Self-Consistent Sources for Integrable Equations via Deformations of Binary Darboux Transformations

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the sine-Gordon, nonlinear Schrodinger, KdV, Boussinesq, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP systems. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.


Introduction
The study of soliton equations with self-consistent sources has been pursued in particular in the work of Mel'nikov [1][2][3][4][5][6]. Mathematically, such systems of equations arise via a multiscaling limit of familiar integrable systems (see, e.g., [5,7]), or via a symmetry constraint imposed on a higher than two-dimensional integrable system (see [8,9], and also, e.g., [10][11][12][13] for related work). Several of these systems appeared, independently from those more mathematical explorations, in various physical contexts. For example, the nonlinear Schrödinger (NLS) equation with self-consistent sources describes the nonlinear interaction of an electrostatic high-frequency wave with the ion acoustic wave in a plasma (cold ions, warm electrons) in a small amplitude limit [14,15]. In nonlinear optics it describes the interaction of self-induced transparency and NLS solitons [16,17]. By now quite a number of publications have been devoted to such equations.
In this work, we show that self-consistent source extensions arise via a simple deformation of the "potential" that appears in the binary Darboux transformation method (see, e.g., [18]). Moreover, the present work provides a more universal approach to such extensions, generalizations to matrix versions of such equations, and a corresponding solution generating method. This is achieved in the framework of bidifferential calculus [19][20][21].
In Section 2 we consider the example of the potential KP (pKP) equation with self-consistent sources. The underlying structure is then abstracted to bidifferential calculus in Section 3. The resulting system allows to generate self-consistent source extensions of other integrable equations and supplies them with a solution-generating method. Examples are treated in the further sections. In Appendix A, we apply the aforementioned deformation to the matrix pKP hierarchy. The first non-trivial member then turns out to be a (2+1)-dimensional version [1] of the Yajima-Oikawa system [22]. Finally, Section 11 contains some concluding remarks.
Remark 2.1. An attempt to iterate this procedure by using φ, q and r instead of φ 0 , θ and η, leads back to the φ 0 we started with to generate φ. Iteration of the binary Darboux transformation method involves the construction of still other solutions of the linear systems. But we will not need these further steps here, since we consider a vectorial binary Darboux transformation, so that there is no need for iteration.
Next we look for choices of ω, such that Ω can be eliminated in half of the equations (2.7) and (2.8). This requires ω x = 0 and leaves us with the choices ω y = 0 or ω t = 0. In the first case, we only keep the equations for q y and r y . In the second case, we only keep those for q t and r t . This results in the two versions of pKP with self-consistent sources that appeared in the literature. (2.10) In terms ofq with a suitable choice of Q(t) and R(t), this system can be written as where ω t has been absorbed. The latter is rather what we should call a system with selfconsistent sources. Its scalar version has been studied in [1,4,7,[23][24][25][26][27]. The "noncommutative" generalization (2.12) already appeared in [28]. In our framework, the modification (2.10) is important. By actually solving (2.10), we obtain solutions of the self-consistent source system (2.12), which depend on arbitrary functions of t. The appearance of arbitrary functions of a single variable in solutions is a generic feature of systems with self-consistent sources.
Again, introducing new variables as in (2.11), now with a suitable Q(y) and R(y), ω y can be absorbed. The scalar version of this system has been studied in [1,26].
Together with (2.5), this provides us with solutions of the scalar versions of (2.7) and (2.8).
If ω depends only on t or y, we obtain solutions of (2.10) and (2.13), respectively. Remark 2.4. As formulated above, the number of sources appears to be n. However, this is only so if ω t , respectively ω y , has maximal rank. If the rank is N < n, then only N sources appear on the right hand side of (2.9).
The above procedure provides us with a hetero binary Darboux transformation from the pKP equation and its associated linear system to any of the pKP systems with self-consistent sources (modified by ω). 1 In the framework of bidifferential calculus, we can abstract the underlying structure from the specific example (here pKP) and then obtain corresponding self-consistent source extensions of quite a number of other integrable equations.
