Notes on Riemann geometry and integrable systems. Part IV
The connection between multidimensional soliton equations and three-dimensional Riemann space is discussed.
gr-qc↗
arXiv subjects
R. Myrzakulov
Publications and source records attributed to R. Myrzakulov.
Deformation surfaces, integrable systems and Chern - Simons theory
A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern-Simons action via reduction of the gauge connection in Hermitian symmetric spaces. In this paper we show that the methods developed in studying classical non-Abelian pure Chern-Simons actions, can be naturally implemented by means of a geometrical interpretation of such systems. The Chern-Simons equation of motion turns out to be related to time evolving 2-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss-Mainardi-Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.
nlin.SI↗
On the Soliton Geometry in Multidimensions
Some aspects of the multidimensional soliton geometry are considered.
math.DG↗
Soliton equations in N-dimensions as exact reductions of the Self-Dual Yang-Mills equation V. Simplest (2+1)-dimensional soliton equations
Some aspects of the multidimensional soliton geometry are considered. It is shown that some simples (2+1)-dimensional equations are exact reductions of the Self-Dual Yang-Mills equation or its higher hierarchy.
math-ph↗
Soliton equations in N-dimensions as exact reductions of Self-Dual Yang-Mills equation IV. The mM-LXII and Bogomolny equations
Some aspects of the multidimensional soliton geometry are considered. The relation between soliton equations in 2+1 dimensions and the Self-Dual Yang-Mills and Bogomolny equations are discussed.
math-ph↗
Soliton equations in N-dimensions as exact reductions of Self-dual Yang-Mills equation III. Soliton geometry in 2+1 and 3+1 dimensions
Some aspects of the relation between differential geometry of curves and surfaces and multidimensional soliton equations is discussed. The connection between multidimensional soliton equations and Self-dual Yang-Mills equation is studied.
math.DG↗
The L-equivalent counterpart of the M-III equation
The connection between differential geometry of curves and the (2+1)-dimensional integrable spin system - the M-III equation is established. Using the presented geometrical formalism the L-equivalent counterpart of the M-III equation is found.
math.DG↗
Soliton equations in 2+1 dimensions and Differential geometry of curves/surfaces
A connection between differential geometry and soliton equations is discussed
math.DG↗
Singularity Structure Analysis, Integrability, Solitons and Dromions in (2+1)-Dimensional Zakharov Equations
The (2+1)-dimensional integrable Zakharov equations and their reductions are considered
solv-int↗
Soliton equations in 2+1 dimensions: reductions, bilinearizations and simplest solutions
Soliton equations in 2+1 and their 1+1 = 2+0 reductions are considered.
solv-int↗
On Some Sigma Models with Potentials and the Klein-Gordon Type Equations
The gauge equivalent counterparts of the some (1+1)-, or (2+0)-dimensional sigma models with potentials are found. The gauge equivalence between the some soliton equations of spin-phonon systems and the Yajima-Oikawa and Ma equations are also established.
hep-th↗
Integrable geometry and Soliton equations in 2+1 dimensions
Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.
math.DG↗
A (2+1)-dimensional integrable spin model(the M-XXII equation) and Differential geometry of curves/surfaces
Using the differential geometry of curves and surfaces the Lakshmanan equivalent counterpart of the M-XXII equation is found ... .
math-ph↗
Motion of Curves and Surfaces and Nonlinear Evolution Equations in (2+1) Dimensions
It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical invariants then define topological conserved quantities. Underlying evolution equations are shown to be associated with a triad of linear equations. Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov -Strachan (2+1) dimensional nonlinear Schrödinger equations respectively.
solv-int↗
The gauge equivalence of the Zakharov equations and (2+1)-dimensional continuous Heisenberg ferromagnetic models
The gauge equivalence between the (2+1)-dimensional Zakharov equations and (2+1)-dimensional integrable continuous Heisenberg ferromagnetic model is established. Also their integrable reductions are shown explicitly.
cond-mat↗
Gauge Equivalence between Two-Dimensional Heisenberg Ferromagnets with Single-Site Anisotropy and Zakharov Equations
Gauge equivalence between the two-dimensional continuous classical Heisenberg ferromagnets(CCHF) of spin 1/2 - the M-I equation with single site anisotropy and the Zakharov equation(ZE) is established for the easy axis case. The anisotropic CCHF is shown to be gauge equivalent to the isotropic CCHF.
cond-mat↗
On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schrödinger equations
Using a moving space curve formalism, geometrical as well as gauge equivalence between a (2+1) dimensional spin equation (M-I equation) and the (2+1) dimensional nonlinear Schrödinger equation (NLSE) originally discovered by Calogero, discussed then by Zakharov and recently rederived by Strachan, have been estabilished. A compatible set of three linear equations are obtained and integrals of motion are discussed. Through stereographic projection, the M-I equation has been bilinearized and different types of solutions such as line and curved solitons, breaking solitons, induced dromions, and domain wall type solutions are presented. Breaking soliton solutions of (2+1) dimensional NLSE have also been reported. Generalizations of the above spin equation are discussed.
solv-int↗