Exact travelling-wave solutions of the (2 + 1)-dimensional sine - Gordon equation possessing a velocity smaller than the velocity of the linear waves in the correspondent model system are obtained. The dependence of their dispersion relations and allowed areas for the wave parameters on the wave amplitude are discussed. The obtained waves contain as particular cases static structures consisting of elementary cells with zero topological charge. The self-consistent parameters of one static structure are calculated. The obtained structures require minima spatial system sizes for their existence. As an illustration the obtained results are applied for a description of structures in spin systems with an anisotropy created by a magnetic field or by a crystal anisotropy field.

On travelling waves and double-periodic structures in two-dimensional sine - Gordon systems

Topological defects in incommensurate magnetic and crystal structures and quasi-periodic solutions of the elliptic sine-Gordon equation

The possibilities of applying magnets with full or partial magnetic moment compensation in various spin groups to improve the performance of magnetic electronic devices using spin current (spintronics) are discussed. The effects of an exchange enhancement of the spin dynamics in antiferromagnets are well known. Over the past few years, antiferromagnetic spintronics has turned into an independent, rapidly developing field of applied physics of magnetism. This article provides for a detailed analysis of the possibility of using another class of magnetic materials, such as ferrimagnets close to the spin compensation point, in which the indicated acceleration effects are also detected. A comparative analysis of these two classes of magnets is conducted. The nonlinear spin dynamics of ferrimagnets are examined using a nonlinear sigma-model for the antiferromagnetic vector, describing the difference in spin densities of various spin groups. The simple conclusion derived based on this model is presented, and its real parameters for popular ferrimagnets, amorphous alloys of iron, and rare earth elements, are discussed. The different nonlinear effects of spin dynamics, ranging from homogeneous spin vibrations in small particles to the dynamics of solitons, domain walls, ferrimagnetic skyrmions, and vortices, are analyzed. The possibility of exciting such dynamic modes using spin torque, and their application in ultrafast spintronics is considered.

Ultrafast spin dynamics and spintronics for ferrimagnets close to the spin compensation point (Review)

The phenomenological theory of superconductors with a many-component order parameter (OP) is developed. On the basis of a generalized Ginzburg-Landau functional, equations for a two-component-OP superconductor are derived. It is shown that such a superconductor is specified by three length dimensionality parameters—penetration depth λ, correlation length ζ, and width d of the boundary between two superconducting-phase domains. With λ ≫ d ≫ ζ, the equations for the OP of a superconductor in a magnetic field can be explored analytically. The transition from the superconducting to the mixed phase may occur not only by the formation of ordinary Abrikosov vortices, but also owing to vortices that have two cores, each transferring a half-integral flux quantum. The total flux transferred by a vortex certainly constitutes an integral quantum. The cores of such a dimer are interconnected by two domain walls, which exercise confinement within the dimer. The distance between the cores in the dimer is of the...

Vortex structure in superconductors with a many-component order parameter

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case.We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation.We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.

https://www.reading.ac.uk/web/FILES/maths/Pelloni-Pinotsis-nonlin.pdf

The elliptic sine-Gordon equation in a half plane

The integration procedure based on the inverse scattering method is developed for a 2D elliptic sine-Gordon equation with a periodic-wave type asymptotic behaviour on one space variable The soliton solutions are found in explicit form

https://iopscience.iop.org/article/10.1088/0266-5611/5/6/006/pdf

A.B. Borisov

Papers

Topological defects in incommensurate magnetic and crystal structures and quasi-periodic solutions of the elliptic sine-Gordon equation

Inverse problem for an elliptic sine-Gordon equation with an asymptotic behaviour of the cnoidal-wave type

Vortices in the sine-Gordon system and solution of the boundary value problem by the inverse scattering transform

Multi-vortex-like solutions of the sine-Gordon equation

Solitons in a ferrimagnet