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 e...

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Date:	2015
Main Authors:	Chvartatskyi, O., Sydorenko, Y.
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Language:	English
Published:	Інститут математики НАН України 2015
Series:	Symmetry, Integrability and Geometry: Methods and Applications
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Cite this:	Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy / O. Chvartatskyi, Y. Sydorenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 85 назв. — англ.

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spelling	irk-123456789-1470162019-02-13T01:24:40Z Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy Chvartatskyi, O. Sydorenko, Y. 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. 2015 Article Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy / O. Chvartatskyi, Y. Sydorenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 85 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q51; 35Q53; 35Q55; 37K35 DOI:10.3842/SIGMA.2015.028 http://dspace.nbuv.gov.ua/handle/123456789/147016 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Chvartatskyi, O. Sydorenko, Y.
spellingShingle	Chvartatskyi, O. Sydorenko, Y. Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Chvartatskyi, O. Sydorenko, Y.
author_sort	Chvartatskyi, O.
title	Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
title_short	Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
title_full	Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
title_fullStr	Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
title_full_unstemmed	Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
title_sort	darboux transformations for (2+1)-dimensional extensions of the kp hierarchy
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147016
citation_txt	Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy / O. Chvartatskyi, Y. Sydorenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 85 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT chvartatskyio darbouxtransformationsfor21dimensionalextensionsofthekphierarchy AT sydorenkoy darbouxtransformationsfor21dimensionalextensionsofthekphierarchy
first_indexed	2025-07-11T01:09:23Z
last_indexed	2025-07-11T01:09:23Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 028, 20 pages Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy Oleksandr CHVARTATSKYI † and Yuriy SYDORENKO ‡ † Mathematisches Institut, Georg-August-Universität Göttingen, 37073 Göttingen, Germany E-mail: alex.chvartatskyy@gmail.com ‡ Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine E-mail: y sydorenko@franko.lviv.ua Received September 23, 2014, in final form March 27, 2015; Published online April 10, 2015 http://dx.doi.org/10.3842/SIGMA.2015.028 Abstract. 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. Key words: KP hierarchy; symmetry constraints; binary Darboux transformation; Davey– Stewartson equation; KP equation with self-consistent sources 2010 Mathematics Subject Classification: 35Q51; 35Q53; 35Q55; 37K35 1 Introduction In the past years, a lot of attention have been given to the study of Kadomtsev–Petviashvili hierarchy (KP hierarchy) and its generalizations from both physical and mathematical points of view [1, 13, 55, 57, 61, 74]. KP equation with self-consistent sources and related k-constrained KP (k-cKP) hierarchy also present an interest [4, 9, 10, 33, 40, 42, 52, 53, 63, 64, 65, 78]. The latter hierarchy contains, in particular, nonlinear Schrödinger equation, Yajima–Oikawa equation, extension of the Boussinesq equation and KdV equation with self-consistent sources. A modified k-constrained KP (k-cmKP) hierarchy was proposed in [8, 43, 59]. The k-cKP hierarchy was extended to 2+1 dimensions ((2+1)-dimensional k-cKP hierarchy) in [48, 54, 73]. A powerful solution generating method for