Multisoliton configurations of a superextended and Moyal-type noncommutative deformed modified 2+1 chiral model have been constructed with the dressing method by Lechtenfeld and Popov several years ago. These configurations have no-scattering property. A two-soliton configuration with nontrivial scattering was constructed soon after that. More multisoliton solutions with general pole data of the superextended and noncommutative Ward model will be constructed in this paper. The method is the supersymmetric and noncommutative extension of Dai and Terng’s in constructing soliton solutions of the Ward model.

1. 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, 4–7].

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 Bäcklund transformations (BTs) and the order k 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 k 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 k limiting method to construct a large family of multisoliton configurations with general pole data in Section 4.

To formulate the noncommutative 𝒩-extended Ward model, we need to introduce the following notations [3, 8].

ℝ2,1=(ℝ3,g) with coordinates (xa)=(x,y,t) and the metric g=diag(1,1,-1).

Grassmann variables (see [11]): some mathematical objects obeying the following anticommutation rules χiχj+χjχi=0 for any 1≤i, j≤n. In particular, when i=j, we have χi2=0. So any function K of the Grassmann variables is a finite polynomial K(χ1,χ2,…,χn)=∑αi=0,1a(α1,α2,…,αn)χ1α1χ2α2⋯χnαn.

The antichiral superspace ℝ3∣𝒩 with coordinates (xa,ηiα) for α=1,2 and i=1,…,(1/2)𝒩≤4; here ηiα are called fermionic coordinates, which are Grassmann variables. The tensor fields on ℝ3∣𝒩 are called superfields.

The noncommutative star product on the antichiral superspace ℝ3∣𝒩 is
(1)(f⋆g)(x,y,t,ηiα)=f(x,y,t,ηiα)exp{i2θ(∂x←∂y→-∂y←∂x→)}g(x,y,t,ηiα).

Note that the time coordinate remains commutative and no derivatives with respect to the Grassmann variables ηiα.

With the notations listed above, we can now state the definition of this model.

Definition 1 (see [<xref ref-type="bibr" rid="B3">3</xref>, <xref ref-type="bibr" rid="B8">8</xref>]).

The U(n)-valued superfield Φ(xa,ηiα) of the noncommutative 𝒩-extended U(n) Ward model satisfies the classical field equations
(2)∂x(Φ†⋆∂xΦ)+∂y(Φ†⋆∂yΦ)-∂t(Φ†⋆∂tΦ)+∂y(Φ†⋆∂tΦ)-∂t(Φ†⋆∂yΦ)=0,∂1i(Φ†⋆∂xΦ)-∂t(Φ†⋆∂2iΦ)+∂y(Φ†⋆∂2iΦ)=0,∂1i(Φ†⋆∂tΦ)+∂1i(Φ†⋆∂yΦ)-∂x(Φ†⋆∂2iΦ)=0,∂1i(Φ†⋆∂2jΦ)+∂1j(Φ†⋆∂2iΦ)=0,
where ∂αi≔∂/∂ηiα, and the unitarity condition
(3)Φ†⋆Φ=Φ⋆Φ†=In.

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.

By [3], model (2) is the compatibility condition for the following linear system of differential equations involving a spectral parameter ζ∈ℂ∪{∞}:
(4)(ζ∂x-∂u)ψ=𝒜ψ,(ζ∂v-∂x)ψ=ℬψ,(ζ∂1i-∂2i)ψ=𝒞iψ,i=1,…,12𝒩,
where the n×n matrix ψ depends on (xa∣ζ,ηiα) and the n×n matrices 𝒜, ℬ, and 𝒞i are superfields on (xa∣ηiα)∈ℝ3∣𝒩 independent of the spectral parameter ζ. Moreover, ψ is subject to the following reality condition:
(5)ψ(xa,ζ,ηiα)[ψ(xa,ζ-,ηiα)]†=In.
Then the superfield Φ(xa,ηiα)=ψ(xa,0,ηiα)-1≔ψ-1(ζ=0) clearly satisfies model (2) and unitary condition (3).

