On the Higher-Order Inhomogeneous Heisenberg Supermagnetic Models

This paper is concerned with the construction of the fifth-order inhomogeneous Heisenberg supermagnetic models. Moreover, the Lax representations of the models are presented. By means of the gauge transformation, we establish their gauge equivalent equations with different quadratic constraints, i.e., the super and fermionic fifth-order inhomogeneous nonlinear Schrödinger equations, respectively. In addition, we investigate their Lax representations and Bäcklund transformations from which the solutions of the super integrable systems have been discussed.


Introduction
The Heisenberg ferromagnet (HF) model [1,2] is an important integrable system, which describes the movement of the magnetization vector of the isotropic ferromagnets. The HF model has been well developed and it is geometrical and gauge equivalent to the nonlinear Schrödinger equation (NLSE) [3,4]. There have been extensive study and application of the HF models and the inhomogeneous integrable equations [5,6], such as deformed HF model [7,8], extended high-order HF model [9][10][11], inhomogeneous deformed HF model [3,12], the multidimensional HF model [13,14], the multi-component extended HF model [15], and integrable counterparts of the Heisenberg soliton hierarchy [16].
Qiao et al. investigated the involutive solutions of the higher-order HF model in terms of the spectral problem nonlinearization approach [17]. Then it is showed the constrained HF hierarchy possess the same r-matrix with the constrained Harry-Dym system [18]. Supersymmetry plays a significant role between theoretical physics and mathematics [19][20][21]. Thus integrable supersymmetric systems have attracted considerable attention in the mathematical as well as physical points of view. A number of famous integrable systems have been generalized to their supersymmetric version, such as the Korteweg-de Vries (KdV) equation [22,23], the Kadomtsev-Petviashvili (KP) equation [24], the HF model [25][26][27] and the inhomogeneous nonlinear Schrödinger type equation [28]. Ma et al. [29] investigate the applications to super integrable systems by means of the supertrace identity on Lie super algebras.
The Heisenberg supermagnet (HS) model can be regarded as the supersymmetric extension of the HF model [11,25]. The HS models and their corresponding gauge equivalence were first developed by Makhankov et al. [25]. Furthermore, the higher order and inhomogeneous HS models have been discussed and their integrable structure and properties have been also derived. Meanwhile, the authors [11,30] constructed the third-order, fourth-order and fifth-order generalized HS models from which their gauge equivalent equations have been presented. Moreover, Yan et al. developed the inhomogeneous thirdorder and fourth-order generalized HS models [26,27], respectively. The corresponding gauge equivalent equations are super and fermionic inhomogeneous NLSEs. Therefore, our purpose of this paper is to develop inhomogeneous deformations of fifth-order HS model and analyze their structure and integral properties. Furthermore, we shall derive the Bäcklund transformations of the super nonlinear evolution equation.
The organization of this paper is as follows. In the second section, the HS model is briefly reviewed and its integrable properties are recalled. In the third section, we construct the inhomogeneous fifth-order HS model. Then the gauge equivalent equations with two quadratic constraints and Bäcklund transformations are derived. In the fourth section, we dedicate to a summary and discussion.

Heisenberg Supermagnet Model
Let us start with a short summary of the HS model that will be useful in what follows. For a more detailed description, we refer the reader to [25].
The HS model is described by where S is a superspin function which can be given by where S 1 , … , S 4 are the bosonic variables and C 5 , … , C 8 are the fermionic variables. T 1 , … , T 4 are bosonic generators of the superalgebra su(2/1) and T 5 , … , T 8 are fermionic generators of the superalgebra su(2/1). The gauge equivalence plays an important role in the integral systems. The fact is that the gauge equivalence exists only for integrable systems possess Lax representations. Understanding the properties of the gauge equivalent counterpart helps us know more about the integrable systems. Under the following two constraints, Makhankov and Pashaev showed the HS model is gauge equivalent to supersymmetric NLSE and Grassman odd NLSE, respectively where (x, t) is a bosonic components and , 1 , 2 are the fermionic ones. The Lax representation of the fifth-order HS model contains no higher than the fourth-order derivatives with respect to x.

Fifth-Order Inhomogeneous Heisenberg Supermagnet Model
We now introduce the Lax representation of the HS model (6), where = ( 1 , 2 , 3 ) ⟂ , j , j=1,2 and 3 are the bosonic and fermionic functions, respectively, and F, G can be presented as SD + DS = D.
where is a spectral parameter.
The Lax pair satisfies the zero-curvature equation By substituting (9) into (10) and taking advantage of the condition (7), we have From the Eq. (10) and contrasting coefficients of the power of , we derive the fifthorder inhomogeneous HS model The corresponding F and G are given by where is a spectral parameter. In order to derive the gauge equivalent equation of (12), one takes where g(x, t) ∈ SU(2∕1).

According to condition
Equation (15) satisfies The orthogonal direct sum decomposition of the super algebra su(2/1) is as follows . The commutation and anticommutator relations are given by Based on (14), (15) and (18), we have Substituting (14) and (19) into (12), we obtain According to the Eqs. (16) and (17), we obtain By substituting (18), (21) into (23) and integrating Eq. (23) in reference to respect to the variable x, we derive where Since J 0 = J (0) 0 + J (1) 0 , it is easy to draw the following conclusion In terms of the gauge transformation ̃ = g , we obtain where F and Ĝ can be written as Substituting (18) and (26) into (28), we obtain where (26) By virtue of the zero-curvature formulation of F and Ĝ , we derive the super fifthorder inhomogeneous NLSE with the constraint (i) where (x, t) is bosonic filed and (x, t) is fermionic one. If one sets = 0 , Eq. (31) reduces to the super Hirota equation [28]. where T and are the bosonic and fermionic functions, respectively. Next we assume that under the transformation the forms Eq. (32) do not change, where ̄ is the conjugate of . Then we obtain (31)  Based on (32) and (34), we derive the Bäcklund transformation Since = 0 , = 0 are the trivial solution of (31), based on the Bäcklund transformation (35), we obtain a new solution of (31) where , and are the bosonic constants and is fermionic one.

Constraint (ii) S 2 = 3S − 2I
Now one turns to the second constraint S 2 = 3S − 2I . One derives S t and [S, S xx ] satisfying SS t S = 2S t and S[S, S xx ]S = 2[S, S xx ] . Thus the deformation term D should satisfy the equation Following the similar procedure as before, we obtain the fifth-order inhomogeneous HS model under the constraint (ii) The corresponding F and G can be expressed as (39)

3
Due to the gauge transformationIn ̃ = g , we obtain where F and G can be written as By means of (40) and (43), we rewrite (46) as follows  (53)