Remark 2.2 shows that there is also a transformation that maps a class of solutions of any of the self-consistent source extensions of the pKP equation to solutions of the source-free pKP equation and its linear system. We expect that this is a general feature of integrable systems with self-consistent sources.
Exact solutions in case of constant seed. Let φ 0 be constant. Special solutions of (2.2) and (2.3) are given by with constant matrices a, b, A, B, P and Q of appropriate size. Then (2.6) is solved by with a constant matrix X that satisfies the Sylvester equation 1 In the scalar case, hetero binary Darboux transformations from a KP equation with self-consistent sources to the KP equation with additional sources have been elaborated in [25].
(2.4) and (2.5) now provide us with explicit solutions of the above matrix pKP equations with self-consistent sources, (2.10) and (2.13). The basic soliton solutions are obtained if P and Q are diagonal with distinct eigenvalues, and if they have disjoint spectra, in which case the Sylvester equation has a unique solution. Also see [29] for the source-free case.
Example 2.5. In the scalar case (m = 1), choosing n = 1, a = A = B = b = 1 and ω = e α /(Q−P ), where α is either a function only of y or of t, we obtain . Then u = φ x , together with q and r, or rather the transformedq andr, is a soliton solution of the scalar KP equation with self-consistent sources. Also see [24,25,30,31].

A framework for generating self-consistent source extensions of integrable equations
Let us recall some basics of bidifferential calculus. A graded associative algebra is an associative algebra Ω = r≥0 Ω r over C, where A := Ω 0 is an associative algebra over C and Ω r , r ≥ 1, are A-bimodules such that Ω r Ω s ⊆ Ω r+s . A bidifferential calculus is a unital graded associative algebra Ω, supplied with two (C-linear) graded derivations d,d : Ω → Ω of degree one (hence dΩ r ⊆ Ω r+1 ,dΩ r ⊆ Ω r+1 ), and such that d 2 =d 2 = dd +dd = 0 . (3.1) In this framework, many integrable equations can be expressed either as with φ 0 ∈ Mat(m, m, A) (the algebra of m × m matrices over A), and possibly with some reduction condition, or as with an invertible g 0 ∈ Mat(m, m, A). The two equations are related by the Miura equation which has both, (3.2) and (3.3), as integrability conditions. (3.2) and (3.3) are generalizations or analogs of well-known potential forms of the self-dual Yang-Mills equation (cf. [20]).
Binary Darboux transformation. Let Ω be a solution of The equation obtained by acting withd on (3.8) is identically satisfied as a consequence of (3.5), (3.6), (3.8), and the equation that results from (3.8) by acting with d on it. Correspondingly, also the equation that results from acting withd on (3.9) is identically satisfied as a consequence of the preceding equations. It follows [20] that is a new solution of (3.2), and Deformation of the potential. Guided by the pKP example in the preceding section, we replace Ω by Ω − ω in the above equations, i.e.,
Theorem 3.1. Let φ 0 be a solution of (3.2) and let θ, η, Ω satisfy the linear equations (3.5), (3.6) and (3.10), respectively. 3 Then From (3.14) and (3.15) we can recover the self-consistent source extensions of the pKP equation, revisited in the preceding section, see below. But now we can choose different bidifferential calculi and obtain self-consistent source extensions also of other integrable equations. A number of examples will be presented in the following sections. In all these examples, we have c = 0, i.e., Γ ω = ω ∆ . (3.16) Remark 3.2. The above equations form a consistent system in the sense that any equation derived from it by acting withd on any of its members yields an equation that is satisfied as a consequence of the system.
Together with (3.14), this constitutes a system of equations for φ, q, r andΩ, for which we now have a solution-generating technique at hand. Also see Remark 2.2.
Generating solutions of the equations for g. If φ 0 and g 0 satisfy the Miura equation (3.4), then φ, q, r together with where I is the identity matrix, satisfȳ If (3.16) holds, i.e., c = 0, then we have and g satisfies the extended Miura equation dg − (dφ) g = (q γ ∆ −1 r) g , (3.17) which implies the following extension of (3.3), Via the extended Miura equation (3.17), we can eliminate φ in (3.15) in favor of g. The resulting equations, together with (3.18), constitute the (Miura-) dual of (3.14) and (3.15). It is another generating system for further self-consistent source extensions of integrable equations.