nonlinear integrable systems is based on the Dar- boux transformations (DT) and the binary Darboux transformations (BDT) [51]. The latter transformations were also applied to k-cKP hierarchy and its (2 + 1)-dimensional extensions (see [58, 82] and [47, 73] respectively). More general (2 + 1)-dimensional extensions of the k- cKP hierarchy and the corresponding solutions were investigated in [11]. The latter hierar- chies 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. Hamiltonian analysis for the above mentioned hierarchies, which is based on group-theoretical and Lie-algebraic methods, was elaborated in [5, 31, 34, 68, 69, 70]. Analytical scheme of the Hamiltonian analysis was described in [28]. The main aim of this work is to present new (2 + 1)-dimensional extensions of k-cKP and modified k-cKP hierarchies. It is organized as follows. In Section 2 we consider reductions (2.2) of Lax operators Lk and Mn involving nonzero integral terms with degenerate kernels. The latter reductions allow us to obtain a hierarchy that is more general then (2 + 1)-dimensional mailto:alex.chvartatskyy@gmail.com mailto:y_sydorenko@franko.lviv.ua http://dx.doi.org/10.3842/SIGMA.2015.028 2 O. Chvartatskyi and Yu. Sydorenko extensions of the k-cKP hierarchy that we considered in [11]. This is shown in Remark 2.5 (Section 2), which describes important special cases of the obtained hierarchy. KP hierarchy as a special case is also included. In Section 2 we also list some nonlinear integrable systems that are provided by Lax pairs (2.2). In particular, we get new generalizations of the N -wave problem, the matrix Davey–Stewartson (DS-III) equation and the matrix KP equation with self-consistent sources (KPSCS). Despite the fact that the latter systems have more compact scalar counterparts, we also present their matrix versions due to the recent interest in matrix and, more generally, noncommutative integrable systems (see, e.g., [14, 15, 32, 44, 71, 72, 75]). In Section 3 we present a solution generating technique (dressing method) for hierarchies (2.2), (2.6) using DTs and BDTs. New (2 + 1)-dimensional extensions of the modified k-cKP hierar- chies and corresponding solution generating methods are discussed in Section 4. Some matrix integrable equations from the respective hierarchy are also listed. This includes new exten- sions of the matrix Chen–Lee–Liu equation and the modified KP equation with self-consistent sources. A short summary of the obtained results and some problems for future investigation are presented in Conclusions. 2 New (2 + 1)-dimensional generalizations of the k-constrained KP hierarchy Further we will use the calculus of the integro-differential (pseudo-differential) operators of the form L = l∑ i=−∞ fiD i, l ∈ Z (see, e.g., [13]). Coefficients fi, i ∈ Z, are matrix-valued functions and symbol D := ∂ ∂x denotes the derivative with respect to x. Composition (multiplication) of integro-differential operators is generated by the commutation rule: Dnf := ∞∑ j=0 ( n j ) f (j)Dn−j , f (j) := ∂jf ∂xj , n ∈ Z, (2.1) where ( n 0 ) := 1, ( n j ) := n(n−1)···(n−j+1) j! . Dnf stands for the composition of the operator Dn and the operator of multiplication by matrix-valued function f , whereas by curly brackets we will denote the action of the operator on the function, i.e., Dn{f} := f (n) = ∂nf ∂xn . More generally, we will use notations L{f} and Lf in the same manner. Consider Sato–Zakharov–Shabat dressing operator: W = I + w1D −1 + w2D −2 + · · · with (N ×N)-matrix-valued coefficients wi. Introduce two differential operators βk∂τk −JkDk and αn∂tn −J̃nDn, αn, βk ∈ C, n, k ∈ N, where Jk and J̃n are N ×N commuting matrices (i.e., [J̃n,Jk] = 0). It is evident that dressed operators have the form: Lk := W ( βk∂τk − JkD k ) W−1 = βk∂τk −Bk − u−1D −1 − u−2D−2 − · · · , Mn := W ( αn∂tn − J̃nDn ) W−1 = αn∂tn −An − v−1D−1 − v−2D−2 − · · · , Bk = k∑ j=0 ujD j , uk = Jk, An = n∑ i=0 viD i, vn = J̃n, where uj and vi are matrix-valued functions of dimension N×N . Impose the following reduction on the integral parts of operators Lk and Mn: Lk = βk∂τk −Bk − qM0D −1r>, Mn = αn∂tn −An − q̃M̃0D −1r̃>, (2.2) Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy 3 where q and r are matrix-valued functions with dimension N ×m; q̃ and r̃ are matrix-valued functions with dimension N × m̃. M0 and M̃0 are constant matrices with dimensions m ×m and m̃× m̃ respectively. Reductions (2.2) generalize the corresponding analogues obtained in [11]. It will be shown at the end of this section (reductions (2.11) and Remark 2.5). Moreover, Lax pairs given by Lk and Mn (2.2) remain covariant under the action of Darboux and Binary Darboux Transfor- mations (see Section 3), which allows to construct families of solutions for the corresponding integrable systems. Lax pairs (2.2) can be also considered as matrix (2 + 1)-dimensional gene- ralizations of the respective operators appearing in the study of dispersive analogues of Benny’s equations [25]. For technical purposes we will use the following statement: Proposition 2.1. For matrix-valued functions h1 and h2 and differential operator A = l∑ i=0 fiD i, l ∈ N, with matrix-valued coefficients fi the following formulae hold: Ah1D −1h2 = ( Ah1D −1h2 ) ≥0 +A{h1}D−1h2, (2.3) h1D −1h2A = ( h1D −1h2A ) ≥0 + h1D −1[Aτ{h>2 }]>, (2.4) D−1h1h2D −1 = D−1{h1h2}D−1 −D−1D−1{h1h2}. (2.5) Symbol > in the latter stands for the matrix transpose and Aτ denotes the transpose of A, i.e., Aτ := l∑ i=0 (−1)iDif>i . Subscript ≥0 denotes the differential part of the respective operator (e.g., ( l∑ i=−∞ fiD i ) ≥0 = l∑ i=0 fiD i). Proof. All three formulae are consequences of the commutation rule (2.1). Let us check (2.4) and (2.5). It is enough to prove (2.4) for A = flD l. Using (2.1) we get h1D −1h2A− h1D−1 [ Aτ { h>2 }]> = h1D −1h2flD l − h1D−1(−1)n(h2fl) (l) = h1 ∞∑ i=0 (−1)i(h2fl) (i)Dl−i−1 − h1 ∞∑ j=0 (−1)j+l(h2ul) (j+l)D−1−j = h1 l−1∑ i=0 (−1)i(h2fl) (i)Dl−i−1 = ( h1D −1h2A ) ≥0. (2.5) follows from the following computations: D−1h1h2D −1 +D−1D−1{h1h2} = ∞∑ i=0 (−1)i(h1h2) (i)D−i−2 + ∞∑ j=0 (−1)j(h1h2) (j−1)D−j−1 = (h1h2) (−1)D−1 = D−1{h1h2}D−1. � Remark 2.2. Formulae (2.3)–(2.5) can be found in [11, 79]. Since the respective references do not contain the proof, we decided to present it in order to make the paper self-contained. The following statement follows from Proposition 2.1. Proposition 2.3. Assume that the following equations hold: Lk{q̃} = q̃Λq̃, Lτk{r̃} = r̃Λr̃, Mn{q} = qΛq, M τ n{r} = rΛr, (2.6) 4 O. Chvartatskyi and Yu. Sydorenko where Λq, Λr and Λq̃, Λr̃ are constant matrices with dimensions (m×m) and (m̃× m̃) respec- tively, that satisfy equations: Λq̃M̃0 − M̃0Λ > r̃ = 0, ΛqM0 −M0Λ > r = 0. Then Lax equation [Lk,Mn] = 0 holds if and only if equation [Lk,Mn]≥0 = 0 is satisfied. Proof. From the equality [Lk,Mn] = [Lk,Mn]≥0 + [Lk,Mn]<0 we obtain that Lax equation [Lk,Mn] = 0 is equivalent to the following one: [Lk,Mn]≥0 = 0, [Lk,Mn]<0 = 0. Thus, it is sufficient to prove that equalities Lk{q̃} = q̃Λq̃, Lτk{r̃} = r̃Λr̃, Mn{q} = qΛq, M τ n{r} = rΛr imply [Lk,Mn]<0 = 0. From the form of operators Lk, Mn (2.2) we obtain: [Lk,Mn]<0 = [ q̃M̃0D −1r̃>, βk∂τk −Bk ] <0 + [ qM0D −1r>, q̃M̃0D −1r̃> ] <0 + [ αn∂tn −An,qM0D −1r> ] <0 . (2.7) After direct computations for each of the three items at the right-hand side of formula (2.7) we get:[ q̃M̃0D −1r̃>, βk∂τk −Bk ] <0 = − ( βkq̃τk −Bk{q̃} ) M̃0D −1r̃> − q̃M̃0D −1(βkr̃>τk + ( Bτ k{r̃} )>) ,[ αn∂tn −An,qM0D −1r> ] <0 = (αnqtn −An{q})M0D −1r> + qM0D −1(αnr>tn + (Aτn{r})> ) ,[ qM0D −1r>, q̃M̃0D −1r̃> ] <0 = qM0D −1{r>q̃}M̃0D −1r̃> − qM0D −1D−1 { r>q̃ } M̃0r̃ > − q̃M̃0D −1{r̃>q}M0D −1r> + q̃M̃0D −1D−1 { r̃>q } M0r >. (2.8) The latter formulae are consequences of (2.3)–(2.5). From formulae (2.7), (2.8) using (2.6) we get [Lk,Mn]<0 = Mn{q}M0D −1r> − qM0D −1(M τ n{r})> − Lk{q̃}M̃0D −1r̃> + q̃M̃0D −1(Lτk{r̃})> = q ( ΛqM0 −M0Λ > r ) D−1r> − q̃ ( Λq̃M̃0 − M̃0Λ > r̃ ) D−1r̃> = 0. From the last formula we obtain that equality [Lk,Mn] = 0 is equivalent to condition (2.6). � Consider some nonlinear systems that hierarchy given by (2.2) and (2.6) contains. In all examples listed below we assume that equations (2.6) hold. Due to Proposition 2.3 it implies the equivalence of equations [Lk,Mn] = 0 and [Lk,Mn]≥0 = 0. For simplicity we set Λq = Λr = 0, Λq̃ = Λr̃ = 0. 