The noncommutative star product can be replaced by ordinary product via the Moyal-Weyl map [3, 6, 7]; that is, f⋆g→f^g^, where f^ and g^ are the corresponding operators of f and g 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.

3. <inline-formula>
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</mml:math></inline-formula>-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
(6)ψ(∂u-ζ∂x)ψ†=𝒜,ψ(∂x-ζ∂v)ψ†=ℬ,ψ(∂2i-ζ∂1i)ψ†=𝒞i,i=1,…,12𝒩,
and build a multisoliton solution ψm with m simple poles at position μ1,…,μm with Imμk<0 by left-multiplying an (m-1) simple pole solution ψm-1 with a single pole factor of the form
(7)In+μm-μ-mζ-μmPm(xa,ηiα),
where the n×n matrix Pm is a Hermitian projection of rank rm; that is, Pm†=Pm and Pm2=Pm; this is obtained by reality condition (5), and therefore one can decompose Pm=Tm(Tm†Tm)-1Tm†, where Tm is an n×rm(n≥2,rm≤n) matrix depending on xa and ηiα and the rm columns of Tm span the image of Pm. Therefore the iteration ψ1↦⋯↦ψm yields the multiplicative ansatz
(8)ψm=∏l=0m-1(In+μm-l-μ-m-lζ-μm-lPm-l),
which, via a partial fraction decomposition, may be written in the additive form
(9)ψm=In+∑k=1mΛmkSk†ζ-μk,
where Λmk and Sk are some n×rk matrices depending on xa and ηiα.

It was shown in [3] that ψm is the solution of the linear system (6) if
(10)Sk=Sk(wk,ηki),Tk={∏l=1k-1(In+μk-l-μ-k-lμ-k-μk-lPk-l)}Sk
with wk=x+μ-ku+μ-k-1v and ηki=ηi1+μ-kηi2 for k=1,…,m. The associated superfield is
(11)Φm=ψm-1(ζ=0)=∏k=1m(In-ρkPk)withρk=1-μkμ-k.
Thus, the solutions of (2) described by the simple pole ansatz are parametrized by the set {Sk}1m of matrix-valued functions of wk and ηki and by the pole position μk.

Example 2.

One-soliton configuration
(12)ψ1=In+μ-μ-ζ-μP1,Φ1=ψ1-1(ζ=0)=In-μ--μμ-P1,
with μ∈ℂ∖ℝ, P1=T1(T1†T1)-1T1†, and T1=T1(w,ηi), is an n×r matrix, where
(13)w=x+μ-u+μ--1v,ηi=ηi1+μ-ηi2,i=1,…,12𝒩.
This configuration will describe a moving soliton if the matrix T1 depends on w rationally [3, 8].

We now change the form of the one-soliton solution ψ1 in (12) into the following one:
(14)ψ1=In+μ-μ-ζ-μP⊥,
where P⊥≔P1 and thus P=In-P⊥=In-P1=P1⊥ built from the n×r matrix T≔T1⊥; then T1†T=T1†T1⊥=0. Since T1=T1(w,ηi), we know T1†=T1†(w-,η-i); hence ∂wT1†=0 and ∂ηiT1†=0. Thus 0=∂w(T1†T)=(∂wT1†)T+T1†(∂wT)=T1†(∂wT) implies ∂wT=0. Similarly, ∂ηiT=0; hence T=T(w-,η-i)=T(x+μu+μ-1v,ηi1+μηi2). If we suppose
(15)w=μx+μ2u+v,ηi=ηi1+μηi2,i=1,…,12𝒩.
Then T=T(w,ηi) is an n×r matrix function of w and ηi and depends on w 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).

Theorem 3.