Transformations. The equations for ∆ and λ in (3.7) and the linear equation (3.5) are invariant under a transformation with an invertible L. Correspondingly, the equations for Γ and κ in (3.7) and the linear equation with an invertible M . Choice of the graded algebra. In this work, we specify the graded algebra Ω to be of the form where Λ is the exterior (Grassmann) algebra of the vector space C 2 . It is then sufficient to define d andd on A, since they extend to Ω in a straightforward way, treating the elements of Λ as constants. Moreover, d andd extend to matrices over Ω. We choose a basis ξ 1 , ξ 2 of Λ 1 .
Recovering the pKP case. Let A 0 be the space of smooth complex functions on R 3 . We extend it to A = A 0 [∂], where ∂ := ∂ x is the partial differentiation operator with respect to x. On A we define (cf. [20]) The maps d andd extend to linear maps on Ω = A⊗Λ, and moreover to matrices over Ω. Choosing where I n is the n × n identity matrix, (3.7) is satisfied, and c = 0, i.e., (3.16), becomes ω x = 0. The second equation in (3.11) leads to (3.10) becomes (2.6) with c 1 = 0, c 2 = ω y , c 3 = ω t . Taking ω x = 0 into account, we recover all equations in Section 2. For example, (3.15) yields (2.7) and (2.8), and (3.14) becomes (2.9). A corresponding extension of the whole pKP hierarchy is presented in Appendix A.

Matrix KdV equation with self-consistent sources
Let A 0 be the space of smooth complex functions on R 2 . We extend it to A = A 0 [∂], where ∂ := ∂ x is the partial differentiation operator with respect to the coordinate x. On A we define (cf. [20]) The maps d andd extend to linear maps on Ω = A⊗Λ, and moreover to matrices over Ω. Choosing with constant matrices P, Q, (3.7) is satisfied, and (3. 16) becomes ω x = 0. The choices for κ and λ considerably simplify the subsequent equations. 4 The second equation in (3.11) yields The linear equations (3.5) and (3.6) read and (3.10) takes the form According to Section 3, it follows that φ, q and r, given by (3.13), satisfy where we introduced u = 2 φ x . From this we obtain the following two versions of a matrix KdV equation with self-consistent sources.
1. Setting γ 1 = 0, i.e., and disregarding the equations for q t and r t , (4.2) reduces to Here ω t can be absorbed by a redefinition of either q or r, which exchanges P and Q in the corresponding linear equation due to a necessary application of (4.3). The scalar version appeared in [3], also see [33][34][35][36].
. The plots for r 1 and r 2 look like those for q 1 and q 2 .
2. γ 2 = 0, i.e., constant ω. In this case (4.2) yields where we introducedq = 6 q (Q 2 ω − ωP 2 ). This system is also obtained from (2.13) by assuming that φ, q, r do not depend on y, whereas ω y is non-zero, but constant. We note that (4.5) is a system of evolution equations for all dependent variables. In this respect it is very different from all other examples of systems with self-consistent sources presented in this work, with the exception of (5.1).
Exact solutions for vanishing seed. Let φ 0 be constant, i.e., u 0 = 0. Then we have the following solutions of the linear equations, with constant matrices a i , b i . A corresponding solution of (4.1) is with constant matrices A ij that satisfy the Sylvester equations Then (3.13) yields exact solutions of (4.4), respectively (4.5), under the corresponding condition for ω.
Example 4.1. In case of the first version of a KdV equation with self-consistent sources, for the sake of definiteness we choose ω(t) = β(t) I n . Then (4.3) leads to Q = ±P . The two cases turn out to be equivalent, hence we choose Q = P . Now (3.13) determines a class of exact solutions of (4.4). Fig. 1 shows an example of a modulated 2-soliton solution. Introducingr = β r, this becomes a solution of (4.4) without the presence of ω t .
