Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem. Furthermore, we recover a system that contains two types of the KP equation with self-consistent sources as special cases. Darboux and binary Darboux transformations are applied to generate solutions of the proposed hierarchies.

A powerful solution generating method for nonlinear systems from the above mentioned hierarchies is based on the Darboux Transformations (DT) and the Binary Darboux Transformations (BDT) [18]. In [19,20] the latter transformations were applied to generate solutions of the k-cKP hierarchy. Solutions of (2+1)-dimensional the k-cKP hierarchy were obtained via BDTs in [16,21]. More general (2+1)-dimensional extensions of the k-cKP hierarchy and the corresponding solutions were investigated in [22]. The latter hierarchies cover matrix generalizations of the Davey-Stewartson (DS) and Nizhnik-Novikov-Veselov (NNV) systems, (2+1)-dimensional extensions of the Yajima-Oikawa and modified Korteweg-de Vries equations. The Inverse Spectral Transform Method for (2+1)-dimensional equations, including DS and NNV systems, was presented in [23].
The aim of this paper is to introduce hierarchies that extend above mentioned cases. It leads to generalizations of the corresponding integrable systems. In particular, we got an equation that contains both types of the KP equation with self-consistent sources as special cases (formula (2.19)). The same holds for the corresponding modified version (4.8).
This work is organized as follows. In Section 2 we present a new (2+1)-dimensional generalizations of the KP hierarchies and enumerate some integrable systems that the latter hierarchy contains. In particular it includes N-wave problem with self-consistent sources, a generalization of the DS-III system and extended KP equation with self-consistent sources. In Section 3 we present solution generating technique (dressing method) for the obtained hierarchies via DTs and BDTs. In Section 4 we present new (2+1)-dimensional extensions of the modified KP hierarchies and propose solution generating method via BDTs. A short summary of the obtained results and some problems for future investigation are presented in Conclusions.

A new (2+1)-dimensional generalization of the kconstrained KP hierarchy
For further purposes we will use the following well-known formulae for integral operator h 1 D −1 h 2 constructed by matrix-valued functions h 1 and h 2 and the differential operator A with matrix-valued coefficients in the algebra of pseudodifferential operators: Consider Sato-Zakharov-Shabat dressing operator: with N × N -matrix-valued coefficients w i . Introduce two differential operators J k D k and α n ∂ tn −J n D n , α n ∈ C, n, k ∈ N, where J k andJ n are N × N commuting matrices (i.e., [J n , J k ] = 0). It is evident that dressed operators have the form: and Impose the following reduction on the integral part of operators L k and M n : where u j and v i are matrix-valued functions of dimension N × N ; q and r are matrix-valued functions of dimension N × m;q andr are matrix-valued functions with dimension N ×m.
M 0 andM 0 are constant matrices with dimensions m × m andm ×m respectively. We shall also assume that functions q, r,q andr satisfy equations: The following proposition holds for operators in (2.7): is satisfied in case the following equations hold: where Λ q , Λ r , Λq, Λr are constant matrices with dimensions (m×m) and (m×m) respectively that satisfy equations: (2.10) After direct computations for each of the three items at the right-hand side of formula (2.10) we get: The latter formulae are consequences of (2.1)-(2.3). From formulae (2.10)-(2.13) using (2.8) we have (2.14) From the last formula we obtain that equality [L k , M n ] = 0 is a consequence of (2.8).
Lax equation [L 1 , M 2 ] = 0 is equivalent to the following generalization of the DS-III equation: If we setq = 0,r = 0 we recover DS-III system: In this case we obtain the following pair of operators: Equation [L 3 , M 2 ] = 0 is equivalent to the following system: The latter consists of several special cases: (a) c 1 = c 2 = 1. In this case the latter system can be rewritten in the following way: In the scalar case (N = 1) under the Hermitian conjugation reduction: . the latter equation reads: This system is a generalization of the KP equation with self-consistent sources (KPSCS). In particular, if we setM 0 = 0 we recover KPSCS of the first type. In case M 0 = 0 we obtain KPSCS of the second type.
New (2+1)-dimensional k-constrained KP hierarchy connected with the Lax pair (2.7) also are closely related to the bidirectional generalization of (2+1)-dimensional matrix kconstrained KP hierarchy ((2+1)-BDk-cKP hierarchy) that was introduced in [22] and its generalizations. Namely, let us put in formulae (2.7): I.e.,m =m 1 + m(l + 1) and matricesq andr consist of N ×m 1 -matrix-valued blocksq 1 andr 1 and N × m-matrix-valued blocks q[j] and r[j], j = 0, l.M 0 is a block-diagonal matrix and I l+1 ⊗ M 1 stands for the tensor product of the l + 1-dimensional identity matrix I l+1 and matrix M 0 . Then we obtain the following pair of operators in (2.7): If we assume that equations with constant matrices Λq and Λr are satisfied, then the following proposition holds. Proof. The proof is similar to the proof of the Proposition 2.1 and the proof of the Theorem 1 in [22].
3. n = 0. The differential part of M 0,l (2.22) is equal to zero in this case (A 0 = 0) and we get a new generalization of DS-III hierarchy.
3 Dressing methods for the new (2+1)-dimensional generalization of k-constrained KP hierarchy