1. k = 1, n = 1. We shall use the following notation β := β1, α := α1, τ := τ1, t := t1. Then (2.2) reads: L1 = β∂τ − JD + [J,Q]− qM0D −1r>, M1 = α∂t − J̃D + [J̃ , Q]− q̃M̃0D −1r̃>, where matrices J and J̃ commute. According to Proposition 2.3 the commutator equation [L1,M1] = 0 is equivalent to the system: β[J̃ , Qτ ]− α[J,Qt] + JQxJ̃ − J̃QxJ + [[J,Q], [J̃ , Q]] + [ J, q̃M̃0r̃ >]+ [ qM0r >, J̃ ] = 0, βq̃τ − J q̃x + [J,Q]q̃− qM0S1 = 0, −βr̃>τ + r̃>x J + r̃>[J,Q] + S2M0r > = 0, αqt − J̃qx + [J̃ , Q]q− q̃M̃0S2 = 0, −αr>t + r>x J̃ + r>[J̃ , Q] + S1M̃0r̃ > = 0, Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy 5 S1,x = r>q̃, S2,x = r̃>q. The latter system is a generalization of the N -wave problem [2, 76, 77, 84]. In case we set Q = 0 we obtain a noncommutative generalization of the nonlinear system of four waves [38, 39]. Under the Hermitian conjugation reduction r̃ = ¯̃q, M0 =M∗0, M̃0 = M̃∗0, r = q̄, Q = −Q∗, α, β ∈ R, J = J∗, J̃ = J̃∗ the latter system reads: β[J̃ , Qτ ]− α[J,Qt] + JQxJ̃ − J̃QxJ + [[J,Q], [J̃ , Q]] + [J, q̃M̃0q̃ ∗]− [q̃M̃0q̃ ∗, J̃ ] = 0, S1,x = q∗q̃, βq̃τ − J q̃x + [J,Q]q̃− qM0S1 = 0, αqt − J̃qx + [J̃ , Q]q− q̃M̃0S ∗ 1 = 0. 2. k = 1, n = 2. L1 = β1∂τ1 − qM0D −1r>, M2 = α2∂t2 − cD2 + v − q̃M̃0D −1r̃>, c ∈ C. Lax equation [L1,M2] = 0 is equivalent to the following generalization of the DS-III equation: β1q̃τ1 = qM0S1, β1r̃ > τ1 = S2M0r >, S1,x = r>q̃, α2qt2 − cqxx + vq = q̃M̃0S2, α2r > t2 + cr>xx − r>v = S1M̃0r̃ >, S2,x = r̃>q, β1vτ1 = 2 ( qM0r >) x . If we set q̃ = 0, r̃ = 0 we recover the system α2qt2 − cqxx + vq = 0, α2r > t2 + cr>xx − r>v = 0, β1vτ1 = 2 ( qM0r >) x , which under reduction α2 ∈ iR, β1, c ∈ R,M0 =M∗0, v = v∗, r = q̄ becomes the matrix version of the DS-III system (see [26]): α2qt2 − cqxx + vq = 0, β1vτ1 = 2 ( qM0q ∗) x . 3. k = 3, n = 2. In this case we obtain the following pair of operators: L3 = β3∂τ3 − c1 ( D3 − wD − u ) − qM0D −1r>, M2 = α2∂t2 − c2 ( D2 − v ) − q̃M̃0D −1r̃>. Equation [L3,M2] = 0 is equivalent to the following system: c1c2(2w − 3v) = 0, −α2c1wt2 − 3 2 c1c2vxx + 3c1 ( q̃M̃0r̃ >) x + 2c1c2ux = 0, β3c2vτ3 − c1c2vxxx + 3c1 ( q̃xM̃0r̃ >) x + c1c2wvx − c1 [ w, q̃M̃0r̃ >] + c1c2[u, v] + c1c2uxx − α2c1ut2 − 2c2 ( qM0r >) x = 0, β3q̃τ3 − c1q̃xxx + c1wq̃x + c1uq̃− qM0S1 = 0, S1,x = r>q̃, − β3r̃>τ3 + c1r̃ > xxx − c1 ( r̃>w ) x + c1r̃ >u+ S2M0r > = 0, S2,x = r̃>q, α2qt2 − c2qxx + c2vq− q̃M̃0S2 = 0, α2r > t2 + c2r > xx − c2r>v − S1M̃0r̃ > = 0. (2.9) The latter consists of several special cases: a) c1 = c2 = 1. In this case the latter system can be rewritten in the following way: −3 2 α2vt2 − 3 2 vxx + 3 ( q̃M̃0r̃ >) x + 2ux = 0, 6 O. Chvartatskyi and Yu. Sydorenko( β3vτ3 − 1 4 vxxx + 3 2 vvx ) x − 3α2vt2t2 + ( [u, v]− [ w, q̃M̃0r̃ >]) x + 3 2 ( q̃xxM̃0r̃ > − q̃M̃0r̃ > xx + α ( q̃M̃0r̃ >) t2 ) x − 2 ( qM0r >) xx = 0, β3q̃τ3 − q̃xxx + 3 2 vq̃x + uq̃− qM0S1 = 0, S1,x = r>q̃, −β3r̃>τ3 + r̃>xxx − 3 2 ( r̃>v ) x + r̃>u+ S2M0r > = 0, S2,x = r̃>q, α2qt2 − qxx + vq− q̃M̃0S2 = 0, α2r > t2 + r>xx − r>v − S1M̃0r̃ > = 0. In the scalar case (N = 1) under the Hermitian conjugation reduction: α2 ∈ iR, r = q̄, M0 = M∗0 (M2 = M∗2 ) and β3 ∈ R, M̃0 = −M̃∗0, r̃ = ¯̃q, w = w∗, w∗x = u + u∗, v = v∗ (L3 = −L∗3). the latter equation reads:( β3vτ3 − 1 4 vxxx + 3 2 vvx ) x − 3α2vt2t2 + 3 2 ( q̃xxM̃0q̃ ∗ − q̃M̃0q̃ ∗ xx + α ( q̃M̃0q̃ ∗) t ) x − 2 ( qM0q ∗) xx = 