scholarly journals Surfaces and Curves Induced by Nonlinear Schrödinger-Type Equations and Their Spin Systems

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1827
Author(s):  
Akbota Myrzakul ◽  
Gulgassyl Nugmanova ◽  
Nurzhan Serikbayev ◽  
Ratbay Myrzakulov
Keyword(s):  
Differential Geometry ◽  
Spin Systems ◽  
Heisenberg Ferromagnet ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Space Curves ◽  
The Third ◽  
Integrable Spin ◽  
Frenet Equation ◽  
Basis Vectors

In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.

scholarly journals THOULESS NUMBER AND SPIN DIFFUSION IN QUANTUM HEISENBERG FERROMAGNETS

1993 ◽  
Vol 07 (27) ◽  
pp. 1747-1759 ◽  
Author(s):  
PETER KOPIETZ
Keyword(s):  
Spin Diffusion ◽  
Nearest Neighbor ◽  
Arbitrary Dimension ◽  
Quantum Spin ◽  
Spin Systems ◽  
Heisenberg Ferromagnet ◽  
Electronic Systems ◽  
Dimensionless Frequency ◽  
Spin Models ◽  
Quantum Spin Models

Using an analogy between the conductivity tensor of electronic systems and the spin stiffness tensor of spin systems, we introduce the concept of the Thouless number g0 and the dimensionless frequency-dependent conductance g(ω) for quantum spin models. It is shown that spin diffusion implies the vanishing of the Drude peak of g(ω), and that the spin diffusion coefficient Ds is proportional to g0. We develop a new method based the Thouless number to calculate D s , and present results for D s in the nearest-neighbor quantum Heisenberg ferromagnet at infinite temperatures for arbitrary dimension d and spin S.

Construction of Integrable Spin Systems with Long-Range Interactions

1992 ◽  
Vol 61 (9) ◽  
pp. 3071-3076 ◽  
Author(s):  
Kazuhiro Hikami ◽  
P. P. Kulish ◽  
Miki Wadati
Keyword(s):  
Long Range ◽  
Spin Systems ◽  
Long Range Interactions ◽  
Integrable Spin

scholarly journals Geometric Flows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Ferromagnet Equation

2021 ◽  
Vol 2090 (1) ◽  
pp. 012068
Author(s):  
Gulgassyl Nugmanova ◽  
Aigul Taishiyeva ◽  
Ratbay Myrzakulov ◽  
Tolkynai Myrzakul
Keyword(s):  
Heisenberg Ferromagnet ◽  
Geometric Flows ◽  
Space Curves ◽  
Holm Equation ◽  
Two Component

Abstract In this paper, we study the generalized Heisenberg ferromagnet equation, namely, the M-CVI equation. This equation is integrable. The integrable motion of the space curves induced by the M-CVI equation is presented. Using this result, the Lakshmanan (geometrical) equivalence between the M-CVI equation and the two-component Camassa-Holm equation is established.

scholarly journals ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

2012 ◽  
Vol 19 (04) ◽  
pp. 1250028
Author(s):  
TING SU ◽  
HUIHUI DAI ◽  
XIAN GUO GENG
Keyword(s):  
Optical Fibers ◽  
Variable Coefficient ◽  
High Order ◽  
Explicit Solutions ◽  
Compatibility Conditions ◽  
Dressing Method ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Hirota Equations

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.

Curvature Expressions for the Large Displacement Analysis of Planar Beam Motions

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Yinhuan Zheng ◽  
Ahmed A. Shabana ◽  
Dayu Zhang
Keyword(s):  
Differential Geometry ◽  
Finite Element ◽  
Beam Element ◽  
Elastic Force ◽  
Large Displacement ◽  
Displacement Analysis ◽  
The Third ◽  
Shear Deformable Beam ◽  
Approach Method ◽  
Shear Deformable

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

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