## Abstract

In this paper, we construct Grammian-like quasideterminant solutions of a non-Abelian Hirota–Miwa equation. Through continuum limits of this non-Abelian Hirota–Miwa equation and its quasideterminant solutions, we construct a cascade of non-commutative differential-difference equations ending with the non-commutative KP equation. For each of these systems, the quasideterminant solutions are constructed as well.

## 1. Introduction

Recently, non-commutative versions of some well-known soliton equations have been extensively studied. These systems include the KP equation, the KdV equation, the Hirota–Miwa equation, the modified KP equation and the two-dimensional Toda lattice equation (Kupershmidt 2000; Paniak 2001; Hamanaka 2003; Hamanaka & Toda 2003; Wang & Wadati 2003*a*,*b*, 2004; Sakakibara 2004; Dimakis & Müller-Hoissen 2005; Nimmo 2006; Gilson & Nimmo 2007; Gilson *et al*. 2007, 2008; Li & Nimmo 2008). Generally speaking, determinants are ubiquitous as solutions of commutative integrable systems while quasideterminants are often solutions of non-commutative integrable systems (Gelfand & Retakh 1991; Etingof *et al*. 1998; Gelfand *et al*. 2005). In many cases, the non-commutative version of the equation is obtained as the compatibility of the same Lax pair for the commutative version. However, in the non-commutative case, the assumption that the coefficients in the Lax pair commute is relaxed.

This paper is concerned with the non-Abelian version of the Hirota–Miwa equation introduced in Nimmo (2006). Given the system of linear equations(1.1)where *ϕ* and *U*_{ij} belong to an associative algebra and depend on variables *n*_{1}, *n*_{2}, *n*_{3}, *T*_{i} denotes the shift operator in variable *n*_{i}, defined by *T*_{i}(*X*)=*X*(*n*_{i}+1) and *a*_{i} are non-zero scalar constants, called lattice parameters. We will also use the shorthand subscript notation *X*_{,i}=*T*_{i}(*X*). This linear system is compatible if and only if(1.2)(1.3)(1.4)for each *i*, *j*, *k*∈{1, 2, 3}. In particular, these give *U*_{ii}=0 and *U*_{ij}=−*U*_{ji} for *i*, *j*∈{1, 2, 3}. Equations (1.2)–(1.4) are referred to as the non-Abelian Hirota–Miwa equation (Nimmo 2006; Gilson *et al*. 2007) since in the commutative case if we make the ansatz(1.5)where *T*_{ij}(*τ*) denotes *T*_{i}(*T*_{j}(*τ*)), then all of the equations in (1.2)–(1.4) reduce to just one equation, the well-known Hirota–Miwa equation (Hirota 1982; Miwa 1982)(1.6)

In the commutative case, it is well known that the Miwa (1982) transformationprovides the link between the discrete variables *n*_{i} and the continuous variables *x*_{k} in the KP hierarchy. Using this transformation, we can view *τ*=*τ*(*n*_{1}, *n*_{2}, *n*_{3}, …; *x*_{1}, *x*_{2}, *x*_{3}, …) as a function of both discrete and continuous variables such that(1.7)and the continuum limits, as lattice parameters *a*_{i}→0, are obtained by using Taylor's theorem. In this way we have, for example,where for a shorter notation, we have written *x*_{1}, *x*_{2}, *x*_{3} as *x*, *y*, *t*, respectively.

Taking successive continuum limits, one obtains a cascade of differential-difference equations starting from the Hirota–Miwa equation and ending with the KP equation. To be precise, as *a*_{3}→0, the leading-order term in (1.6) is at *O*(*a*_{3}) and is(1.8)which we call the *Δ*^{2}∂KP equation. Next, as *a*_{2}→0, the leading-order term in (1.8) is at and is(1.9)which is nothing but the differential-difference KP equation. Here we denote (1.9) as the *Δ*∂^{2}KP equation to stress that it has one discrete variable and two continuous ones. Finally, as *a*_{1}→0, the leading-order term in (1.8) is at and is(1.10)the Hirota form of the KP equation. The whole procedure is illustrated in figure 1.