Example 4.2. In case of the second KdV equation with self-consistent sources, let n = 1. Then we obtain the following 2-soliton solution of (4.5), with constants µ i , ν i and

Matrix Boussinesq equation with self-consistent sources
Let A 0 be the space of smooth complex functions on R 2 , with coordinates x and t, and A = A 0 [∂], where ∂ := ∂ x is the partial differentiation operator with respect to x. On A we define a bidifferential calculus via with constant matrices P, Q, (3.7) is satisfied, and (3. 16) becomes ω x = 0. The second equation in (3.11) yields The linear equations (3.5) and (3.6) read and (3.10) takes the form Our general results in Section 3 imply that φ, q and r, given by (3.13), satisfy 5 From this we obtain the following two versions of a matrix Boussinesq equation with self-consistent sources. Here we set u := 2φ x .
The scalar version appeared in [37]. 5 Up to the term involving Q and P , the first equation is obtained as a reduction of (2.9), exchanging y and t.
. In contrast to most systems with self-consistent sources obtained in this work, here we have evolution equations for all dependent variables. Also see (4.5).
Solutions with vanishing seed. If u 0 = φ 0,x /2 = 0, solutions of the linear equations are given by where a i , b i are constant and A corresponding solution of the equations for Ω is then given by where A kj are constant matrices subject to Now (3.13) yields exact solutions of the above Boussinesq systems with self-consistent sources. In case of the first version, we still have to take the constraint Q 3 ω = −ω P 3 into account.

Matrix sine-Gordon equation with self-consistent sources
Let A 0 be an associative algebra, where the elements depend on variables x and y.
where J is an idempotent operator that determines an involution * via Jf =: f * J, for f ∈ A 0 , i.e., f * = Jf J. A bidifferential calculus is then determined on A by setting In order to eliminate explicit appearances of the operator J in the equations resulting from the linear equations (3.5) and (3.6), we write 6 where ϕ, P, Q, κ i , λ i can now be taken to be matrices over A 0 . We will assume that P and Q are invertible. Then we obtain 1) 6 Only in this section we carry the full freedom in κ and λ along with us, in particular in order to demonstrate that there are special non-zero choices, here given by κi = λi = 0, which considerably simplify the equations.
where θ and η can now be restricted to be matrices over A 0 . The equations (3.7) impose the following conditions, Next we set Ω =Ω J , ω =ω J , q =q J , r = Jr .

Then (3.16) becomes
Qω =ω * P , (6.2) and the involution property implies Q * Qω =ω P * P . The second equation in (3.11) reads From (3.10) we obtain the Sylvester equation Given solutions θ and η of the linear equations, and a solution Ω of the latter consistent system, it follows from our general results in Section 3 that which is obtained from (3.14), and the system which results from (3.15). { , } denotes the anti-commutator. The extended Miura equation obtained from (3.17), turns the last system intõ and (3.18) has the form The latter system is solved by if g 0 is a solution of the source-free version of (6.6). Equations with self-consistent sources are now obtained as follows.
1. γ 2 = 0. In this case we have To turn this into a concrete system of PDEs, we have to specify A 0 and the operator J. We note that for a non-zero choice of κ and λ, corresponding to κ i = λ i = 0, i = 1, 2, the above equations attain a particularly simple form.
Scalar sine-Gordon equations with self-consistent sources. Let C ∞ (R 2 ) be the algebra of smooth complex functions of real variables x and y. Let σ j , j = 1, 2, 3, be the Pauli matrices, and A 0 = {a I 2 + b σ 2 | a, b ∈ C ∞ (R 2 )}, which is a commutative algebra. We choose J = σ 3 . Note that, for f ∈ A 0 , also f * ∈ A 0 . Let g = e i σ 2 u/2 , with a complex function u. Then we have g * = e −i σ 2 u/2 and, for vanishing sources, (6.6) becomes the complex sine-Gordon equation u xy = sin u.