Let ψ(xa,ζ,ηiα) be a solution of (4) and Φ=ψ-1(ζ=0) the associated superfield; choose μ∈ℂ∖ℝ such that ψ is holomorphic and nondegenerate at ζ=μ. Let
(16)ψμ,P(ζ)≔In+μ-μ-ζ-μP⊥
be a solution of (4) with a simple pole at ζ=μ, where P=P(xa,ηiα) is the Hermitia projection built from an n×r matrix T=T(w,ηi) depending on w rationally. Let P~=P~(xa,ηiα) be the Hermitian projection built from the n×r matrix T~=ψ(xa,μ,ηiα)T. Then

ψ~(xa,ζ,ηiα)=ψμ,P~(ζ)ψ(xa,ζ,ηiα)ψμ,P(ζ)-1 is holomorphic and nondegenerate at ζ=μ,μ-;

ψ1=ψμ,P~ψ=ψ~ψμ,P is a new solution to the linear system (4) with some new superfields 𝒜1, ℬ1, and 𝒞1i, and the new associated superfield is Φ1(xa,ηiα)=Φ(P~+(μ/μ-)P~⊥).

Proof.

(1) The residues of the right hand side of ψ~(ζ)=ψμ,P~(ζ)ψ(ζ)ψμ,P(ζ)-1 at μ and μ- are
(17)(μ-μ-)P~⊥ψ(μ)(In-P⊥),(μ-μ-)(In-P~⊥)ψ(μ-)P⊥,
respectively, and both equal zero by the definition of P~. Thus we have two factorizations of ψ1=ψμ,P~ψ=ψ~ψμ,P.

(2) It suffices to show that 𝒜1:=(ζ∂xψ1-∂uψ1)ψ1-1 is independent of the spectral parameter ζ. Using ψ1=ψμ,P~ψ, we have
(18)𝒜1=(ζ∂xψμ,P~-∂uψμ,P~)ψμ,P~-1+ψμ,P~(ζ∂xψ-∂uψ)ψ-1ψμ,P~-1.
Since (ζ∂xψ-∂uψ)ψ-1 is constant in ζ by assumption, (18) is holomorphic at ζ∈ℂ∪{∞}∖{μ,μ-}. On the other hand, using ψ1=ψ~ψμ,P, we have
(19)𝒜1=(ζ∂xψ~-∂uψ~)ψ~-1+ψ~(ζ∂xψμ,P-∂uψμ,P)ψμ,P-1ψ~-1.
Since ψμ,P is a solution of (4), (ζ∂xψμ,P-∂uψμ,P)ψμ,P-1 is independent of ζ. Together with (1), we deduce that (19) is holomorphic at ζ=μ,μ-. Thus 𝒜1 is holomorphic on ℂ∪{∞} and hence independent of ζ by Liouville's theorem. Likewise ℬ1≔(ζ∂vψ1-∂xψ1)ψ1-1 and 𝒞1i≔(ζ∂1iψ1-∂2iψ1)ψ1-1 are also independent of ζ.

The associated superfield is Φ1=ψ1-1(ζ=0)=Φ(P~+(μ/μ-)P~⊥), and then the proof is completed.

Let ψ1=ψμ,P♯ψ,Φ1=ψμ,P♯Φ denote the 𝒩-extended noncommutative algebraic BT generated by ψμ,P. 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 higher-order pole data, which will be constructed in the next section.

Example 4.

Two-soliton configurations with two simple poles.

Let μ1, μ2 be two distinct complex numbers and μ1≠μ-2, and letTk(wk,ηki) be an n×1 matrix function depending on wk rationally, where
(20)wk=μkx+μk2u+v,ηki=ηi1+μkηi2,i=1,…,12𝒩.
Let Pk=Tk(Tk†Tk)-1Tk†,k=1,2; then ψμ1,P1 and ψμ2,P2 are one-soliton solutions and ψ2=ψμ2,P2♯ψμ1,P1=ψμ2,P~2ψμ1,P1 is a two-soliton solution of (4) with two simple poles at ζ=μ1 and ζ=μ2, where P~2 is the Hermitian projection built from the n×1 matrix T~2=ψμ1,P1(μ2)T2=(In+((μ1-μ-1)/(μ2-μ1))P1⊥)T2. The associated superfield is Φ2=(P1+(μ1/μ-1)P1⊥)(P~2+(μ2/μ-2)P~2⊥).