The main aim of this paper is to show that the same cascade of continuum limits also exists in the non-Abelian case and thereby we construct non-commutative versions of the differential-difference KP equations (1.8) and (1.9). We also give solutions of these systems in terms of quasideterminants.

The paper is organized as follows. In §2, we review results from Nimmo (2006) on the Darboux transformations for the non-Abelian Hirota–Miwa equation and consider the binary Darboux transformation and the quasi-Grammian solutions of (1.2)–(1.4), which it constructs. Using the ansatz that is suggested by this binary Darboux transformation, in §3, we first reformulate the results presented in a way that facilitates taking continuum limits. Then, by taking the successive continuum limits described above, we first obtain a non-commutative version of (1.8). Quasi-Wronskian solutions and quasi-Grammian solutions for this system are also obtained. Then a second continuum limit yields a non-commutative version of (1.9) and its quasi-Wronskian and quasi-Grammian solutions are presented. Finally, we show that the continuum limit of this non-commutative differential-difference KP is nothing but the non-commutative KP equation. Conclusions and discussions are given in §4.

## 2. The non-Abelian Hirota–Miwa equation

In this section, we will present quasi-Grammian solutions to the non-Abelian Hirota–Miwa equation (1.2)–(1.4) constructed by using binary Darboux transformations.

Let us first review some results on the non-Abelian Hirota–Miwa equation. Quasi-Casoratian solutions were constructed for the non-Abelian Hirota–Miwa equation by repeating Darboux transformations in Nimmo (2006) and Gilson *et al*. (2007). The linear system (1.1) considered here is a generalization of the one used in Nimmo (2006) and Gilson *et al*. (2007), but it is not difficult to prove that (1.2)–(1.4) has the Darboux transformation(2.1)(2.2)where *θ* is a particular solution of the linear system (1.1). This suggests the ansatz , where is a seed solution. Then iterating the Darboux transformation gives Casoratian-type quasideterminant solutions(2.3)where ** θ**=(

*θ*

_{1}, …,

*θ*

_{n}) is a vector solution of (1.1). In §3, we will rewrite this Darboux transformation using a different parametrization and obtain a different Casoratian-type solution.

To construct binary Darboux transformations for (1.2)–(1.4), we must define an appropriate adjoint system. The formal adjoint of the linear system (1.1) has the formand we will choose *V*_{ij} in order that the compatibility condition of this adjoint system is equivalent to (1.2)–(1.4). The compatibility condition for this system of linear equations can be deduced from (1.2)–(1.4) by simply changing *T*_{i} to and *U*_{ij} to *V*_{ij} in this system and we get(2.4)(2.5)(2.6)This is equivalent to (1.2)–(1.4) when , where ^{*} is an involutive anti-automorphism defined on satisfying (*AB*)^{*}=*B*^{*}*A*^{*} and (*A*^{*})^{*}=*A*. Thus we obtain the adjoint linear system in its final form(2.7)

An eigenfunction potential *Ω*(*ϕ*, *ψ*) may be defined by(2.8)for all *i*, where . It is straightforward to show that the exactness condition (*Ω*_{,i}),_{j}=(*Ω*,_{j}),_{i} when *ϕ* and *ψ* satisfy (1.1) and (2.7), respectively. Following the standard construction, a binary Darboux transformation is then defined by(2.9)(2.10)(2.11)where *θ* and *ρ* are particular solutions of the linear system (1.1) and its adjoint (2.7).

There is also a ‘vector’ form of this transformation, equivalent to an iterated transformation that gives solutions(2.12)where is a seed solution of (1.2)–(1.4); ** θ** is as defined above; and

*P*=(

*ρ*

_{1}, …,

*ρ*

_{n}) is a vector solution of the adjoint linear problem (2.7) with potential . This solution may also be expressed as a quasideterminant(2.13)

This result can be proved by induction, but here we choose to verify the solution by direct substitution into (1.2)–(1.4). We write(2.14)where(2.15)Observe that (1.2) and (1.3) are identically satisfied by the ansatz (2.14), irrespective of the form of *G*, and all non-trivial versions of (1.4), in which *i*, *j* and *k* are distinct, give the same equation(2.16)Using (2.8), we obtain for any *i*,*j*,*k*Then substituting these expressions into (2.16), and making use of the linear equations (1.1) and (2.7), it is straightforward to show that (2.16) is identically satisfied. This completes the verification of the solutions (2.13).