1. γ 2 = 0. Setting κ 2 = λ 2 = 0, from (6.8) we obtain Then P y = Q y = 0, so that P and Q are only allowed to depend on x. With the decompositioñ with row vectors f i and column vectors h i , and assuming Q * = Q and P * = P , so that Q =Q ⊗ I 2 and P =P ⊗ I 2 , the linear equations forq andr can be written as Of course, the constraint (6.2) has to be taken into account. Due to the latter,ω x can be absorbed by a redefinition of eitherq orr, while only exchanging P and Q in the respective linear equation. Then we obtain which establishes contact with the sine-Gordon equation with self-consistent sources considered, e.g., in [38].
2. γ 1 = 0. Setting κ 1 = λ 1 = 0, (6.9) becomes u xy − sin u = 2 qω * y (P * ) −1r In terms of the new variablesq =q e −i σ 2 u/2 andr = e i σ 2 u/2r , this becomes Then P and Q can only depend on y. We can absorbω y P −1 by a redefinition ofq, but in the linear equation forq we have to replace Q by P * (via an application of (6.2)). Then, writinĝ where denotes the transpose), and setting P = (P * ) , we obtain and thus the sine-Gordon equation with self-consistent sources considered in [39,40]. Of course, we still have to respect the constraint (6.2).
With the above choice for A 0 and J, the systems (6.8) and (6.9) can thus be regarded as matrix generalizations of the above sine-Gordon equations with self-consistent sources. The corresponding systems involving ϕ instead of g are Miura-duals of them. Remark 6.1. A simpler choice for A 0 and the involution * is the algebra of smooth complex functions on R 2 together with complex conjugation. In the scalar case (m = 1), we then set g = e i u/2 , with real u. This leads to similar results, but there is a restriction to self-consistent source extensions of the real sine-Gordon equation. The slightly more complicated setting we chose above is more flexible.
Exact solutions in case of trivial seed. We set κ 1 = κ 2 = λ 1 = λ 2 = 0, and P = I 2 ⊗P , Q =Q ⊗ I 2 , with constant complex matricesP andQ. Then we have P = P * and Q = Q * , and special solutions of the linear equations (6.1) with ϕ 0 = 0 are given by where the constant complex matrices A 0 , A 1 , B 0 , B 1 are subject to the Sylvester equations and ω has to satisfy (6.2). Now (6.7), with constant g 0 , leads to a class of exact solutions of the above scalar sine-Gordon equations with self-consistent sources.

Matrix Nonlinear Schrödinger equation with self-consistent sources
Let A be the algebra of M × M matrices of smooth functions of coordinates x and t on R 2 . We define a bidifferential calculus on A via where J = I is a constant M × M matrix, i.e., J x = J t = 0, satisfying J 2 = I (also see [41]). d and d extend to matrices over A and to the corresponding graded algebra. We set m = 1 and where P, Q, κ i , λ i are matrices over A 0 . Then the equations (3.7) lead to

16) becomes
Q ω = ωP , (7.1) and the second equation in (3.11) reads The linear equations (3.5) and (3.6) take the form and (3.10) results in Now φ, q, r given by (3.13) solve and Next we choose J = block-diag(I M 1 , −I M 2 ), where M 1 +M 2 = M , and use the block decompositions where u, v, q, r are, respectively, M 1 × M 1 , M 2 × M 2 , M 1 × M 2 and M 2 × M 1 matrices. q 1 and q 2 have size M 1 × (M · n) and M 2 × (M · n), respectively. Then we obtain the following AKNS equations with self-consistent sources. Here and in the following we set κ i = λ i = 0, in which case P and Q have to be constant.
i q t + q xx − 2 q r q = i q 1 ω t r 2 , −i r t + r xx − 2 r q r = i q 2 ω t r 1 , and u = −q r, v = −r q.
i q t + q xx − 2 q r q = q (q 2 ω x r 2 ) − (q 1 ω x r 1 ) q + (q 1 ω x r 2 ) x + q 1 ω x P r 2 , −i r t + r xx − 2 r q r = −r (q 1 ω x r 1 ) + (q 2 ω x r 2 ) r − (q 2 ω x r 1 ) x + q 2 ω x P r 1 , and In both cases, Q, P and ω still have to satisfy (7.1). Via a Hermitian conjugation reduction, see below, these systems become matrix versions of two kinds of (scalar) Nonlinear Schrödinger (NLS) equations with self-consistent sources. The second system seems to be new, even in the scalar case.