4. <inline-formula>
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<mml:mrow>
<mml:mi>𝒩</mml:mi></mml:mrow>
</mml:math></inline-formula>-Extended Multisoliton Configurations with Higher-Order Pole Data

A solution of (6) with a double pole at ζ=-i is constructed in [8] by making a second-order pole at ζ=-i in the dressing ansatz, that is, considering the following dressing transformation:
(21)ψ=In-2iζ+iP⟼ψ~=(In-2iζ+iP~)(In-2iζ+iP),
where the Hermitian projection P is known and P~=T~(T~†T~)-1T~† is yet to be determined. Demanding that ψ~ is again a solution of (6) with some new superfields 𝒜~, ℬ~, and 𝒞~i, which are independent of ζ, they obtained the following equations:
(22)(In-P~)(∂z-T~+(∂z-P)T~)=0,(In-P~)(∂tT~-2i(∂zP)T~)=0,(In-P~)(12(∂1i+i∂2i)T~+(∂1iP)T~)=0.
After constructing a projection P~ via a solution T~ of (22), ψ~ and the associated superfield Φ~=ψ~-1(ζ=0)=(In-2P)(In-2P~) 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~=T+T⊥(T⊥†T⊥)-1g with T⊥ orthogonal to T.

Below we will present a supersymmetric extension of our noncommutative limiting method as in the bosonic case to construct multisoliton configurations with a higher-order pole at ζ=μ. 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 μ∈ℂ∖ℝ, f, g be two holomorphic functions of (w,ηi) and depend on the bosonic variable w rationally:
(23)T1=(1f(w,ηi)),T2=(1f(wϵ,ηϵi)+ϵg(wϵ,ηϵi)),
where w=μx+μ2u+v and ηi=ηi1+μηi2, wϵ=(μ+ϵ)x+(μ+ϵ)2u+v and ηϵi=ηi1+(μ+ϵ)ηi2, i=1,…,(1/2)𝒩, for any ϵ∈ℂ with |ϵ| small, P1=T1(T1†T1)-1T1†, and P2=T2(T2†T2)-1T2†. The Taylor expansion of f(wϵ,ηϵi) in ϵ is
(24)f(wϵ,ηϵi)=f(w,ηi)+∂f∂w(wϵ-w)+∑i=1(1/2)𝒩∂f∂ηi(ηϵi-ηi)+O(ϵ2)=f(w,ηi)+[∂f∂w(x+2μu)+∑i=1(1/2)𝒩∂f∂ηiηi2]ϵ+O(ϵ2),
where ∂f/∂ηi is right derivative of f with respect to the Grassmann variable ηi. Note that the products in the expansion are the ordinary products and that we omit the variables (w,ηi) of ∂f/∂w and ∂f/∂ηi for simplicity. Clearly, g(wϵ,ηϵi) has similar expansion; therefore
(25)f(wϵ,ηϵi)+ϵg(wϵ,ηϵi)=f(w,ηi)+[∑i=1(1/2)𝒩∂f∂ηiηi2+g(w,ηi)∂f∂w(x+2μu)+∑i=1(1/2)𝒩∂f∂ηiηi2+g(w,ηi)]ϵ+O(ϵ2).
Then
(26)T~2=ψμ1,P1(μ+ϵ)T2=(In+μ1-μ-1ϵP1⊥)×(1f+[∂f∂w(x+2μu)+∑i=1(1/2)𝒩∂f∂ηiηi2+g]ϵ+O(ϵ2))=(1f)+(μ-μ-)P1⊥×(0∂f∂w(x+2μu)+∑i=1(1/2)𝒩∂f∂ηiηi2+g)+O(ϵ);T^2=limϵ→0T~2=(1f)+(μ-μ-)P1⊥×(0∂f∂w(x+2μu)+∑i=1(1/2)𝒩∂f∂ηiηi2+g).
Let P~2=T~2(T~2†T~2)-1T~2† and P^2=T^2(T^2†T^2)-1T^2†, then
(27)ψ^2=ψμ,P^2ψμ,P1=limϵ→0ψμ+ϵ,P~2ψμ,P1=limϵ→0ψμ+ϵ,P2♯ψμ,P1
is a two-soliton solution of (4) with a double pole at ζ=μ, and the associated superfield is
(28)Φ^2=ψ^2-1(ζ=0)=(P1+μμ-P1⊥)(P^2+μμ-P^2⊥).