## 3. Continuum limits of the non-Abelian Hirota–Miwa equation

In this section, we will find the proper continuum limits of the non-Abelian Hirota–Miwa equation as the lattice parameters tend to zero. As well as constructing the nonlinear equations, the corresponding linear equations (Lax pairs), Darboux and binary Darboux transformations and solutions will be obtained also in the same limit.

The first step is to reformulate the results for the non-Abelian Hirota–Miwa equation in a way that facilitates these continuum limits. In particular, we re-express many terms using the forward and backward difference operators and . We make an ansatz of the form (2.14) suggested by the binary Darboux transformation, in which is any seed solution of (1.2)–(1.4). In fact, in order to successfully balance leading-order terms, it turns out to be necessary to choose the seed solution in the form . Thus we write the solution of (1.2)–(1.4) as(3.1)Under this assumption, the system (1.2)–(1.4) reduces to the single equation(3.2)and its linear system and adjoint linear system may be written as(3.3)and(3.4)respectively.

For the purpose of taking continuum limits, it is more convenient to reformulate the Darboux transformation (2.1)–(2.2) using the parametrization (2.14) given by the binary Darboux transformation. Doing this, we find that(3.5)(3.6)Superficially, these expressions seem strange as the r.h.s. depends on index *i* but the l.h.s. does not. It is, however, easy to check using the linear problem (3.3) that the r.h.s. in each case is independent of the choice of *i*. Iterating the Darboux transformation in this form again gives quasi-Casoratian solutions but different from the ones given in (2.3),(3.7)where ** θ** is as in (2.3). Again, it may be shown that the r.h.s. of (3.7) is independent of the choice of

*i*.

### (a) Non-commutative *Δ*^{2}∂KP equation

To take the first continuum limit, let *a*_{3}→0 and note that, for example, *Δ*_{3}*G*=*G*_{x}+*O*(*a*_{3}).

*Nonlinear equation*. The leading-order terms in (3.2) give(3.8)which we call the non-commutative *Δ*^{2}∂KP equation.

*Lax pair*. The leading-order terms in (3.3) give the Lax pair(3.9)(3.10)Similarly, the leading-order terms in (3.4) give the adjoint form(3.11)(3.12)The compatibility conditions of both (3.9)–(3.10) and (3.11)–(3.12) are (3.8).

*Darboux transformation*. Similarly from (3.5)–(3.6), we get the Darboux transformation for (3.9)–(3.10), either expressed in terms of the *x*-derivative or difference operators *Δ*_{i}(3.13)(3.14)for *i*=1, 2.

*Binary Darboux transformation*. It is clear that the binary Darboux transformation always keeps the same form as (2.9)–(2.11) whatever the continuum limits are. The only difference is the definition of the eigenfunction potential *Ω*. Here and hereafter, we will give only the definition of *Ω*. The leading-order terms in (2.8) give the eigenfunction potential(3.15)

(3.16)

*Solutions*. From the continuum limit of the quasi-Casoratian solutions (3.7) for *i*=3, we get an alternative quasi-Wronskian expression and so the Darboux transformations give solutions(3.17)for *i*=1, 2. The quasi-Grammian solutions are given by (2.15) with *Ω* defined by (3.15)–(3.16).

### (b) Non-commutative *Δ*∂^{2}KP equation

To take the second continuum limit, let *a*_{2}→0 and note that, for example, and .

*Nonlinear equation*. The leading-order terms in (3.8) give the non-commutative differential-difference version of the potential KP equation(3.18)

*Lax pair*. The leading-order terms in (3.9) and (3.10) give the Lax pair(3.19)(3.20)The leading-order terms in (3.11)–(3.12) give the adjoint Lax pair(3.21)(3.22)The compatibility conditions of both (3.19)–(3.20) and (3.21)–(3.22) are (3.18).

