Abstract
By taking values in a commutative subalgebra , we construct a new generalized -Heisenberg ferromagnet model in (1+1)-dimensions. The corresponding geometrical equivalence between the generalized -Heisenberg ferromagnet model and -mixed derivative nonlinear Schrödinger equation has been investigated. The 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.
1. Introduction
The 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]. The 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 [14–18]. 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]. After 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 , 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 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.
This 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. The last section will be devoted to a summary and discussion.
2. -Heisenberg Ferromagnet Model
The Heisenberg ferromagnet (HF) model in (1+1)-dimensions [4] is an important integrable equation which reads as
where denotes the spin vector, S = (S1, S2, S3) and satisfies the constraint .
The matrix form of the HF model can be expressed as
Where , and are Pauli matrices.
Let take values in a commutative subalgebra and . From the equation (2), we obtain
Where , is an identity matrix. Suppose can be expressed as
Then can be divided into parts
where
and when , , is a identity matrix. Then we may derive the following theorems.
Theorem 1. The 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.
The integrability condition of (7) is as the following linear systems
where
Substituting (5) and (6) into (3), we obtain the following corollary:
Corollary 2. The vector form of the -HF model:here we use the property
This proves that the -HF is geometrical equivalent to the following -NLSE.
Theorem 3. The following identity holds
Proof. From NLS equation, we obtainwhereBy choosing the coefficients of for the identity (13), (13) leads to the -NLS equation (12).
The Lax pair of (12) can be represented as
where
and
where is a identity matrix.
3. Generalized -Heisenberg Ferromagnet Model in (1+1)-Dimensions
Let us consider the integrable deformed HF model [28]
where is a deformation parameter.
By expanding , we obtain the generalized -Heisenberg ferromagnet model in (1+1)-dimensions
where is a deformation parameter. When , Eq. (19) reduces to the -HF model (10). The Lax representation of the generalized -HF equation (19) is given by
where are spectral parameters.
In order to derive the geometrical equivalent counterpart of (19), we introduce the multi-component Serret-Frenet equation
By introducing the multi-component Hasimoto function
where
here
Identifying in (19) with the tangent vector of a curve , we obtain
Then we have
By the equation
one finds that the time evolution equation satisfies the following equation
Substituting (26) and (27) into (28) and taking , we derive the -mixed derivative NLSE equation
Taking , the -mixed derivative NLSE equation (29) degenerates into -NLSE equation (12). Then we obtain the Lax representation of the -mixed derivative NLSE equation
where
and
4. 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]. The 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. There 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.
4.1. -Inhomogeneous M-I Equation
Let take values in a commutative algebra, we have . By means of multi-component generalization, we obtain the generalized -inhomogeneous Heisenberg ferromagnet model in (2+1)-dimensions
where
and the parameters satisfy
When , Eq. (33) reduced to the integrable inhomogeneous Myrzakulov-I equation [13].
The linear problem of the multi-component HF models (33) in (2+1)-dimensions can be expressed as
where
and
Then the Lax representation of Eq. (33) is given by
Now one considers the the geometrical equivalent counterpart of the multi-component Eq. (33). Let us introduce the multi-component Serret-Frenet equation
Then we derive the multi-component Hasimoto function
In order to derive the geometrical equivalent counterpart of (33), we identify in the vector form of the -generalized inhomogeneous HF model in (2+1)-dimensions (33) with the tangent vector of a curve . Then we have
Thus we obtain
By the equation
Substituting (43) and (44) into (28) and taking , we derive the -NLS equation
where
When , the -NLS equation (45) degrades into the (2+1)-dimensional focusing nonlinear Schrödinger equation equation [13]. The Lax representation of the -HLS equation can be expressed as
where
where are spectral parameters.
4.2. -Ishimori Equation
Based on the multi-component generalization, we construct the multi-component Ishimori equation in (2+1)-dimensions
The Lax representation of (49) is given by
where
In terms of gauge transformation
The functions and can be written as
where satisfy the following equations:and
here
Then it follows that
where
Thus we obtain the gauge equivalent counterpart of Eq. (49) which can be considered as the -Davey-Stewartson equation
Its Lax reprensentation is given by
with
where
Similarly,
where
with
Here and are the solution of the following equations
and are given by
5. Summary and Discussion
Considering the commutative subalgebra , 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. The introduction of new degrees of freedom may emerge from multiscale procedures or regularizations of gradient catastrophes. Their physical meaning and application should be of interest. The methods in the paper may clearly be applied to the other generalized Heisenberg supermagnetic models. Therefore, other types of generalized -HF models still deserve further study.
Data Availability
All data included in this study are available upon request by contact with the corresponding author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is partially supported by National Natural Science Foundation of China (Grant Nos. 11965014, 11605096, and 11601247), National Undergraduate Training Program for Innovation (Grant No. 201810126023). We thank Professor Ke Wu and Professor Wei-Zhong Zhao (CNU, China) for long-term encouragements and supports. The authors thank the valuable suggestions of the referees.