Next, we use a systematic limiting method and our extended algebraic BTs in Section 3 to construct multisoliton solutions of (4) with pole data (μ,k) for any μ∈ℂ∖ℝ and k≥2.

Let μ∈ℂ∖ℝ be a constant and {aj(w,ηi)}j=0∞ a sequence of n×1 matrix functions depending on w rationally. For any ϵ∈ℂ with |ϵ| small, let w=μx+μ2u+v, ηi=ηi1+μηi2, wϵ=(μ+ϵ)x+(μ+ϵ)2u+v, and ηϵi=ηi1+(μ+ϵ)ηi2, for i=1,…,(1/2)𝒩. Here we only consider the case of 𝒩=8 since it is easily truncated to any smaller even number 𝒩 of supersymmetries (see [8]); thus
(29)wϵ-w=(x+2μu)ϵ+uϵ2,ηϵi-ηi=ηi2ϵ,i=1,…,4.
Let Fk,ϵ=∑j=0kaj(wϵ,ηϵi)ϵj for k≥1. The Taylor expansion of aj(wϵ,ηϵi) at (w,ηi) is
(30)aj(wϵ,ηϵi)=aj(w,ηi)+∂aj∂w(wϵ-w)+∑i=14∂aj∂ηi(ηϵi-ηi)+12![∑i,k=14∂2aj∂ηi∂ηk∂2aj∂w2(wϵ-w)2+2∑i=14∂2aj∂w∂ηi(wϵ-w)(ηϵi-ηi)+∑i,k=14∂2aj∂ηi∂ηk(ηϵk-ηk)(ηϵi-ηi)]+13![∂3aj∂w3(wϵ-w)3+3∑i=14∂3aj∂w2∂ηi(wϵ-w)2(ηϵi-ηi)+3∑i,k=14∂3aj∂w∂ηi∂ηk·(wϵ-w)(ηϵk-ηk)(ηϵi-ηi)+∑i,k,l=14∂3aj∂ηi∂ηk∂ηl(ηϵl-ηl)×(ηϵk-ηk)(ηϵi-ηi)∂3aj∂w3]+14![∂4aj∂w4(wϵ-w)4+4∑i=14∂4aj∂w3∂ηi(wϵ-w)3(ηϵi-ηi)+6∑i,k=14∂4aj∂w2∂ηi∂ηk(wϵ-w)2×(ηϵk-ηk)(ηϵi-ηi)+4∑i,k,l=14∂4aj∂w∂ηi∂ηk∂ηl(wϵ-w)×(ηϵl-ηl)(ηϵk-ηk)(ηϵi-ηi)+∑i,k,l,m=14∂4aj∂ηi∂ηk∂ηl∂ηm×(ηϵm-ηm)(ηϵl-ηl)×(ηϵk-ηk)(ηϵi-ηi)∂4aj∂w4]+⋯,
where the sums are taken over different indices when there are more than one index and the partial derivatives of aj(w,ηi) with respect to the Grassmann variables are given by the following remark.

Remark 6.