We can set 1 = 1, since it can be absorbed into ω.
Though ω x can be absorbed (e.g., by redefinitions of r 1 and r 2 , before the reduction), we cannot simultaneously absorb Q on the right hand side of the first equation. The last system is therefore of a different nature than the familiar integrable equations with self-consistent sources. In contrast to (7.8), the equations for q 1 and q 2 are nonlinear in the system (7.9).
In both cases, Q and ω also have to satisfy Q ω + ω Q † = 0, as a consequence of (7.1). This severely restricts Q if ω = 0. If ω = β I nM , with a scalar β, then Q † = −Q. For diagonal Q, this restricts its eigenvalues to be imaginary, a restriction that also appears in [43], in the focusing NLS case ε = −1. ε = 1 is the defocusing case.
The above reduction conditions (7.7) for q, r, u, v, q j , r j can be expressed as follows, For the solution generating procedure to respect the reduction conditions, we still have to require Exact solutions for vanishing seed. Let φ 0 = 0. The linear equations (7.2) are then solved by A corresponding solution of (7 .3) is where the constant matrices A, B have to satisfy the Sylvester equations If Q and −Q † have no eigenvalue in common, the solutions A and B are unique and Hermitian. But Qω + ωQ † = 0 then implies ω = 0. Thus, in order to obtain solutions of (7.8) or (7.9) with ω = 0, the spectrum condition for Q needs to be violated. Requiring A † = A and B † = B, then Ω is Hermitian, since ϑ(Q) † = −ϑ(−Q † ), and φ satisfies the reduction condition. If ω and Q satisfy Qω + ωQ † = 0, and if ω x = 0 or ω t = 0, then provides us with exact solutions of the above matrix NLS equations with self-consistent sources. In the defocusing NLS case, it is more interesting to start with a constant density seed solution (see, e.g., [43]).

Matrix Davey-Stewartson equation with self-consistent sources
Let A 0 be the algebra of M × M matrices of smooth complex functions on R 3 , and where ∂ is again the partial differentiation operator with respect to x. We define a bidifferential calculus on A via The linear equations (3.5) and (3.6) read From (3.10) we obtain Given solutions θ and η of the linear equations, and a solution Ω of the latter consistent system, then which is obtained from (3.14), and which results from (3.15). The equations (8.5) and (8.6) include the following two systems with self-consistent sources.
1. ω y = 0. Then (8.5) and two of the equations (8.6) read 2. ω t = 0. In this case we obtain Hermitian conjugation reductions. (8.5) and (8.6), and its special cases (8.7) and (8.8), are compatible with the reduction conditions (7.10). Using the same block-decomposition as in (7.4), we have again the conditions (7.7), considered in Section 7. The last two equations in (8.6) are then redundant, and we obtain the following reduced systems with self-consistent sources. Without restriction of generality, we can set 1 = 1.
1. ω y = 0. From (8.7) we obtain the following matrix Davey-Stewartson (DS) equation with self-consistent sources: 2. ω t = 0. From (8.8) we obtain another matrix Davey-Stewartson (DS) equation with selfconsistent sources: In order that the solution generating procedure respects the reduction conditions, we also have to require with Λ defined in (7.10).
Remark 8.1. In the scalar case, i.e., M 1 = M 2 = 1, the inhomogeneous linear equations for u and v in (8.9) integrate to u y − u x = −ε |q| 2 = v y + v x , dropping "constants" of integration. This is solved by u = w y + w x and v = w y − w x , with a function w, and we obtain w xx − w yy = ε |q| 2 . The first of equations (8.9) then takes the form i q t + q xx + q yy + 2 (w xx + w yy ) q = −i ε q 1 ω t q † 2 . Passing over to "light cone variables", we recover the DS equation with self-consistent sources treated in [44] (also see [45]).
dy, from (8.9) we obtain i q t + q xx + q yy + s y q + 2ε |q| 2 q = −i q 1 ω t q † 2 , s xx − s yy = 4ε (|q| 2 ) y + h(x) , q 1,y − q 1,x = q q 2 , q 2,y + q 2,x = ε q † q 1 . (8.12) Here h(x) is an arbitrary function of x, which can be eliminated by a redefinition of s. In this way we recover the system of Example 3 in [46].

