Noncommutative Multisoliton Solutions of a Supersymmetric Chiral Model
Multisoliton configurations of a superextended and Moyal-type noncommutative deformed modified 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.
A generalization of the modified chiral model  (also called the Ward model in ) with supersymmetries and a Moyal deformation of this model is introduced in . Since the -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  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 .
In a recent paper , we extended the algebraic Bäcklund transformations (BTs) and the order limiting method of Dai and Terng  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 -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  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.
2. Noncommutative -Extended Ward Model
To formulate the noncommutative -extended Ward model, we need to introduce the following notations [3, 8].(1) with coordinates and the metric .(2)Grassmann variables (see ): some mathematical objects obeying the following anticommutation rules for any , . In particular, when , we have . So any function of the Grassmann variables is a finite polynomial .(3)The antichiral superspace with coordinates for and ; here are called fermionic coordinates, which are Grassmann variables. The tensor fields on are called superfields.(4)The noncommutative star product on the antichiral superspace is Note that the time coordinate remains commutative and no derivatives with respect to the Grassmann variables .
With the notations listed above, we can now state the definition of this model.
Definition 1 (see [3, 8]). The -valued superfield of the noncommutative -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.
By , model (2) is the compatibility condition for the following linear system of differential equations involving a spectral parameter : where the matrix depends on and the matrices , , and are superfields on independent of the spectral parameter . Moreover, is subject to the following reality condition: Then the superfield 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, , where and 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.
3. -Extended Multisoliton Configurations with Simple Pole Data
Dressing method is employed in  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 with by left-multiplying an simple pole solution with a single pole factor of the form where the matrix is a Hermitian projection of rank ; that is, and ; this is obtained by reality condition (5), and therefore one can decompose , where is an matrix depending on and and the columns of span the image of . Therefore the iteration yields the multiplicative ansatz which, via a partial fraction decomposition, may be written in the additive form where and are some matrices depending on and .
It was shown in  that is the solution of the linear system (6) if with and for . The associated superfield is Thus, the solutions of (2) described by the simple pole ansatz are parametrized by the set of matrix-valued functions of and and by the pole position .
Example 2. One-soliton configuration with , , and , is an matrix, where This configuration will describe a moving soliton if the matrix depends on rationally [3, 8].
We now change the form of the one-soliton solution in (12) into the following one: where and thus built from the matrix ; then . Since , we know ; hence and . Thus implies . Similarly, ; hence . If we suppose 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).
Theorem 3. Let be a solution of (4) and the associated superfield; choose such that is holomorphic and nondegenerate at . Let be a solution of (4) with a simple pole at , where is the Hermitia projection built from an matrix depending on rationally. Let be the Hermitian projection built from the matrix . Then(1) is holomorphic and nondegenerate at ;(2) is a new solution to the linear system (4) with some new superfields , , and , and the new associated superfield is .
Proof. (1) The residues of the right hand side of at and are
respectively, and both equal zero by the definition of . Thus we have two factorizations of .
(2) It suffices to show that is independent of the spectral parameter . Using , we have Since is constant in by assumption, (18) is holomorphic at . On the other hand, using , we have Since is a solution of (4), is independent of . Together with (1), we deduce that (19) is holomorphic at . Thus is holomorphic on and hence independent of by Liouville's theorem. Likewise and are also independent of .
The associated superfield is , and then the proof is completed.
Let denote the -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 , 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 , be two distinct complex numbers and , and let be an matrix function depending on rationally, where Let ; then and are one-soliton solutions and is a two-soliton solution of (4) with two simple poles at and , where is the Hermitian projection built from the matrix . The associated superfield is .
4. -Extended Multisoliton Configurations with Higher-Order Pole Data
A solution of (6) with a double pole at is constructed in  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 is yet to be determined. Demanding that is again a solution of (6) with some new superfields , , and , which are independent of , they obtained the following equations: After constructing a projection via a solution of (22), and the associated superfield are derived.
Their construction of the solution is inspired by the known form of in the bosonic case [4, 5], that is, making the ansatz 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 higher-order 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 , , be two holomorphic functions of and depend on the bosonic variable rationally: where and , and , , for any with small, , and . The Taylor expansion of in is where is right derivative of with respect to the Grassmann variable . Note that the products in the expansion are the ordinary products and that we omit the variables of and for simplicity. Clearly, has similar expansion; therefore Then Let 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 and .
Let be a constant and a sequence of matrix functions depending on rationally. For any with small, let , , , and , for . Here we only consider the case of since it is easily truncated to any smaller even number of supersymmetries (see ); thus Let for . The Taylor expansion of at is where the sums are taken over different indices when there are more than one index and the partial derivatives of with respect to the Grassmann variables are given by the following remark.
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 , represent that the sums are taken over the permutation group of and , respectively.
Substituting (29) into the Taylor expansion of , we get the Taylor expansion of in . Assume that , then
Expanding in , we obtain where are given as follows:
From the expressions, we know that ’s are matrix functions of and depend on rationally.
For , let , . Define and by induction as follows: Let , ; then , where implicit “” products are still assumed between classical fields and their components.
Theorem 7. Let be a sequence of matrix functions of and and depend on rationally. Let , , , , and be defined as above. Then one has(1), where ,, and ;(2) is a multisoliton solution of (4) with only a pole at of order ;(3)the associated superfield is .
Proof. We prove the theorem by induction on . For , the theorem is clearly true. Suppose the theorem is true for , we will prove that (1)–(3) hold for .(1)By Theorem 3 and induction hypothesis, In the last step we have used . Thus all terms of negative powers of vanish in the Laurent series expansion of in ; hence Therefore we obtain (2)By the inductive definition before this theorem, By Theorem 3, is a solution of (4) for small . By continuity, so is . The coefficient of of is . To show that has a pole at of multiplicity , it suffices to show that As in the nonsupersymmetric scenario, we choose the operator formalism to prove (44). So now the Hermitian projections , (under the Moyal-Weyl map) are -valued matrices, and the products in (45) are the ordinary products instead of star products. We write . So (42) for can be written as By the induction hypothesis, , then the above formula implies that If (44) does not hold, that is, , then , but by induction hypothesis, and clearly , so we have which contradicts (47). Therefore (2) holds for .(3)The expression for the associated superfield is straightforward; that is The proof is completed.
Note that all ’s are of rank one in the above construction. But the same limiting method also produces multisoliton solutions of the form with . Similar computations give the construction of multisoliton solutions with pole data and arbitrary rank data .
Like the bosonic case, we can also associate to each multisoliton solution of (4) with pole data a generalized algebraic BT. Using these generalized BTs, we can construct multisoliton solutions with general pole data .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Key Project of Jiangsu Second Normal University (JSNU-Z-4469).
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