We can write
(31)aj(w,ηi)=aj,0(w)+∑i=14aj,i(w)ηi+∑k,i=14aj,ki(w)ηkηi+∑l,k,i=14aj,lki(w)ηlηkηi+∑m,l,k,i=14aj,mlki(w)ηmηlηkηi;
hence the right derivatives of aj(w,ηi) with respect to one or several of the Grassmann variables are computed as follows:
(32)∂aj∂ηi=aj,i+∑k=1k≠i4aj,kiηk+∑l,k=1l,k≠i4aj,lkiηlηk+∑m,l,k=1m,l,k≠i4aj,mlkiηmηlηk,∂2aj∂ηi∂ηk=aj,ki-aj,ik+∑l=1l≠i,k4(aj,lki-aj,lik)ηl+∑m,l=1m,l≠i,k4(aj,mlki-aj,mlik)ηmηl,∂3aj∂ηi∂ηk∂ηl=∑σsignσaj,σ(l)σ(k)σ(i)+∑m=1m≠i,k,l4∑σsignσaj,mσ(l)σ(k)σ(i)ηm,∂4aj∂ηi∂ηk∂ηl∂ηm=∑τsignτaj,τ(m)τ(l)τ(k)τ(i),
where ∑σ, ∑τ represent that the sums are taken over the permutation group of {l,k,i} and {m,l,k,i}, respectively.

Substituting (29) into the Taylor expansion of aj(wϵ,ηϵi), we get the Taylor expansion of aj(wϵ,ηϵi) in ϵ. Assume that aj(wϵ,ηϵi)=∑p=0∞bj,pϵp, then
(33)bj,0=aj(w,ηi),bj,1=∂aj∂w(x+2μu)+∑i=14∂aj∂ηiηi2,bj,2=∂aj∂wu+12∂2aj∂w2(x+2μu)2+∑i=14∂2aj∂w∂ηi(x+2μu)ηi2+12∑i,k=14∂2aj∂ηi∂ηkηk2ηi2,bj,3=∂2aj∂w2(x+2μu)u+∑i=14∂2aj∂w∂ηiuηi2+16∂3aj∂w3(x+2μu)3+12∑i=14∂3aj∂w2∂ηi(x+2μu)2ηi2+12∑i,k=14∂3aj∂w∂ηi∂ηk(x+2μu)ηk2ηi2+16∑i,k,l=14∂3aj∂ηi∂ηk∂ηlηl2ηk2ηi2,bj,4=12∂2aj∂w2u2+12∂3aj∂w3(x+2μu)2u+∑i=14∂3aj∂w2∂ηi(x+2μu)2uηi2+12∑i,k=14∂3aj∂w∂ηi∂ηkuηk2ηi2+124∂4aj∂w4(x+2μu)4+16∑i=14∂4aj∂w3∂ηi(x+2μu)3ηi2+14∑i,k=14∂4aj∂w2∂ηi∂ηk(x+2μu)2ηk2ηi2+16∑i,k,l=14∂4aj∂w∂ηi∂ηk∂ηl×(x+2μu)ηl2ηk2ηi2+124∑i,k,l,m=14∂4aj∂ηi∂ηk∂ηl∂ηm·ηm2ηl2ηk2ηi2;⋮

Expanding Fk,ϵ in ϵ, we obtain
(34)Fk,ϵ=∑j=0kaj(wϵ,ηϵi)ϵj=∑j=0k∑p=0∞bj,pϵj+p=c0+c1ϵ+⋯+ckϵk+O(ϵk+1),
where cp=∑j=0pbj,p-j are given as follows:
(35)c0=a0(w,ηi),(36)c1=∂a0∂w(x+2μu)+∑i=14∂a0∂ηiηi2+a1(w,ηi),(37)c2=∂a0∂wu+12∂2a0∂w2(x+2μu)2+∑i=14∂2a0∂w∂ηi(x+2μu)ηi2+12∑i,k=14∂2a0∂ηi∂ηkηk2ηi2+∂a1∂w(x+2μu)+∑i=14∂a1∂ηiηi2+a2(w,ηi),(38)c3=∂2a0∂w2(x+2μu)u+∑i=14∂2a0∂w∂ηiuηi2+16∂3a0∂w3(x+2μu)3+12∑i=14∂3a0∂w2∂ηi(x+2μu)2ηi2+12∑i,k=14∂3a0∂w∂ηi∂ηk(x+2μu)ηk2ηi2+16∑i,k,l=14∂3a0∂ηi∂ηk∂ηlηl2ηk2ηi2+∂a1∂wu+12∂2a1∂w2(x+2μu)2+∑i=14∂2a1∂w∂ηi(x+2μu)ηi2+12∑i,k=14∂2a1∂ηi∂ηkηk2ηi2+∂a2∂w(x+2μu)+∑i=14∂a2∂ηiηi2+a3(w,ηi),⋮

From the expressions, we know that cj’s are n×1 matrix functions of (w,ηi) and depend on w rationally.

