Heisenberg Ferromagnet Equation Research Articles

Breather–breather interactions, rational and semi-rational solutions of an integrable spin system in optical fibers

This paper investigates an integrable spin system (or Heisenberg ferromagnet-type equation or Nurshuak-IIA equation) which is an integrable generalization of the Heisenberg ferromagnet equation that raises from magneto-optical devices. The N-fold Darboux transformation of the Nurshuak-IIA equation will be constructed. With the selection of several parameters, a variety of the solutions will be produced, including breather–breather interactions, higher-order rogue waves and semi-rational solutions.

Magnetic soliton and breather interactions for the higher-order Heisenberg ferromagnetic equation via the iterative N-fold Darboux transformation

This paper focuses on a higher-order Heisenberg ferromagnetic equation, which may describe the motion of the magnetic vector of isotropic ferromagnetism. The iterative N-fold Darboux transformation is first constructed to generate the dark and anti-dark magnetic solitons on the non-zero constant backgrounds, bright and dark breathers on the trigonometric function and non-zero constant backgrounds as well as breathers on the trigonometric function and vanishing backgrounds. We discover that the soliton structures of three different components can generate rotation with different constant seed solutions. Meanwhile, the trajectory curve and the direction of the magnetic vector are also discussed from the perspective of magnetism, we find that for constant seed solutions, the motion of the magnetic vector is limited to the hemisphere, while for trigonometric seed solutions, the motion of the magnetic vector can be distributed throughout the whole sphere. These novel phenomena may be helpful to understand the dynamics of the magnetic vector in the magnetic materials.

Rogue wave excitations and hybrid wave structures of the Heisenberg ferromagnet equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque.

In this paper, we focus on the localized rational waves of the variable-coefficient Heisenberg spin chain equation, which models the local magnetization in ferromagnet with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. First, we establish the iterative generalized (m,N-m)-fold Darboux transformation of the Heisenberg spin chain equation. Then, the novel localized rational solutions (LRSs), rogue waves (RWs), periodic waves, and hybrid wave structures on the periodic, zero, and nonzero constant backgrounds with the time-dependent coefficients α(t) and β(t) are obtained explicitly. Additionally, we provide the trajectory curves of magnetization and the variation of the magnetization direction for the obtained nonlinear waves at different times. These phenomena imply that the LRSs and RWs play the crucial roles in changing the circular motion of the magnetization. Finally, we also numerically simulate the wave propagations of some localized semi-rational solutions and RWs.

The new soliton solution types to the Myrzakulov-Lakshmanan-XXXII-equation

<abstract> <p>Our attention concenters on deriving diverse forms of the soliton arising from the Myrzakulov-Lakshmanan XXXII (M-XXXII) that describes the generalized Heisenberg ferromagnetic equation. This model has been solved numerically only using the N-fold Darboux Transformation method, not solved analytically before. We will derive new types of the analytical soliton solutions that will be constructed for the first time in the framework of three impressive schemas that are prepared for this target. These three techniques are the Generalized Kudryashov scheme (GKS), the (G'/G)-expansion scheme and the extended direct algebraic scheme (EDAS). Moreover, we will establish the 2D, 3D graphical simulations that clear the new dynamic properties of our achieved solutions.</p> </abstract>

Dynamical analysis of multi-soliton and breather solutions on constant and periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnet equation

Dynamical analysis of multi-soliton and breather solutions on constant and periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnet equation

Multiple nonlinear wave solutions of a generalized Heisenberg ferromagnet model and their interactions

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers.

Continuum limits for rational, breather solutions, and gauge equivalent model of the generalized discrete nonlinear Schrödinger equation

In this paper, we focus on an integrable higher‐order discrete nonlinear Schrödinger (dNLS) equation, which yields the Lakshmanan–Porsezian–Daniel (LPD) equation in the continuum limit. The breather solutions of the higher‐order dNLS are obtained for the first time. We show that the rational solutions and breather solutions of the higher‐order dNLS equation yield the counterparts of the integrable fourth‐order NLS equation under proper continuous limits. Most importantly, we also explore the gauge equivalent equation of the higher‐order dNLS equation from the point of view of the continuous limit theory. We make integrable discretization for the linear spectrum problem of the integrable generalized Heisenberg ferromagnetic equation that is just the gauge equivalent structure of the LPD equation. Compatibility condition of the discrete linear spectrum problem leads to a discrete integrable generalized Heisenberg Ferromagnetic equation. It is revealed that the discrete linear spectrum problem of the discrete integrable generalized ferromagnetic equation converges to the corresponding spectrum problem of the integrable generalized Feisenberg spin model in the continuum limit.In addition, with the help of the method of linear combinations for the conserved fluxes and densities, the conservation laws of the integrable higher‐order dNLS equation converge to the homologous results of the LPD equation.

Integrable decomposition for the (2+1)-dimensional AKNS hierarchy and its applications

In this paper, we are concerned with the integrable decomposition for the (2+1)-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. By utilizing recursive relations and symmetric reductions, we propose that the (n2 − n1 + 1)-flow of the (2+1)-dimensional AKNS hierarchy can be decomposed into the corresponding n1-flow and n2-flow of the (1+1)-dimensional AKNS hierarchy, both in the coupled and reduced cases. As an appropriate generalization, the integrable decompositions for the standard (2+1)-dimensional Heisenberg ferromagnet equation, the standard (2+1)-dimensional modified Heisenberg ferromagnet equation, and their two coupled generalizations are investigated. With no loss of generality, one-soliton solutions and their dynamic projections for the relevant gauge equivalent structures are discussed and illustrated through some figures.

Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics

We present a class of periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation, which was recently introduced by the authors together with Langmann as a classical, continuum limit of an Inozemtsev-type spin chain. These exact analytic solutions are constructed via a spin-pole ansatz written in terms of certain elliptic functions. The dynamical parameters in our solutions solve an elliptic spin Calogero-Moser (CM) system subject to certain constraints. In the course of our construction, we establish a novel Bäcklund transformation for this constrained elliptic spin CM system.

Location controllable rogue wave and mixed interaction solutions on the sine-cosine function period background for the Heisenberg ferromagnetic equation

Location controllable rogue wave and mixed interaction solutions on the sine-cosine function period background for the Heisenberg ferromagnetic equation

Breathers, rogue waves and semi-rational solutions for a generalized Heisenberg ferromagnet equation

We focus on a generalized Heisenberg ferromagnet (HF) equation which names the Myrzakulov XXXII (M-XXXII) equation in this paper. The N-fold Darboux transformation (DT) is constructed to derive various nonlinear waves such as breathers, rogue waves and their hybrids. Furthermore, the dynamic characteristics of these solutions are demonstrated by graphical simulation.

Spin generalizations of the Benjamin–Ono equation

We present new soliton equations related to the A-type spin Calogero–Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin–Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.

The non-chiral intermediate Heisenberg ferromagnet equation

We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter δ > 0, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit δ → ∞ it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of an Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.

From discrete nonlocal nonlinear Schrödinger equation to coupled discrete Heisenberg ferromagnet equation

The investigation of nonlocal reverse space, reverse time, and reverse space–time integrable equations becomes an interesting and hot topic. In the paper Ma and Zhu (2016), we have shown that the nonlocal focusing nonlinear Schrödinger (NLS) equation, the nonlocal defocusing NLS equation and their discrete versions are, respectively gauge equivalent to a Heisenberg ferromagnet (HF)-like equation, and a modified HF-like equation and their corresponding discrete cases. In this paper, we further show that the HF-like equation and the modified HF-like equation essentially are the coupled HF equation and the coupled HF-type equation in Minkowski space. We give a connection of the discrete coupled HF equation and the discrete coupled HF-type equation in Minkowski space and the corresponding coupled HF equation and coupled HF-type equation in Minkowski space. This means that the nonlocal focusing and defocusing NLS equations are gauge equivalent to the coupled HF and coupled HF-type equation in Minkowski space. The solution of the coupled HF-type equation in Minkowski space is constructed.

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

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.

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

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.

The visco modified Heisenberg ferromagnet equation and physical applications

In this paper, we introduce three classes of a spacelike curve evolution in terms of Frenet frame for the visco modified Heisenberg ferromagnet (visco MHF) equation in Minkowski 3-space. By using those frames and the visco MHF equation, we obtain the visco modified nonlinear heat system and the visco modified Schrödinger equation. Also, we study the Lorentz force equations of the electric fields for the t,n,b-magnetic curves of the magnetic vector fields with respect to Frenet frame of the spacelike curve. From this, we discuss the geometric phases, the Frenet geometric electric flow phases and the Frenet magnetic flow densities for Lorentz forces connected with the visco MHF equation.

ЭКВИВАЛЕНТНОСТЬ УРАВНЕНИЯ ХАНТЕРА-САКСОНА И ОБОБЩЕННОГО УРАВНЕНИЯ ФЕРРОМАГНЕТИКА ГЕЙЗЕНБЕРГА

Integrable systems play an important role in modern mathematics, theoretical and mathematical physics. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions of the HSE and the GHFE will be presented.

Local and nonlocal solitons in a coupled real system of Landau–Lifshitz equations

We consider the gauge equivalence of all NLS-type reductions with a coupled system of Landau–Lifshitz equations. In the local case it is shown to reduce to the continuum limit of the classical isotropic Heisenberg ferromagnet equation. All nonlocal reductions, however, are shown to be equivalent to a two-sublattice antiferromagnetic system. Nonlocal solitons are shown to represent excitations from nonzero ground state magnetization and antiferromagnetic vectors, indicative of a ferrimagnetic system. An alternate geometric interpretation is given by means of a coupled Hasimoto transformation, establishing the connection between the nonlocal reductions of the AKNS system and the motion of a curve in ℂ3. Through Darboux transformations, the one-soliton solutions of the coupled LL equations are obtained.

Binormal schrodinger system of Heisenberg ferromagnetic equation in the normal direction with Q-HATM approach

In this paper, we first study the applications of the wave propagation flow in the normal direction, which is assumed to be the path of the propagated light radiated by Heisenberg ferromagnetic equation. Then the Coriolis phase is mainly used to demonstrate the relationship between the geometric magnetic phase and parallel transportation of the wave propagation field of the evolving light radiating in the normal orientation with Heisenberg ferromagnetic equation. Moreover, we investigate the geometric magnetic interpretation of the binormal evolution of the wave propagation field in the normal direction by considering the nonlinear fractional system with the repulsive type. Finally, we obtain numerical fractional solutions for the nonlinear fractional systems with the repulsive type by using the [Formula: see text]-Homotopy analysis transform ([Formula: see text]-HATM) method.