0, β3q̃τ3 − q̃xxx + 3 2 vq̃x + uq̃− qM0S1 = 0, S1,x = q∗q̃, α2qt2 − qxx + vq− q̃M̃0S ∗ 1 = 0. (2.10) This system is a generalization of the KP equation with self-consistent sources (KPSCS). In particular, if we set M̃0 = 0, q̃ = 0 we recover KPSCS of the first type( β3vτ3 − 1 4 vxxx + 3 2 vvx ) x − 3α2vt2t2 = 2 ( qM0q ∗) xx , α2qt2 − qxx + vq = 0. In case M0 = 0, q = 0 in (2.10) we obtain KPSCS of the second type( β3vτ3 − 1 4 vxxx + 3 2 vvx ) x − 3α2vt2t2 = −3 2 ( q̃xxM̃0q̃ ∗ − q̃M̃0q̃ ∗ xx + α ( q̃M̃0q̃ ∗) t ) x , β3q̃τ3 − q̃xxx + 3 2 vq̃x + uq̃ = 0. KPSCS and the respective matrix (1 + 1)-dimensional counterpart (KdV equation with self- consistent sources) have been investigated recently via Darboux transformations [45, 83] and the inverse scattering method [6]. b) c1 = 0, c2 = 1. In this case (2.9) becomes the following: β3vτ3 = 2 ( qM0r >) x , β3q̃τ3 − qM0S1 = 0, −β3r̃>τ3 + S2M0r > = 0, α2qt2 − qxx + vq− q̃M̃0S1 = 0, S1,x = r>q̃, α2r > t2 + r>xx − r>v − S2M̃0r̃ > = 0, S2 = r̃>q. In case α2 ∈ iR, β3 ∈ R, M0 = M∗0, q = r̄, M̃0 = 0, q̃ = r̃ = 0 the latter becomes the noncommutative generalization of the DS-III system. Now we will show that (2 + 1)-BDk-cKP hierarchy presented in [11] can be recovered from Lax operators (2.2). At first, let us put in formulae (2.2): q̃ := (q̃1, clq[0], clq[1], . . . , clq[l]), r̃ := (r̃1, r[l], r[l − 1], . . . , r[0]), M̃0 = diag(M̃1, Il+1 ⊗M0), (2.11) Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy 7 where q[j] = (Lk) j{q}, r[j] = (Lτk)j{r}, j = 0, l. I.e., m̃ = m̃1 + m(l + 1) and matrices q̃ and r̃ consist of N × m̃1-matrix-valued blocks q̃1 and r̃1 and (N ×m)-matrix-valued blocks q[j] and r[j], j = 0, l. M̃0 is a block-diagonal matrix and Il+1⊗M0 stands for the tensor product of the (l+1)-dimensional identity matrix Il+1 and matrixM0. Then we get the following operators in (2.2): Lk = βk∂τk −Bk − qM0D −1r>, Bk = k∑ j=0 ujD j , uj = uj(x, τk, tn), βk ∈ C, Mn = Mn,l = αn∂tn −An − q̃1M̃1D −1r̃>1 − cl l∑ j=0 q[j]M0D −1r>[l − j], l = 1, . . . An = n∑ i=0 viD i, vi = vi(x, τk, tn), αn ∈ C. (2.12) The following proposition holds: Proposition 2.4. Assume that equations Mn,l{q} = cl(Lk) l+1{q}, M τ n,l{r} = cl(L τ k)l+1{r}, Lk{q̃1} = q̃1Λq̃1 , Lτk{r̃1} = r̃1Λr̃1 (2.13) with constant matrices Λq̃1 and Λr̃1 are satisfied, where the latter solve Λq̃1M̃1 − M̃1Λ > r̃1 = 0. Then Lax equation [Lk,Mn,l] = 0 holds if and only if its differential part is equal to zero, i.e., [Lk,Mn,l]≥0 = 0. Proof. The proof is similar to the proof of the Proposition 2.3 and the proof of the Theorem 1 in [11]. � Remark 2.5. Setting q̃1 = 0 and r̃1 = 0 in (2.12) we recover (2 + 1)-BDk-cKP hierarchy (Lax pairs (2.12) with equations (2.13)) that contains the following subcases: 1. βk = 0, cl = 0. Under this assumption we obtain matrix k-constrained KP hierarchy [58]. We shall point out that the case βk = 0 and cl 6= 0 also leads to matrix k-constrained KP hierarchy. 2. cl = 0, N = 1, vn = uk = 1, vn−1 = uk−1 = 0. In this way we recover (2+1)-dimensional k- cKP hierarchy [73]. 3. n = 0. The differential part of M0,l (2.12) is equal to zero in this case (A0 = 0) and we get a new generalization of DS-III hierarchy. 4. cl = 0. We obtain (tA, τB)-matrix KP hierarchy that was investigated in [36]. 5. If l = 0 we recover (γA, σB)-matrix KP hierarchy [37]. 6. Case N = 1, q = 0, r = 0 leads to KP hierarchy. 