For k≥1, let Tk,ϵ=Fk-1,ϵ, Pk,ϵ=Tk,ϵ(Tk,ϵ†Tk,ϵ)-1Tk,ϵ†. Define ψk,ϵ and ψ^k by induction as follows:
(39)ψ1,ϵ=ψμ+ϵ,P1,ϵ,ψ^1=limϵ→0ψ1,ϵ,ψk,ϵ=ψμ+ϵ,Pk,ϵ♯ψ^k-1,ψ^k=limϵ→0ψk,ϵ.
Let T~k,ϵ=ψ^k-1(μ+ϵ)Tk,ϵ, P~k,ϵ=T~k,ϵ(T~k,ϵ†T~k,ϵ)-1T~k,ϵ†; then ψk,ϵ=ψμ+ϵ,P~k,ϵψ^k-1, where implicit “⋆” products are still assumed between classical fields and their components.

Theorem 7.

Let a0,a1,… be a sequence of n×1 matrix functions of w and ηi and depend on w rationally. Let Tk,ϵ, Pk,ϵ, ψk,ϵ, ψ^k, and T~k,ϵ be defined as above. Then one has

T~k,ϵ=T^k+ϵSk,1+ϵ2Sk,2+⋯, where T^k=c0+∑j=1k-1(μ-μ-)jPk-1,jcj,Pl,j=∑l≥i1>⋯>ij≥1P^i1⊥⋯P^ij⊥, and P^i=T^i(T^i†T^i)-1T^i†;

ψ^k=ψμ,P^k⋯ψμ,P^1 is a multisoliton solution of (4) with only a pole at ζ=μ of order k;

the associated superfield is Φ^k=(P^1+(μ/μ-)P^1⊥)⋯(P^k+(μ/μ-)P^k⊥).

Proof.

We prove the theorem by induction on k. For k=1, the theorem is clearly true. Suppose the theorem is true for k, we will prove that (1)–(3) hold for k+1.

By Theorem 3 and induction hypothesis,
(40)T~k+1,ϵ=ψ^k(μ+ϵ)Tk+1,ϵ=(In+μ-μ-ϵP^k⊥)ψ^k-1(μ+ϵ)Fk,ϵ=(In+μ-μ-ϵP^k⊥)ψ^k-1(μ+ϵ)×(Fk-1,ϵ+ak(wϵ,ηϵi)ϵk)=(In+μ-μ-ϵP^k⊥)×(ψ^k-1(μ+ϵ)Tk,ϵ+ψ^k-1(μ+ϵ)ak(wϵ,ηϵi)ϵk)=(In+μ-μ-ϵP^k⊥)(T~k,ϵ+(In+μ-μ-ϵP^k-1⊥)⋯(In+μ-μ-ϵP^1⊥)×ak(wϵ,ηϵi)ϵkT~k,ϵ+(In+μ-μ-ϵP^k-1⊥))=(In+μ-μ-ϵP^k⊥)(T^k+ϵSk,1+ϵ(μ-μ-)k-1×Pk-1,k-1ak(w,ηi)+O(ϵ2)T^k+ϵSk,1+ϵ(μ-μ-)k-1)=T^k+(μ-μ-)P^k⊥(Sk,1+(μ-μ-)k-1×Pk-1,k-1ak(w,ηi)(μ-μ-)k-1)+O(ϵ).

