Three Types Generalized -Heisenberg Ferromagnet Models

By taking values in a commutative subalgebra g 푙(푛, C), we construct a new generalized -Heisenberg ferromagnet model in (1+1)-dimensions. e corresponding geometrical equivalence between the generalized -Heisenberg ferromagnet model and -mixed derivative nonlinear Schrödinger equation has been investigated. e Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized -inhomogeneous Heisenberg ferromagnet model and -Ishimori equation in (2+1)-dimensions. We also discuss the integrable properties of the multi-component systems. Meanwhile, the generalized Zn-nonlinear Schrödinger equation, Zn-Davey–Stewartson equation and their Lax representation have been well studied.


Introduction
e Heisenberg ferromagnet (HF) model is one of the most investigated integrable systems which plays an important role in the two-dimensional (2D) gravity theory [1] and anti-de Sitter/conformal field theories [2,3]. It is proved that the HF model is gauge and geometric equivalent to the nonlinear Schrödinger (NLS) equation [4,5]. (1+1)-dimensional generalized HF models involving inhomogeneous and higher order deformed HF models have been analyzed [6,7]. e deformed HF models in (2+1)-dimensions also have been investigated, such as the higher order HF models [8,9], the HF models with self-consistent potentials [10], the Ishimori equation [11], and inhomogeneous HF models [12,13].
Multi-component version of the integrable systems has deserved much attention due to its wide application in multiple orthogonal polynomials, representation theory, random matrix model, the related Riemann-Hilbert problems, and Brownian motions [14][15][16][17][18]. Many important integrable systems have been extended to their multi-component counterparts, such as multi-component KP [19,20], multi-component Toda systems [14], and multi-component BKP [21]. A er considering commutative subalgebra of diagonal matrices, Bogdanov et al. [22] constructed the generalized multicomponent KP hierarchy which involves independent generalized scalar KP hierarchies. Starting from the maximal commutative subalgebra of g푙(푚, ℂ), one [23,24] constructed a new -Kadomtsev-Petviashvili (KP) hierarchy and investigated the existence of -functions. Meanwhile, the relation between dispersionless reduced -KP hierarchy and Frobenius manifold has been discussed. Recently, Li et al. [25] constructed the extended multi-component Toda hierarchy and extended multi-component bigraded Toda hierarchy. By virtue of taking values in a matrix-valued differential algebra set, they also establish a class of Hirota quadratic equation, which may be useful in Gromov-Witten theory and noncommutative symplectic geometry. In [25], one has defined the new multi-component sinh-Gordon systems by considering commutative subalgebra of g푙(푛, C) and established their Bäcklund transformations. A natural problem then arises as to how to construct the corresponding extended HF models. With this motivation, this paper will be devoted to constructing three types commutative multi-component generalized HF models by taking values in commutative subalgebra. Furthermore their corresponding geometrical and gauge equivalent counterparts shall be discussed.
is paper is organized as follows. In Section 2, we present a brief review of some elementary facts about the -HF model and -NLS equation. Section 3 is devoted to constructing the generalized -HF models and establishing the geometrical equivalence with the -mixed derivative NLSE. In Section 4, we investigate the generalized -inhomogeneous HF models and their structure and integrability. In addition, we deduce the multi-component Ishimori equation and discuss its corresponding gauge equivalent counterpart. e last section will be devoted to a summary and discussion. Let take values in a commutative subalgebra 푍 = ℂ[훤]/ 훤 and 훤 = 훿 푖 푗,푗+1 푖 푗 ∈ g푙(푛, ℂ). From the equation (2), we obtain Where ̃ 2 = , is an identity matrix. Suppose ̃ can be expressed as en 푆 (푥, 푡) can be divided into parts where and when 푘 = 0, 0 = , is a identity matrix. en we may derive the following theorems.

Theorem 1. e following equation holds
Proof. By choosing the coefficient of for two sides of the identity (3), (3) leads to (7), which will be referred to as the -HF model. e integrability condition of (7) is as the following linear systems where (1) S = S × S , Substituting (5) and (6) into (3), we obtain the following corollary: Corollary 2. e vector form of the -HF model: here we use the property is proves that the -HF is geometrical equivalent to the following -NLSE. where By choosing the coefficients of for the identity (13), (13) leads to the -NLS equation (12). ☐ e Lax pair of (12) can be represented as where and where is a identity matrix.

Generalized -Heisenberg Ferromagnet Model in (1+1)-Dimensions
Let us consider the integrable deformed HF model [28] where is a deformation parameter.

Generalized -Heisenberg Ferromagnet Model in (2+1)-Dimensions
Many (2+1)-dimensional integrable inhomogeneous Heisenberg ferromagnet equations have been of interest, for instance, Inhomogeneous M-I equation [13] and the Ishimori equation [11]. e Ishimori equation [11] is a well-known (2+1)-dimensional integrable extension of the HF model, which involves an infinite dimensional symmetry algebra with a loop algebra structure and is solved by the inverse scattering transform approach. ere is geometrical and gauge equivalence between the Ishimori equation and Davey-Stewartson equation [29,30]. In this section, we shall derive the multi-component counterparts of two types deformed HF models in (2+1)-dimensions.

Data Availability
All data included in this study are available upon request by contact with the corresponding author.

Summary and Discussion
Considering the commutative subalgebra g푙(푛, ℂ), we have constructed three types generalized -HF models in (1+1) and (2+1)-dimensions. From the geometrical and gauge equivalence point of view, we also establish the corresponding equivalent counterparts of three types generalized -Heisenberg ferromagnet models. e introduction of new degrees of freedom may emerge from multiscale procedures or regularizations of gradient catastrophes.
eir physical meaning and application should be of interest. e methods in the paper may clearly be applied to the other generalized Heisenberg supermagnetic models. erefore, other types of generalized -HF models still deserve further study.