Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation

Two families of solutions of a generalized non-Abelian Toda lattice are considered. These solutions are expressed in terms of quasideterminants, constructed by means of Darboux and binary Darboux transformations. As an example of the application of these solutions, we consider the 2-periodic reduction to a matrix sine-Gordon equation. In particular, we investigate the interaction properties of polarized kink solutions.


Introduction
There has been great interest in non-commutative versions of some well-known soliton equations, such as the KP equation, the KdV equation and the Hirota-Miwa equation (Kupershmidt 2000;Paniak 2001;Hamanaka 2003;Hamanaka & Toda 2003;Wang & Wadati 2003a,b;Sakakibara 2004;Wang et al. 2004;Dimakis & Müller-Hoissen 2005;Nimmo 2006; . Often, these non-commutative versions are obtained simply by removing the assumption that the coefficients in the Lax pair of the commutative equation commute. The non-Abelian Toda lattice U n;x C U n V nC1 K V n U n Z 0; ð1:1Þ V n;t C U nK1 K U n Z 0 ð1:2Þ was first studied in Mikhailov (1979). A Darboux transformation for this system was given by Salle (1982). In Nimmo & Willox (1997), the following generalization: U n;x C U n V nC1 K V n U n Z 0; ð1:3Þ V n;t C a n U nK1 K U n a nC1 Z 0; ð1:4Þ was studied and the Darboux and binary Darboux transformations were obtained. We note that, in general, a n is not a scalar, but is independent of t. In the case that U n , V n and a n are scalars, it is easy to show, by setting a nC1 U n Z e Kq n and eliminating V n , that (1.3) and (1.4) become the standard two-dimensional Toda lattice equation q n;xt Ke Kq nK1 C 2e Kq n Ke Kq nC1 Z 0: ð1:5Þ Introducing new variables X n where U n Z X n X K1 nC1 ; V n Z X n;x X K1 n ; ð1:6Þ (1.3) and (1.4) can be rewritten as X n;x X K1 n À Á t C a n X nK1 X K1 n K X n X K1 nC1 a nC1 Z 0: ð1:7Þ From now on, we will refer to (1.7) as the non-Abelian Toda lattice. One type of quasideterminant solution of (1.7) was found in Etingof et al. (1997). We will show how these (quasi-wronskian) solutions arise from the Darboux transformation and consider a second type of quasideterminant, which we call quasi-grammian, solutions obtained using the binary Darboux transformation. It is well known that the 2-periodic reduction of the standard two-dimensional Toda lattice leads to the scalar sine-Gordon equation. In the same way, the 2-periodic reduction of the non-Abelian Toda lattice (1.7) leads to a noncommutative sine-Gordon equation. This equation has been studied already in a number of papers (Andreev 1990;Etingof et al. 1997;Cabrera-Carnero & Moriconi 2003;Grisaru & Penati 2003;Grisaru et al. 2004;Zuevsky 2004;Lechtenfeld et al. 2005;Hamanaka 2006) concerning both the matrix and the Moyal product versions. Here we consider only the matrix version in detail.
Recently, a matrix KdV equation was considered in Goncharenko (2001). A multisoliton solution was found by using the inverse scattering method. In particular, the properties of one-and two-soliton solutions expressed in terms of projection matrices were investigated. We will apply some of these ideas to the matrix sine-Gordon equation to study the interaction of its kink solutions.
The paper is organized as follows. In §2, some properties of quasideterminants used in the paper are described. In §3, we present quasi-wronskian solutions to the non-Abelian Toda lattice constructed by iterating Darboux transformations and in §4, we present quasi-grammian solutions to the system using the related binary Darboux transformation. In the rest of the paper, we consider the 2-periodic reduction to a matrix sine-Gordon equation. In particular, we consider the matrix kink solutions obtained from the quasi-grammian solutions, we show that kink solutions for the matrix sine-Gordon equation emerge intact from interaction apart from change of polarization and phase.

Preliminaries
In this short section we recall some of the key elementary properties of quasideterminants. The reader is referred to the original papers (Gelfand & Retakh 1991;Etingof et al. 1997;Gelfand et al. 2005) for a more detailed and general treatment.

