provided by finite sets and lattices. As an application, it is explained how theExpand
Dynamical evolution in non-commutative discrete phase space and the derivation of classical kinetic equations
By considering a lattice model of extended phase space, and using techniques of non-commutative differential geometry, we are led to: (a) the concept of vector fields as generators of motion andExpand
Correct Rules for Clifford Calculus on Superspace
Abstract.In this paper an extension of Clifford analysis to superspace is given, inspired by the abstract framework of radial algebra. This framework leads to the introduction of the so-calledExpand
BICOMPLEXES, INTEGRABLE MODELS, AND NONCOMMUTATIVE GEOMETRY
We discuss a relation between bicomplexes and integrable models, and consider corresponding noncommutative (Moyal) deformations. As an example, a noncommutative version of a Toda field theory isExpand
Bicomplexes and integrable models
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 are discussed inExpand
Obituary: Aristophanes Dimakis
The theoretical physicist and mathematician Aristophanes Dimakis passed away on July 8, 2021, at the age of 68, in Athens, Greece. We briefly review his life, career and scientific achievements. WeExpand

References

SHOWING 1-10 OF 30 REFERENCES
The soliton connection
It is pointed out that the linear scattering problem for a non-linear evolution equation which admits soliton solutions may be described in terms of a linear connection on a principal SL(2, ℝ). TheExpand
Non-commutative geometry and kinetic theory of open systems
The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase spaceExpand
Stochastic differential calculus, the Moyal *-product, and noncommutative geometry
A reformulation of the Itô calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator dExpand
Hamiltonian methods in the theory of solitons
The Nonlinear Schrodinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic ExamplesExpand
Integrable Discretizations of Chiral Models
A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprisesExpand
Noncommutative Differential Calculus: Quantum Groups, Stochastic Processes, and the Antibracket
We explore a differential calculus on the algebra of smooth functions on a manifold. The former is `noncommutative' in the sense that functions and differentials do not commute, in general. RelationsExpand
Soliton Connection of the sinh-Gordon Equation
It is demonstrated that the sinh-Gordon equation can be written as covariant exterior derivatives of Lie algebra valued differential forms and, moreover, that these nonlinear differential equationsExpand
Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation
An explicit nonlinear transformation relating solutions of the Korteweg‐de Vries equation and a similar nonlinear equation is presented. This transformation is generalized to solutions of aExpand
Solitons and SL(2, R)
Abstract The known relationship between non-linear partial differential equations which have soliton solutions, and SL (2, R), is developed to the point where it provides a framework for discussingExpand
Solitons: An Introduction
Preface 1. The Kortewag-de Vries equation 2. Elementary solutions of the Korteweg-de Vries equation 3. The scattering and inverse scattering problems 4. The initial-value problem for the Korteweg-deExpand
...
1
2
3
...