*Darboux transformation*. Similarly, from (3.13)–(3.14), we get the Darboux transformation for (3.19)–(3.20), either expressed in terms of *x*-derivative or the difference operator *Δ*_{1}(3.23)

(3.24)

*Binary Darboux transformation*. Considering the expansion of (3.16) for *i*=2 as *a*_{2}→0 givesFrom this, it follows that and so the eigenfunction potential *Ω* satisfies(3.25)

(3.26)

(3.27)

*Solutions*. The Darboux transformations give solutions(3.28)and the quasi-Grammian solutions are given by (2.15) with *Ω* defined by (3.25)–(3.27).

### (c) Non-commutative KP equation

To take the final continuum limit, let *a*_{1}→0 and note that here and .

*Nonlinear equation*. The leading-order terms in (3.18) give the non-commutative potential KP equation(3.29)

*Lax pair*. The leading-order terms in (3.19)–(3.20) give the Lax pair(3.30)(3.31)Similarly, the leading-order terms in (3.21)–(3.22) give the adjoint form(3.32)(3.33)The compatibility conditions of both (3.30)–(3.31) and (3.32)–(3.33) are (3.29).

*Darboux transformation*. The leading orders in (3.23)–(3.24) give the Darboux transformation for (3.29)(3.34)(3.35)

*Binary Darboux transformation*. Considering the expansion of (3.16) for *i*=2 as *a*_{1}→0 givesFrom this, it follows that and so the eigenfunction potential *Ω* satisfies(3.36)

(3.37)

(3.38)

In Gilson & Nimmo (2007), the following non-commutative KP equation is considered:(3.39)Its quasideterminant solutions are obtained through Darboux transformation and binary Darboux transformation. In fact, using the scaling transformations *y*→−*y*, *t*→−4*t* and writing in terms of *U*=−2*G*, (3.29) can be transformed into (3.39).

*Solutions*. Finally, we recover the quasideterminant solutions of the non-commutative KP equation, either quasi-Wronskian (Etingof *et al*. 1998; Gilson & Nimmo 2007) or quasi-Grammian (Gilson & Nimmo 2007)

## 4. Conclusions and discussion

In this paper, we first obtained the quasi-Grammian solutions of the non-Abelian Hirota–Miwa equation by using an iterated binary Darboux transformation. The binary Darboux transformation leads to an ansatz (2.14) for *U*_{ij,} which reduces the non-Abelian Hirota–Miwa equations to a single equation symmetric in the three discrete variables *n*_{i}. Then by considering the continuum limits of this equation and its Darboux transformations and binary Darboux transformations, we obtain a cascade of non-commutative differential-difference equations and their quasideterminant solutions.

There is a second parametrization, , which can be seen from the quasi-Wronskian solution constructed by its Darboux transformation. A natural question is ‘Can we obtain continuum limits of the non-Abelian Hirota–Miwa equation under the second parametrization?’

Using this parametrization, the linear equations areand in the limit *a*_{3}→0, this system becomesThe algebraic compatibility of these equations gives(4.1)and it may be shown also that the only other compatibility condition, *ϕ*_{,12}=*ϕ*_{,21,} is identically satisfied provided (4.1) is satisfied. This, as expected, agrees with the leading-order terms in the continuum limit of *U*_{12}+*U*_{23}−*U*_{13}=0.

Now let and , then (4.1) can be rewritten as(4.2)On the other hand, exactly the same equation arises if we write *u*=−*a*_{1}*Δ*_{1}(*G*) and *v*=−*a*_{2}*Δ*_{2}(*G*) in the non-commutative *Δ*^{2}∂KP equation (3.8). Thus we see that, in the first continuum limit, the second parametrization gives a system equivalent to that obtained from the first. However, it is not at all clear how to take the second continuum limit, *a*_{2}→0. For this reason, we have not pursued the second parametrization any further.

## Acknowledgments

This work was supported in part by a Royal Society China fellowship and the National Natural Science Foundation of China (grant no. 10601028). K.M.T. wishes to acknowledge the NBHM for the project and J.J.C.N. wishes to acknowledge the support of a Royal Society of London International Outgoing Short Visit award.

## Footnotes

- Received October 29, 2008.
- Accepted January 15, 2009.

- © 2009 The Royal Society