(a ) Quasideterminants
An n!n matrix A over a ring R (non-commutative, in general) has n 2 quasideterminants written as jAj i, j for i, jZ1, ., n, which are also elements of R. They are defined recursively by . ;n : ð2:1Þ In the above equation, r j i represents the i th row of A with the jth element removed; c i j is the jth column with the i th element removed; and A i, j the submatrix obtained by removing the ith row and the j th column from A. Quasideterminants can be also denoted as shown below by boxing the entry about which the expansion is made The case nZ1 is rather trivial; let AZ(a), say, and then there is one quasideterminant jAj 1;1 Z j a jZ a. For nZ2, let then there are four quasideterminants Note that if the entries in A commute, the above becomes the familiar formula for the inverse of a 2!2 matrix with entries expressed as ratios of determinants. Indeed, this is true for any size of square matrix; if the entries in A commute then jAj i;j Z ðK1Þ iCj detðAÞ detðA i;j Þ : ð2:2Þ In this paper, we will consider only quasideterminants that are expanded about a term in the last column, most usually the last entry. For a block matrix where d2R, A is a square matrix over R of arbitrary size and B, C are column and row vectors over R of compatible lengths, respectively, we have (b ) Non-commutative Jacobi identity There is a quasideterminant version of Jacobi's identity for determinants, called the non-commutative Sylvester's theorem by Gelfand & Retakh (1991). The simplest version of this identity is given by As a direct result, we have the homological relation Given an (nCk)!n matrix A, denote the ith row of A by A i , the submatrix of A having rows with indices in a subset I of {1, 2, ., nCk} by A I and A {i }/ by Aî. Given i; j 2 f1; 2; .; nC kg and I such that #IZnK1 and j;I, one defines the (right) quasi-Plücker coordinates for any column index s2{1, ., n}. The final equality in (2.5) comes from an identity of the form (2.3) and proves that the definition is independent of the choice of s.
Remark 2.1. A useful consequence of (2.5) is the identity that shows that quasideterminants of this form may be inverted very simply.

Solutions obtained by Darboux transformations
The non-Abelian Toda lattice (1.3) and (1.4) has Lax pair f n;x Z V n f n C a n f nK1 ; ð3:1Þ Let q n;i ; iZ 1; :::; N be a particular set of eigenfunctions of the linear system and introduce the notation Q n Z ðq n;1 ; .; q n;N Þ. The Darboux transformation, determined by particular solution q n , for the non-Abelian Toda lattice is This may be iterated by defining q n ½k Z f n ½kj f n /q n;k : ð3:9Þ In particular, In what follows, we will show by induction that the results of N-repeated Darboux transformation f n [NC1] and X n [NC1] can be expressed as in closed form as quasideterminants The initial case NZ1 follows directly from (3.10) and (3.11). Also using the non-commutative Jacobi identity (2.4) and the homological relation and then using the quasi-Plucker coordinate formula (2.5), we get

Solutions obtained by binary Darboux transformation
The linear equations (3.1) and (3.2) have the formal adjoints Following the standard construction of a binary Darboux transformation, one introduces a potential U n Z Uðf n ; j n Þ satisfying the three conditions:

5Þ
A binary Darboux transformation is then defined by f n ½N C 1 Z f n ½N K q n ½N Uðq n ½N ; r n ½N Þ K1 Uðf n ½N ; r n ½N Þ; ð4:6Þ where f n [1]Zf n , j n [1]Zj n , X n [1]ZX n and q n ½N Z f n ½N j f n /q n;N ; r n ½N Z j n ½N j j n /r n;N : ð4:9Þ Using the notation Q n Z ðq n;1 ; .; q n;N Þ and P n Z ðr n;1 ; .; r n;N Þ, it is easy to prove by induction that for NR1, f n ½N C 1 Z UðQ n ; P n Þ Uðf n ; P n Þ Q n f n ; ð4:10Þ and Uðf n ½N C 1; j n ½N C 1Þ Z UðQ n ; P n Þ Uðf n ; P n Þ UðQ n ; j n Þ Uðf n ; j n Þ : ð4:12Þ We may thus after N binary Darboux trasnformations obtain X n ½N C 1 ZK UðQ n ; P n Þ P † n Q n KI In fact, we can prove the above results by induction.

Matrix sine-Gordon equation and its kink solutions
It is well known in the commutative case that one may obtain reductions by imposing periodic conditions on the q n . Similarly in the non-Abelian case, one can make periodic reductions of (1.7). From now on, we only consider the case that X n is a d!d matrix and a n Z I d!d and so (1.7) is The simplest such reduction has period 2, that is, we take X n ZX nC2 and (5.1) gives the system We call this a non-Abelian sinh-Gordon equation since in the commutative case, it will be seen that X 0 Z X K1 1 Z F 1 =F 0 and then qZ 2 logðF 1 =F 0 Þ satisfies the standard sinh-Gordon equation q xt Z 4 sinh q: By changing q/iq, we can also obtain the sine-Gordon equation In what follows, we will construct solutions to (5.2) and (5.3) by the reduction of the solutions (4.13) of the non-Abelian Toda lattice (5.1). It is clear that (5.1) has vacuum solution X n ZI and (4.13) gives the quasi-grammian solutions X n ZK UðQ n ; P n Þ P T n Q n KI ; ð5:4Þ where q n,i and r n,i satisfy ðq n Þ x Z q nK1 ; ðq n Þ t Z q nC1 ; ðr n Þ x ZKr nC1 ; ðr n Þ t ZKr nK1 ð5:5Þ and U is defined by (4.3)-(4.5). We choose the simplest non-trivial solutions of (5.5) where A i and B j are d!d matrices and then we obtain The choice of constant of integration as d i, j I is needed to effect the periodic reduction that will be made shortly. This can also be written as Uðq n;j ; r n;i Þ Z p i q j Now using the invariance of a quasideterminant to scaling of its rows and columns (Gelfand et al. 2005), we get It is obvious from this expression for X n that it is 2-periodic when ðq 1 =p 1 Þ 2 Z/Zðq N =p N Þ 2 Z1, i.e. p i ZKq i Zl i for i Z1; .;N . Therefore, the non-Abelian sinh-Gordon equation has the solutions ð5:7Þ From now on, we will assume that A i ZI are real and B j are pure imaginary and written as ir j P j , where r j are real scalars. In this case, it follows that X 0 and X 1 are complex conjugate to one another. For this reason we introduce ð5:8Þ Next, we will derive matrix kink solutions for the matrix sine-Gordon equation using the method applied to study the soliton solutions of the matrix KdV equation in Goncharenko (2001). To get a visual representation of the solution we will consider the matrix W ðx; tÞ defined by

