Noncommutative Multisoliton Solutions of a Supersymmetric Chiral Model

and Applied Analysis 3 Thus, the solutions of (2) described by the simple pole ansatz are parametrized by the set {S k } m 1 of matrix-valued functions of w k and η k and by the pole position μ k . Example 2. One-soliton configuration


Introduction
A generalization of the modified 2 + 1 chiral model [1] (also called the Ward model in [2]) with N ≤ 8 supersymmetries and a Moyal deformation of this model is introduced in [3].Since the N-extended and deformed chiral model can be formulated as the compatibility conditions of a linear system of differential equations, solutions of this model can be generated with the aid of the system just as in the nonsupersymmetric case [1,2,[4][5][6][7].
A powerful method, the so-called dressing method, is employed in [3] to construct multisoliton configurations with only simple poles.By allowing for second-order poles in the dressing ansatz, a two-soliton configuration with genuine soliton-soliton interaction is constructed in [8].
In a recent paper [9], we extended the algebraic B ä cklund transformations (BTs) and the order  limiting method of Dai and Terng [2] to the noncommutative case, with which we constructed multisoliton solutions with general pole data of the noncommutative Ward model [7,10].In this paper, we further extend the above case to the supersymmetric one; that is, we use the supersymmetric and noncommutative extended algebraic BTs and the order  limiting method to construct multisoliton solutions with arbitrary poles and multiplicities of the supersymmetric and noncommutative Ward model.
The plan of the paper is as follows: we review the explicit definition of the N-extended and deformed Ward model and the linear system associated to it in Section 2. In Section 3, we first briefly review the dressing construction in [3] and then give the supersymmetric algebraic BTs for the linear system.We apply these algebraic BTs and the superextended order  limiting method to construct a large family of multisoliton configurations with general pole data in Section 4.

Noncommutative N-Extended Ward Model
To formulate the noncommutative N-extended Ward model, we need to introduce the following notations [3,8].
Definition 1 (see [3,8]).The ()-valued superfield Φ(  ,    ) of the noncommutative N-extended () Ward model satisfies the classical field equations where    := /   , and the unitarity condition To avoid cluttering the formulae we suppress the "⋆" notation for the supersymmetric version of noncommutative multiplication from now on and most products between classical fields and their components are assumed to be star products until mentioned otherwise.
The noncommutative star product can be replaced by ordinary product via the Moyal-Weyl map [3,6,7]; that is,  ⋆  → f ĝ, where f and ĝ are the corresponding operators of  and  under the Moyal-Weyl map, respectively.In later sections, we will mainly use the star-product formulation but use the operator formalism when the former does not work well.

N-Extended Multisoliton Configurations with Simple Pole Data
Dressing method is employed in [3] to construct solutions to the linear system (4).This method is a recursive procedure for generating a new solution from an old one.We briefly review their construction.Rewrite (4) in the form and build a multisoliton solution   with  simple poles at position  1 , . . .,   with Im   < 0 by left-multiplying an (− 1) simple pole solution  −1 with a single pole factor of the form where the  ×  matrix   is a Hermitian projection of rank   ; that is,  †  =   and  2  =   ; this is obtained by reality condition (5), and therefore one can decompose   =   ( †    ) −1  †  , where   is an  ×   ( ≥ 2,   ≤ ) matrix depending on   and    and the   columns of   span the image of   .Therefore the iteration  1  → ⋅ ⋅ ⋅  →   yields the multiplicative ansatz which, via a partial fraction decomposition, may be written in the additive form where Λ  and   are some  ×   matrices depending on   and    .It was shown in [3] that   is the solution of the linear system (6) if with   =  +    +  −1  V and    =  1  +    2  for  = 1, . . ., .The associated superfield is Thus, the solutions of (2) described by the simple pole ansatz are parametrized by the set {  }  1 of matrix-valued functions of   and    and by the pole position   .
Example 2. One-soliton configuration with and  1 =  1 (,   ), is an  ×  matrix, where ( This configuration will describe a moving soliton if the matrix  1 depends on  rationally [3,8]. We now change the form of the one-soliton solution  1 in (12) into the following one: where Then  = (,   ) is an  ×  matrix function of  and   and depends on  rationally.With these notations at hand, we can give the following supersymmetric extension of our noncommutative version of algebraic BT, which will be used to construct multisoliton solutions with only simple poles of the linear system (4).
Proof.(1) The residues of the right hand side of ψ() =  , P()() , () −1 at  and  are respectively, and both equal zero by the definition of P. Thus we have two factorizations of  1 =  , P = ψ , .
Let  1 =  , ♯, Φ 1 =  , ♯Φ denote the N-extended noncommutative algebraic BT generated by  , .If we apply these BTs repeatedly (with distinct poles) to a one-soliton solution, then we obtain a multisoliton solution of (4) with simple poles and the associated superfield.Such configurations coincide with the ones constructed in [3], and we call them multisoliton configurations with simple pole data.In contrast, we call solutions of (4) with higher-order poles and the associated fields multisoliton configurations with higherorder pole data, which will be constructed in the next section.
Example 4. Two-soliton configurations with two simple poles.

N-Extended Multisoliton Configurations with Higher-Order Pole Data
A solution of ( 6) with a double pole at  = − is constructed in [8] by making a second-order pole at  = − in the dressing ansatz, that is, considering the following dressing transformation: where the Hermitian projection  is known and P = T( T † T) −1 T † is yet to be determined.Demanding that ψ is again a solution of ( 6) with some new superfields Ã, B, and C , which are independent of , they obtained the following equations: After constructing a projection P via a solution T of (22), ψ and the associated superfield Φ = ψ−1 ( = 0) = (  − 2)(  − 2 P) are derived.
Their construction of the solution T is inspired by the known form of T in the bosonic case [4,5], that is, making the ansatz T =  +  ⊥ ( † ⊥  ⊥ ) −1  with  ⊥ orthogonal to .Below we will present a supersymmetric extension of our noncommutative limiting method as in the bosonic case to construct multisoliton configurations with a higherorder pole at  = .We give an example of two-soliton configuration with a second-order pole by taking a limit at first.
Example 5. Two-soliton configuration with a second-order pole.
Let  ∈ C \ R, ,  be two holomorphic functions of (,   ) and depend on the bosonic variable  rationally: where  =  +  2  + V and   = Then ) . ( is a two-soliton solution of (4) with a double pole at  = , and the associated superfield is Next, we use a systematic limiting method and our extended algebraic BTs in Section 3 to construct multisoliton solutions of ( 4) with pole data (, ) for any  ∈ C \ R and  ≥ 2.
Remark 6.We can write hence the right derivatives of   (,   ) with respect to one or several of the Grassmann variables are computed as follows: where   = ∑  =0  ,− are given as follows: From the expressions, we know that   's are  × 1 matrix functions of (,   ) and depend on  rationally.
Proof.We prove the theorem by induction on .For  = 1, the theorem is clearly true.Suppose the theorem is true for , we will prove that (1)-( 3) hold for  + 1.