In the last step we have used P^k⊥T^k=0. Thus all terms of negative powers of ϵ vanish in the Laurent series expansion of T~k+1,ϵ in ϵ; hence
(41)T~k+1,ϵ=ψ^k(μ+ϵ)Fk,ϵ=(In+μ-μ-ϵP^k⊥)⋯(In+μ-μ-ϵP^1⊥)×(c0+c1ϵ+⋯+ckϵk+O(ϵk+1))=(In+μ-μ-ϵPk,1+⋯+(μ-μ-)kϵkPk,k)×(c0+c1ϵ+⋯+ckϵk+O(ϵk+1))=c0+(μ-μ-)Pk,1c1+⋯+(μ-μ-)kPk,kck+O(ϵ).
Therefore we obtain
(42)T^k+1=limϵ→0T~k+1,ϵ=c0+∑j=1k(μ-μ-)jPk,jcj.

By the inductive definition before this theorem,
(43)ψ^k+1=limϵ→0ψμ+ϵ,Pk+1,ϵ♯ψ^k=limϵ→0ψμ+ϵ,P~k+1,ϵψ^k=ψμ,P^k+1ψ^k=ψμ,P^k+1ψμ,P^k⋯ψμ,P^1=(In+μ-μ-ζ-μP^k+1⊥)⋯(In+μ-μ-ζ-μP^1⊥)=In+∑j=1k+1(μ-μ-)j(ζ-μ)jPk+1,j.

By Theorem 3, ψμ+ϵ,Pk+1,ϵ♯ψ^k is a solution of (4) for small |ϵ|>0. By continuity, so is ψ^k+1. The coefficient of (ζ-μ)-k-1 of ψ^k+1 is (μ-μ-)k+1Pk+1,k+1. To show that ψ^k+1 has a pole at ζ=μ of multiplicity k+1, it suffices to show that
(44)Pk+1,k+1=P^k+1⊥P^k⊥⋯P^1⊥≠0.
As in the nonsupersymmetric scenario, we choose the operator formalism to prove (44). So now the Hermitian projections P^i,i=1,…,k+1,
(45)Pk+1,k+1=P^k+1⊥P^k⊥⋯P^1⊥,Pl,j=∑l≥i1>⋯>ij≥1P^i1⊥⋯P^ij⊥
(under the Moyal-Weyl map) are U(n)-valued matrices, and the products in (45) are the ordinary products instead of star products. We write Pk,j=Pk-1,j+P^k⊥Pk-1,j-1. So (42) for k+1 can be written as
(46)T^k+1=c0+∑j=1k-1(μ-μ-)jPk-1,jcj+P^k⊥∑j=0k-1(μ-μ-)j+1Pk-1,jcj+1=T^k+(μ-μ-)P^k⊥∑j=0k-1(μ-μ-)jPk-1,jcj+1.
By the induction hypothesis, T^k≠0, then the above formula implies that
(47)ImP^k+1∩ImP^k⊥=0.
If (44) does not hold, that is, P^k+1⊥P^k⊥⋯P^1⊥=0, then P^k⊥⋯P^1⊥⊂ImP^k+1, but P^k⊥⋯P^1⊥≠0 by induction hypothesis, and clearly P^k⊥⋯P^1⊥⊂ImP^k⊥, so we have
(48)0≠P^k⊥⋯P^1⊥⊂ImP^k+1∩ImP^k⊥,
which contradicts (47). Therefore (2) holds for k+1.

The expression for the associated superfield Φ^k+1 is straightforward; that is
(49)Φ^k+1=ψ^k+1-1(ζ=0)=(P^1+μμ-P^1⊥)⋯(P^k+μμ-P^k⊥)(P^k+1+μμ-P^k+1⊥).

The proof is completed.

Note that all P^j’s are of rank one in the above construction. But the same limiting method also produces multisoliton solutions of the form ψμ,P^k⋯ψμ,P^1 with rank(P^1)≥⋯≥rank(P^k). Similar computations give the construction of multisoliton solutions with pole data (μ,k) and arbitrary rank data (n1,…,nk).

Like the bosonic case, we can also associate to each multisoliton solution of (4) with pole data (μ,k) a generalized algebraic BT. Using these generalized BTs, we can construct multisoliton solutions with general pole data (μ1,…,μk,n1,…,nk).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Key Project of Jiangsu Second Normal University (JSNU-Z-4469).

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