9Þ
We choose this dependent variable so that in the scalar case W Z q, the solution of the sine-Gordon equation. For NZ1, (5.8) gives We first assume further that P is a projection matrix (i.e. satisfies P 2 Z P). This choice allows us to calculate the inverse matrices in the above expression explicitly using the formula where a s1 is a scalar and P is any projection matrix.
In this way, we find that and hence where fZ logðr=2Þ=2l. Note also that X X Z I . Taking one final step, we integrate to obtain the one-kink solution to the matrix sine-Gordon equation KfÞÞÞ: ð5:12Þ Remark 5.1. For the one-kink solution (5.12), we call the projection matrix P as its polarization and f its phase. In the scalar case, if we choose PZ1, (5.12) is simply the one-kink solution to the standard sine-Gordon equation.
For NZ2, expanding X by the definition (2.1), we can rewrite X as In the expressions B j Z ir j P j , jZ1, 2, we assume that P j are the rank-1 projection matrices and the d-vectors p j and q j satisfy the condition ðp j ; q j Þ s0, we can solve for L 1 and L 2 by using (5.11) to obtain where g Z ðl 1 C l 2 Þ 2 g 1 g 2 C l 1 l 2 r 1 r 2 a; a Z ðp 1 ; q 2 Þðp 2 ; q 1 Þ ðp 1 ; q 1 Þðp 2 ; q 2 Þ and g j Z expð2l j q j ÞK ir j 2 for j Z 1; 2; q j Z x C 1 l 2 j t: Therefore, We now investigate the behaviour of X as t /GN. We will use the fact that W is invariant under the transformation X / XC for any constant matrix C and assume, without loss of generality, that 0! l 1 ! l 2 . In the calculations that follow, we will demonstrate that kinks emerge from the interaction and undergo phase shifts as in the scalar case. In addition, however, we will see that there are changes of polarization, in other words, amplitudes may also change as a result of the interaction.
First we fix q 1 . Then q 2 Z q 1 C ð1=l 2 2 K 1=l 2 1 Þt and hence as t/KN, The above calculations show that W ðx; tÞ decomposes into the sum of two kink solutions as t/GN and the jth kink solution propagates with the speed 1=l 2 j . The phase shifts D j Z f C j Kf K j for the kink solutions are where b Z 1K 4l 1 l 2 a ðl 1 C l 2 Þ 2 : Remark 5.2. In a similar way to the matrix KdV equation in Goncharenko (2001), we find that the matrix amplitude of the first kink solution changes from 4P 1 to 4P 1 and the matrix amplitude of the other one changes from 4P 2 to 4P 2 as t changes from KN to CN. If ðp 1 ; q 2 ÞZ 0 (P 2 P 1 Z 0) or ðp 2 ; q 1 ÞZ 0 (P 1 P 2 Z 0), there is no phase shift; however, the amplitudes may change. In the case that both P 1 P 2 Z P 2 P 1 Z 0, there is neither phase shift nor change in amplitude and so the kink solutions have trivial interactions.
To illustrate the above, we will consider the case dZ2, i.e. the 2!2 matrix sine-Gordon equation. We choose l 1 Z 1; l 2 Z 2; r 1 Z r 2 Z 1 and The analysis above shows that P K 1 Z P 1 , P C 2 Z P 2 and For convenience, rather than plotting the kink W given by (5.9), we plot the derivative W x and refer to it as a soliton. In figure 1, the asymptotic forms of the matrix soliton 1 are plotted as t/GN. The first plot exhibits the amplitudes given by P K 1 and the second, those of P C 1 . Similarly, in figure 2, we show the same plots for soliton 2.

Conclusions
In this paper, we have considered a generalized non-Abelian Toda lattice and presented quasi-wronskian and quasi-grammian solutions obtained by means of Darboux transformations and binary Darboux transformations, respectively. Then we imposed a 2-periodic reduction on the non-Abelian Toda lattice to derive a non-commutative sine-Gordon equation. Using a method similar to that developed in Goncharenko (2001) for the matrix KdV equation, we obtained kink solutions for the matrix sine-Gordon equation from the quasi-grammian solutions of the non-Abelian Toda lattice. Then we investigated the interaction properties of matrix kink solutions. It is known (Veselov 2003) that the change of matrix amplitude of solitons for the matrix KdV equation gives rise to a Yang-Baxter map. It would be interesting to investigate whether there is a similar result for the matrix sine-Gordon equation.