3 Dressing methods for the new (2 + 1)-dimensional generalizations of k-constrained KP hierarchy 3.1 Dressing via Darboux transformations In this section we will consider Darboux transformations (DT) for the pair of operators (2.2) and its reduction (2.12). At first, we shall start with the linear problem associated with the 8 O. Chvartatskyi and Yu. Sydorenko operator Lk (2.2): Lk{ϕ1} = βk(ϕ1)τk − k∑ j=0 uj(ϕ1) (j) − qM0D −1{r>ϕ1 } = ϕ1Λ1, where ϕ1 is (N × N)-matrix-valued function; Λ1 is a constant matrix with dimension N × N . Introduce the DT in the following way: W1[ϕ1] = ϕ1Dϕ −1 1 = D − ϕ1,xϕ −1 1 . (3.1) The following proposition holds. Proposition 3.1. The operator L̂k[1] := W1[ϕ1]LkW −1 1 [ϕ1] obtained from Lk (2.2) via DT (3.1) has the form L̂k[1] := W1[ϕ1]LkW −1 1 [ϕ1] = βk∂τk − B̂k − q̂1M0D −1r̂>1 , B̂k[1] = k∑ j=0 ûj [1]Dj , where q̂1 = W1[ϕ1]{q}, r̂1 = W−1,τ1 [ϕ1]{r}. ûj [1] are (N ×N)-matrix coefficients depending on function ϕ1 and coefficients ui, i = 0, k. In particular, ûk[1] = uk. Proof. It is evident that the inverse operator to (3.1) has the form W−11 [ϕ1] = ϕ1D −1ϕ−11 . Thus, we have L̂k[1] = W1[ϕ1]LkW −1 1 [ϕ1] = ϕ1Dϕ −1 1 ( βk∂τk −Bk − qM0D −1r> ) ϕ1D −1ϕ−11 = βk∂τk + (L̂k[1])≥0 + (L̂k[1])<0, where (L̂k[1])≥0 = −B̂k[1] = − k∑ j=0 ûj [1]Dj . It remains to find the explicit form of (L̂k)<0. Using formulae (2.3)–(2.5) we have: (L̂k[1])<0 = ( βkϕ1Dϕ −1 1 ϕ1,τkD −1ϕ−11 ) <0 − ( ϕ1Dϕ −1 1 Bk{ϕ1}D−1ϕ−11 ) <0 − ( ϕ1Dϕ −1 1 qM0D −1{r>ϕ1 } D−1ϕ−11 − ϕ1Dϕ −1 1 qM0D −1D−1 { r>ϕ1 } ϕ−11 ) <0 = ( ϕ1Dϕ −1 1 ϕ1Λ1D −1ϕ−11 ) <0 + ϕ1D { ϕ−11 q } M0D −1D−1 { r>ϕ1 } ϕ−11 = −W1{q}M0D −1(W−1,τ1 {r} )> . � It is also possible to generalize the latter theorem to the case of finite number of solutions of linear problems associated with the operator Lk. Namely, let functions ϕs, s = 1,K be solutions of the problems: Lk{ϕs} = βk(ϕs)τk − k∑ j=0 uj(ϕs) (j) − qM0D −1{r>ϕs} = ϕsΛs, s = 1,K. (3.2) For further convenience we shall use the notations ϕs[1] := ϕs, s = 1,K and define the following functions: ϕs[2] = W1[ϕ1[1]]{ϕs[1]}, s = 2,K. (3.3) Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy 9 Now, using functions ϕ1[1], ϕ2[2], we shall define functions ϕs[3], s = 3,K: ϕs[3] := W1[ϕ2[2]]{ϕs[2]} = W1[ϕ2[2]]W1[ϕ1[1]]{ϕs[1]}, s = 3,K. At the p-th step we obtain functions: ϕs[p] := W1[ϕp−1[p − 1]]{ϕs[p − 1]} = W1[ϕp−1[p − 1]] · · ·W1[ϕ2[2]]W1[ϕ1[1]]{ϕs[1]}, s = p,K. Now we shall construct the following generalization of DT (3.1): WK [ϕ1, . . . , ϕK ] = W1[ϕK [K]] · · ·W1[ϕ1[1]] = ( D − ϕK,x[K]ϕ−1K [K] ) · · · ( D − ϕ1,x[1]ϕ−11 [1] ) . (3.4) The following statement holds: Proposition 3.2. The operator L̂k[K] := WK [ϕ1[1], ϕ2[1], . . . , ϕK [1]]LkW −1 K [ϕ1[1], ϕ2[1], . . . , ϕK [1]] = WKLkW −1 K obtained from Lk (2.12) via DT (3.4) has the form L̂k[K] := WKLkW −1 K = βk∂τk − B̂k[K]− q̂KM0D −1r̂>K , B̂k[K] = k∑ j=0 ûj [K]Dj , where q̂K = WK [ϕ1[1], . . . , ϕK [1]]{q}, r̂K = W−1,τK [ϕ1[1], . . . , ϕK [1]]{r}. ûj [K] are (N ×N)-matrix coefficients depending on functions ϕs, s = 1,K and coefficients ui, i = 0, k. In particular, ûk[K] = uk. Proof. The proof can be done via induction by K. Namely, assume that the statement holds for K − 1. I.e., L̂k[K − 1] = WK−1LkW −1 K−1 = βk∂τk − B̂k[K − 1]− q̂K−1M0D −1r̂>K−1, (3.5) with q̂K−1 = WK−1[ϕ1[1], . . . , ϕK−1[1]]{q} and r̂K−1 = W−1,τK−1 [ϕ1[1], . . . , ϕK−1[1]]{r}. The function ϕK [K] = WK−1{ϕK [1]} = WK−1[ϕ1, . . . , ϕK−1]{ϕK [1]} (see formulae (3.3), (3.4)) satis- fies the equation: L̂k[K − 1]{ϕK [K]} = WK−1LkW −1 K−1{WK−1{ϕK [1]}} = = WK−1Lk{ϕK [1]} = ϕK [K]ΛK . Now, it remains to apply Proposition 3.1 to operator L̂k[K−1] (3.5) with the DT W1[ϕK [K]] (see formula (3.1)) and use formula WK = W1[ϕK [K]]WK−1 that immediately follows from (3.4). � Remark 3.3. We shall also point out that in a scalar case (N = 1) the DT WK (3.4) can be rewritten in the following way: WK := 1 W[ϕ1, ϕ2, . . . , ϕK ] ∣∣∣∣∣∣∣∣ ϕ1 . . . ϕK 1 ϕ′1 . . . ϕ′K D . . . . . . . . . . . . ϕ (K) 1 . . . ϕ (K) K DK ∣∣∣∣∣∣∣∣ = DK + K−1∑ i=0 wiD i, (3.6) where W[ϕ1, ϕ2, . . . , ϕK ] denotes the Wronskian constructed by solutions ϕj , j = 1, . . . ,K, of the linear problem (3.2). It acts on the vector-valued function q = (q1, . . . , qm) in the following way: WK{q} = (WK{q1}, . . . ,WK{qm}), where WK{qj} = W[ϕ1,ϕ2,...,ϕK ,qj ] W[ϕ1,ϕ2,...,ϕK ] . 