Remark 8.3.
Assuming that all objects do not depend on the variable x, (8.9) reduces to the NLS equation in Section 7 up to changes in notation (J → −J, and y has finally to be renamed to x). We note that the second of equations (8.4) becomes a constraint: η J θ = 0. As formulated above, our solution generating method does not work for this NLS reduction. This can be corrected by extending the first equations in (8.2) to ∆ = −2I n ∂ + P and Γ = −2I n ∂ + Q, with constant n × n matrices P, Q, and generalizing the subsequent equations accordingly. In particular, we need non-vanishing κ and λ, cf. Section 7. The matrices P and Q are redundant on the level of DS. Dropping in (8.1) the partial derivatives with respect to x, we recover the bidifferential calculus for the NLS system in Section 7, up to the stated changes in notation. Dropping instead the partial derivatives with respect to y, we obtain a bidifferential calculus for the NLS system different from that used in Section 7.
Evaluation of φ = −θ Ω −1 θ † Λ and its decomposition now leads to From q = θ Ω −1 we obtain If det(Ω) nowhere vanishes, this solution is regular. If ω is constant, it describes a single dromion solution of the DS equation [47][48][49][50], or its degeneration to a solitoff [51] or a soliton. A solution of a DS equation with self-consistent sources, with non-constant ω, can change its type. In particular, we find the following.
A nonlinear superposition of n of such elementary solutions is obtained by taking a ± to be a row of n 2 × 2 matrices of the above form, and P i = block-diagonal(λ i1 I 2 , . . . , λ in I 2 ).

Matrix two-dimensional Toda lattice equation with self-consistent sources
Let A 0 be the space of complex functions on R 2 × Z, smooth in the first two variables. We extend it to A = A 0 [S, S −1 ], where S is the shift operator in the discrete variable k ∈ Z. A bidifferential calculus is determined by setting Remark 9.1. In the scalar case, in terms of V := ϕ y , (9.1) without sources becomes (ln(1 + V )) x = ϕ + − ϕ + ϕ − . Differentiating with respect to y, this becomes the two-dimensional Toda lattice equation (ln(1 + V )) xy = V + − 2V + V − [52][53][54]. A self-consistent source extension, and corresponding exact solutions, has been obtained in [55] (also see [56]), using the framework of Hirota's bilinear difference operators [54]. (9.1) leads to a matrix version.
Correspondingly, (9.7) and (9.6) reduce to (9.11) These equations constitute the Miura-dual of the first type of the two-dimensional matrix Toda lattice equation with self-consistent sources. In the scalar case, in terms of u = ln g this takes the form Those of equations (9.5) that do not depend onΩ arẽ q y =q −q − − ϕ yq − ,r y =r + −r +r + ϕ + y . (9.13) (9.12) and (9.13) constitute the second type of the two-dimensional matrix Toda lattice equation with self-consistent sources.
From (9.7) and (9.6) we obtain the corresponding Miura-dual, (9.14) In the scalar case (m = 1), in terms of u = ln g, a = e −y g −1qω x and b = e y gr, (9.14) can be expressed as follows, Such a system appeared in [57].
The above matrix versions of the two-dimensional Toda lattice equations with self-consistent sources are new according to our knowledge.
Explicit solutions for trivial seed. Let ϕ 0 = 0 and g 0 = I. Then (9.2) becomes where the constant matrix X has to satisfy the Sylvester equation Then (9.4) and (9.8) provide us with explicit solutions of the above matrix two-dimensional Toda lattice equations with self-consistent sources.