Dressing via Darboux Transformations
In this section we will consider Darboux Transformations (DT) for the pair of operators (2.7) and its reduction (2.22). At first, we shall start with the linear problem associated with the operator L k (2.7): where ϕ 1 is (N × N )-matrix-valued function; Λ 1 is a constant matrix with dimension N × N . Introduce the DT in the following way: The following proposition holds.
u j [1] are N × N -matrix coefficients depending on function ϕ 1 and coefficients u i , i = 0, k.
Proof. It is evident that the inverse operator to (3.2) has the form W −1 It remains to find the explicit form of (L k ) <0 . Using formulae (2.1)-(2.3) we have: It is also possible to generalize the latter theorem to the case of finite number of solutions of linear problems associated with the operator L k . Namely, let functions ϕ s , s = 1, K be solutions of the problems: For further convenience we shall use the notations ϕ s [1] := ϕ s , s = 1, K and define the following functions: (3.10) The following statement holds: are N × N -matrix coefficients depending on functions ϕ s , s = 1, K and coefficients u i , i = 0, k. In particular,û k [K] = u k .
Proof. The proof can be done via induction by K. Namely, assume that the statement holds for K − 1. I.e., Now, it remains to apply the Proposition 3.1 to operatorL k [K − 1] (3.13) with the DT W 1 [ϕ K [K]] (see formula (3.2)) and use formula W K = W 1 [ϕ K [K]]W K−1 that immediately follows from (3.10).
Dressing methods for integro-differential operator given by the Proposition 3.2 are closely related to the results of papers on constrained KP hierarchies and their (2+1)-dimensional generalizations [15,16,19,21,47].
From Proposition 3.2 we obtain the corollary for Lax pairs consisting of operators L k and M n (2.7). Namely, let functions ϕ s , s = 1, K be solutions of the problems: Then the following statement holds: 2. We obtain the proof of this item from the following formulae we get the form ofq K mentioned in item 3. The form ofr K can be obtained in a similar way.

Dressing via Binary Darboux Transformations
Let N × K-matrix functions ϕ and ψ be solutions of linear problems: Introduce binary Darboux transformation (BDT) in the following way: where C is a K × K-constant nondegenerate matrix. The inverse operator W −1 has the form: The following theorem is proven in [48].
Theorem 3.5. The operatorL k := W L k W −1 obtained from L k in (2.7) via BDT (3.19) has the form u j are N × N -matrix coefficients depending on functions ϕ, ψ and u j . In particular, Solution generating method for the hierarchy (2.7)-(2.8) is given by the corollary, which follows from the previous theorem.
u j are N × N -matrix coefficients depending on functions ϕ, ψ and u j , v i . In particular,

New (2+1)-dimensional generalizations of the modified k-constrained KP hierarchy
Consider the following pair of integro-differential operators: where Λ q , Λ r , Λq, Λr are constant matrices with dimensions (m×m) and (m×m) respectively that satisfy equations: Proof. Operators (4.1) can be rewritten as: It remains to use the Proposition 2.1 to complete the proof.
Lax representation [L 1 , M 2 ] = 0 is equivalent to the following system: In case of the Hermitian conjugation reduction β ∈ R, α ∈ iR, the latter equation reduces to the following: In case we setq = 0, Λ q = 0 we get a matrix (2+1)-dimensional generalization of the Chen-Lee-Liu equation.
Using (4.2) we get that Lax representation [L 3 , M 2 ] = 0 is equivalent to the following system (4.6) Set c 1 = c 2 = 1 in the scalar case (N = 1). Eliminating variables w and v from the first and second equation respectively, we get (4.7) The latter under the Hermitian conjugation reduction

Dressing via Binary Darboux Transformations
In this subsection we consider dressing methods for (2+1)-dimensional extensions of the modified k-constrained KP hierarchy given by the family of Lax pairs (4.1). First of all, we start with the matrix version of the theorem that was proven in [50].

(4.9)
Then the operator L k transformed via where has the form: Proof. Proof is similar to the proof of Theorem 2 in [50].
The following consequence of the latter theorem provides a solution generating method for hierarchy (4.1)-(4.2): with operators L k and M n given by (4.1).
Then operatorsL k = W m L k W −1 m andM n = W m M n W −1 m transformed via W m (4.10)-(4.11) have the form: where

Conclusions
In this work we proposed new integrable generalizations of the KP hierarchy with selfconsistent sources. The obtained hierarchies of nonlinear equations include, in particular, matrix integrable system that contains as special cases two types of the matrix KP equation with self-consistent sources (KPSCS) and its modified version. They also cover new generalizations of the N-wave problem and the DS-III system. Under reductions (2.21) imposed on the obtained hierarchies one recovers (2+1)-BDk-cKP hierarchy. The latter contains (t A , τ B )-and (γ A , σ B )-Matrix KP hierarchies [40,41] (see [22] for details). It leads to the following family of integro-differential operators in (4.1) Lax equation [L k , M n ] = 0 involving the latter operators should lead (under additional reductions) to (2+1)-dimensional generalizations of the corresponding integrable systems that were obtained in [50]. In particular it concerns systems that extend KdV, mKdV and Kaup-Broer equations.
The latter involve fixed solutions of linear problems and an arbitrary seed (initial) solution of the corresponding integrable system. Exact solutions of equations with self-consistent sources (complexitons, negatons, positons) and the underlying hierarchies were studied in [40,51,52]. One of the problems for future interest consists in looking for the corresponding analogues of these solutions in the obtained generalizations. The same question concerns lumps and rogue wave solutions that were investigated in several integrable systems recently [53][54][55][56][57][58][59][60] It is also known that Inverse Scattering and Spectral methods were applied to generate solutions of equations with self-consistent sources [61][62][63]. An extension of these methods to the obtained hierarchies and comparison with results that can be provided by BDTs (e.g., following [64]) presents an interest for us.
The search for the corresponding discrete counterparts of the constructed hierarchies is another problem for future investigation. The latter is expected to contain the discrete KP equation with self-consistent sources [65,66]. One of the possible ways to solve the problem consists in looking for the formulation of the corresponding continuous hierarchy within a framework of bidifferential calculus. The latter framework provides better possibilities to search for the discrete counterparts of the corresponding continuous systems (see, e.g. [67]).