A generalized discrete KP equation with self-consistent sources
Let A 0 be the space of complex functions of discrete variables k 0 , k 1 , k 2 ∈ Z, and S 0 , S 1 , S 2 corresponding shift operators. We extend where c i are constants. Then d andd extend to Ω = A⊗Λ and to matrices over Ω. In the following we will use the notation We set Then (3.7) is satisfied, (3.16) becomesω ,0 =ω, and the second equation in (3.11) leads to The linear equations (3.5) and (3.6) read and from (3.10) we obtaiñ Given solutions θ, η andΩ, according to Section 3, solve the equations They result, respectively, from (3.14) and (3.15). The extended Miura equation (3.17) takes the form Correspondingly, (3.18) becomes and the above equations forq andr transform tõ Settingω ,1 =ω, and retaining only those equations forq andr that do not containΩ, we obtain the following equations with self-consistent sources, respectively, By using these equations, and choosingω ,2 −ω =: K (c 1 − 1)/[c 1 (c 2 − 1)] to be constant, the first of equations (10.3) can be cast into the form 1 τ ,0 τ ,1,2 c 2 τ ,0,1 τ ,2 − c 1 τ ,0,2 τ ,1 − K ρ ,2 σ ,0,1 which implies with an arbitrary scalar c 12 that does not depend on the discrete variable k 0 . Up to differences in notation, the system (10.4) and (10.5) coincides with the discrete KP equation (Hirota bilinear difference or Hirota-Miwa equation) with self-consistent sources considered in [58] (also see [56]). (10.3) thus constitutes a "non-commutative" generalization of the latter.
Some explicit solutions for vanishing seed. We set ϕ 0 = 0 and g 0 = I. Then the linear equations for θ and η are satisfied by with constant matrices a, b, A, B, P and Q. A corresponding solution of the equations forΩ is with a constant matrix X that satisfies the Sylvester equation (10.2) together with g = I − θ (Ω + ) −1 η now provides us with explicit solutions of the above matrix discrete KP equations with self-consistent sources.

Conclusions
The present work clarifies the origin of self-consistent source extensions of integrable equations from the binary Darboux transformation perspective. The essential point is a deformation of the potential Ω that is central in this solution generating method. We presented an abstraction of the underlying structure in the framework of bidifferential calculus. Choosing realizations of the bidifferential calculus then leads to self-consistent source extensions of various integrable equations, and in this work we provided a number of examples, recovering known examples and obtaining generalizations to matrix versions. All this is not at last a demonstration of the power of bidifferential calculus. Generalizing an integrability feature of an integrable system to this framework opens the door toward a large set of integrable systems sharing this feature. It therefore establishes such a feature as a common property of a wide class of integrable systems and provides a universal proof. Our approach also demonstrated that self-consistent source extensions of integrable equations typically 9 admit classes of solutions that depend on arbitrary functions of a single independent variable (which, of course, may be a combination of the independent variables used). We note that the "source-generation method" in [56] essentially consists of promoting constant parameters in soliton solutions of an integrable equation to arbitrary functions of a single variable.
Remark A.1. Via expansion of (A.1) in the indeterminates µ 1 and µ 2 , one obtains in particular the bidifferential calculus given by which underlies the above evolution equations with variables t 1 and t 2 . They will be considered in the next example.
Example A.2. Setting ω t 2 = 0, the equations for q t 2 and r t 2 do not involveΩ. Disregarding the equations for q t 1 and r t 1 , in terms of 10 u := −2 (φ t 1 − q ω t 1 r), we obtain the system 11 u t 1 − u x = −2 (q ω t 1 r) x , q t 2 = q xx − u q , r t 2 = −r xx + r u , (A. 6) where the first equation results from (A.5) for i = 1. Via t 2 → −i t 2 , and with the reduction r = q † and u, ω t 1 Hermitian, the system becomes [60] u t 1 − u x = −2 (q ω t 1 q † ) x , i q t 2 = q xx − u q .
We would rather like to obtain solutions of (A.6) with ω t 1 replaced by −I n , since in this case (A.7) becomes the system studied, e.g., in [62]. Since this can be achieved by a transformation of q and r, this means that, for any matrix function ω(t 1 ), we obtain a class of solutions.