10 O. Chvartatskyi and Yu. Sydorenko The Darboux transformations (3.1), (3.4) are widely used for the solution generating tech- nique involving Lax pairs consisting of differential operators [29, 30, 50, 51, 66]. Corresponding extensions to integro-differential cases of Lax pairs were made in [47, 54, 58, 60, 73] to con- struct solutions of constrained KP hierarchies and their generalizations. Transformations that generalize (3.1), (3.4) also arise in the bidifferential calculus approach to integrable systems and their hierarchies [16, 18]. In contrast to (3.4), formula (3.6) does not require iterative applica- tions of DTs and therefore can be used more effectively in the scalar case. We point out that in the matrix (noncommutative) case (3.6) is not valid anymore. However, the corresponding quasideterminant representations can be used (see, e.g., [29, 30]). From Proposition 3.2 we obtain the corollary for Lax pairs (2.2). Namely, let functions ϕs, s = 1,K be solutions of the problems: Lk{ϕs} = βk(ϕs)τk − k∑ j=0 uj(ϕs) (j) − qM0D −1{r>ϕs} = ϕsΛs, Mn{ϕs} = αn(ϕs)tn − n∑ i=0 vi(ϕs) (i) − q̃M̃0D −1{r̃>ϕs} = ϕsΛ̃s, s = 1,K. Then the following statement holds: Corollary 3.4. Assume that Lax equation with operators Lk and Mn (2.2) holds: [Lk,Mn] = 0. Then: 1. Transformed operators L̂k[K] := WK [ϕ1, . . . , ϕK ]LkW −1 K [ϕ1, . . . , ϕK ], M̂n[K] := WK [ϕ1, . . . , ϕK ]MnW −1 K [ϕ1, . . . , ϕK ], where WK is defined by (3.4), have the form: L̂k[K] := WKLkW −1 K = βk∂τk − B̂k[K]− q̂KM0D −1r̂>K , M̂n[K] := WKMnW −1 K = αn∂tn − Ân[K]− ˆ̃qKM̃0D −1ˆ̃r > K , Ân[K] = n∑ i=0 v̂i[K]Di, B̂k[K] = k∑ j=0 ûj [K]Dj , (3.7) where q̂K = WK [ϕ1[1], . . . , ϕK [1]]{q}, r̂K = W−1,τK [ϕ1[1], . . . , ϕK [1]]{r}, ˆ̃qK = WK [ϕ1[1], . . . , ϕK [1]]{q̃}, ˆ̃rK = W−1,τK [ϕ1[1], . . . , ϕK [1]]{r̃}. 2. The operators L̂k[K] and M̂n[K] (3.7) satisfy Lax equation: [L̂k[K], M̂n[K]] = 0. 3. In case of reduction (2.11) in Lax pair (2.2) we have: ˆ̃qK = (ˆ̃q1,K , clq̂K [0], . . . , clq̂K [l]), ˆ̃rK = (ˆ̃r1,K , r̂K [0], . . . r̂K [l]), where ˆ̃q1,K = WK{q̃1}, ˆ̃r1,K = W−1,τK {r̃1}, q̂[j] = (L̂k[K])j{q̂K}, r̂[j] = (L̂τk[K])j{r̂K}, j = 0, l. Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy 11 Proof. 1. Form (3.7) of operators L̂k[K], M̂n[K] follows from Proposition 3.2. 2. We obtain the proof of this item from the following formulae[ L̂k[K], M̂n[K] ] = [ WKLkW −1 K ,WKMnW −1 K ] = WK [Lk,Mn]W−1K = 0. 3. From formulae: ˆ̃qK = WK{(q̃1, clq[0], . . . , clq[l])} = (WK{q̃1}, clWK{q[0]}, . . . , clWK{q[l]}), WK{q[j]} = WK{(Lk)j{q}} = ( WKLkW −1 K ) {WK{q}} = L̂k[K]{q̂K} we get the form of ˆ̃qK mentioned in item 3. The form of ˆ̃rK can be obtained in a similar way. � 3.2 Dressing via binary Darboux transformations In this section we will show that results of paper [80] on binary Darboux transformations (BDT) for linear integro-differential operators can be extended to families of Lax pairs given by (2.2). Namely, let (N ×K)-matrix functions ϕ and ψ be solutions of linear problems: Lk{ϕ} = ϕΛk, Lτk{ψ} = ψΛ̃k, Λk, Λ̃k ∈ MatK×K(C). Following [80] we introduce BDT in the following way: W = I − ϕ ( C +D−1 { ψ>ϕ })−1 D−1ψ>, (3.8) where C is a K ×K-constant nondegenerate matrix. The inverse operator W−1 has the form: W−1 = I + ϕD−1 ( C +D−1{ψ>ϕ} )−1 ψ>. The following theorem is proven in [80]. Theorem 3.5. The operator L̂k := WLkW −1 obtained from Lk in (2.2) via BDT (3.8) has the form L̂k := WLkW −1 = βk∂τk − B̂k − q̂M0D −1r̂> + ΦMkD −1Ψ>, B̂k = k∑ j=0 ûjD j , where Mk = CΛk − Λ̃>k C, Φ = ϕ∆−1, Ψ = ψ∆−1,>, ∆ = C +D−1 { ψ>ϕ } , q̂ = W{q}, r̂ = W−1,τ{r}. ûj are (N ×N)-matrix coefficients depending on functions ϕ, ψ and uj. In particular, ûk = uk, ûk−1 = uk−1 + [ uk, ϕ ( C +D−1 { ψ>ϕ })−1 ψ> ] . Solution generating method for the hierarchy (2.2)–(2.6) is given by the corollary, which follows from the previous theorem. Corollary 3.6. Let (N ×K)-matrix functions ϕ and ψ satisfy equations: Lk{ϕ} = ϕΛk,1, L τ k{ψ} = ψΛ̃k,1, Λk,1, Λ̃k,1 ∈ MatK×K(C), Mn{ϕ} = ϕΛn,2,M τ k {ψ} = ψΛ̃n,2, Λn,2, Λ̃n,2 ∈ MatK×K(C) 12 O. Chvartatskyi and Yu. Sydorenko with operators Lk and Mn (2.2) satisfying [Lk,Mn] = 0. Then transformed operators L̂k and M̂n satisfy Lax equation [L̂k, M̂n] = 0 and have the form: L̂k := WLkW −1 = βk∂τk − B̂k − q̂M0D −1r̂> + ΦMk,1D −1Ψ>, B̂k = k∑ j=0 ûjD j , M̂n := WMnW −1 = αn∂tn − Ân − ˆ̃qM̃0D −1ˆ̃r > + ΦMn,2D −1Ψ>, Ân = n∑ i=0 v̂iD i,(3.9) where Mk,1 = CΛk,1 − Λ̃>k,1C, Mn,2 = CΛn,2 − Λ̃>n,2C, Φ = ϕ∆−1, Ψ = ψ∆−1,>, ∆ = C +D−1{ψ>ϕ}, q̂ = W{q}, r̂ = W−1,τ{r}, ˆ̃q = W{q̃}, ˆ̃r = W−1,τ{r̃}. ûj are (N ×N)-matrix coefficients depending on functions ϕ, ψ and uj, vi. In particular, ûk = uk, ûk−1 = uk−1 + [ uk, ϕ ( C +D−1 { ψ>ϕ })−1 ψ> ] , v̂n = vn, v̂n−1 = vn−1 + [ vn, ϕ ( C +D−1 { ψ>ϕ })−1 ψ> ] . Proof. From formulae W [Lk,Mn]W−1 = [L̂k, M̂n] = 0 we obtain that Lax equation with transformed operators is satisfied. Form (3.9) of the transformed operators L̂k and M̂n follows from Theorem 3.5. � BDTs were used to generate solutions of the constrained KP hierarchies in [56, 82] and they were also applied to (2 + 1)-BDk-cKP hierarchy in [11]. Theorem 3.5 with Corollary 3.6 extend the respective results to (2 + 1)-dimensional generalizations of the k-constrained KP hierarchy (2.2), (2.6). 4 New (2 + 1)-dimensional generalizations of the modified k-constrained KP hierarchy We will investigate the following (2+1)-dimensional generalizations of the modified k-constrained KP (k-cmKP) hierarchy: Lk = βk∂τk −Bk − qM0D −1r>D, Bk = k∑ j=1 ujD j , uj = uj(x, τk, tn), βk ∈ C, Mn = αn∂tn −An − q̃M̃0D −1r̃>D, An = n∑ i=1 viD i, vi = vi(x, τk, tn), αn ∈ C,(4.1) where uj and vi are matrix-valued functions of dimension N × N ; q and r are matrix-valued functions of dimension N ×m; q̃ and r̃ are matrix-valued functions with dimension N × m̃. M0 and M̃0 are constant matrices with dimensions m×m and m̃× m̃ respectively. Using equality (2.4) we can rewrite operators (4.1) as Lk = βk∂τk −Bk − qM0r > + qM0D −1r>x , Mn = αn∂tn −An − q̃M̃0r̃ > + q̃M̃0D −1r̃>x . (4.2) From the latter it becomes clear that (4.1) can be considered as a reduction in (2.2) (Bk → Bk+qM0r >, r> → −r>x ). However, Lax pairs (4.1) provide us with different (2+1)-dimensional Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy 13 equations (see (4.4) and (4.5)). In addition, they require different kind of BDT (see Theorem 4.2 with its corollary) for the corresponding solution generating technique. Setting q̃ = 0, r̃ = 0, N = 1 in (4.1) we recover (2 + 1)-dimensional k-cmKP hierarchy [47]. The following proposition holds: Proposition 4.1. Lax equation [Lk,Mn] = 0 is satisfied in case the following equations hold: [Lk,Mn]≥0 = 0, Lk{q̃} = q̃Λq̃, ( D−1LτkD ) {r̃} = r̃Λr̃, Mn{q} = qΛq, ( D−1M τ nD ) {r} = rΛr, (4.3) where Λq, Λr, Λq̃, Λr̃ are constant matrices with dimensions (m×m) and (m̃× m̃) respectively that satisfy equations: Λq̃M̃0 − M̃0Λ > r̃ = 0, ΛqM0 −M0Λ > r = 0. Proof. The proof is a direct consequence of reductions (4.2) and Proposition 2.3. � Consider some examples of the hierarchy given by (4.1) and (4.3). 1. k = 1, n = 2. L1 = β1∂τ1 −D − qM0D −1r>D, M2 = α2∂t2 −D2 − vD − q̃M̃0D −1r̃>D. Lax representation [L1,M2] = 0 is equivalent to the following system: α2qt2 − qxx − vqx − q̃M̃0D −1{r̃>qx} = qΛq, −α2rt2 − rxx + v>rx − r̃M̃>0 D−1 { q̃>rx } = rΛr, β1q̃τ1 − q̃x − qM0D −1{r>q̃x} = q̃Λq̃, −β1r̃τ1 + r̃x − rM>0 D−1 { q>r̃x } = r̃Λr̃, vx − β1vτ1 = 2 ( qM0r >) x . In case of the Hermitian conjugation reduction β1 ∈ R, α2 ∈ iR, M∗0 = −M0, M̃∗0 = M̃0, ¯̃r = q̃, r̄ = q, v = −v∗ (L∗1 = −DL1D −1, M∗2 = DM2D −1) the latter equation reduces to the following one: α2qt2 − qxx − vqx − q̃M̃0D −1{q̃∗qx} = qΛq, β1q̃τ1 − q̃x − qM0D −1{q∗q̃x} = q̃Λq̃, vx − β1vτ1 = 2 ( qM0q ∗) x . (4.4) In case we set q̃ = 0, Λq = 0 we get a matrix (2 + 1)-dimensional generalization of the Chen- Lee-Liu equation α2qt2 − qxx − vqx = 0, vx − β1vτ1 = 2 ( qM0q ∗) x . 2. k = 3, n = 2. L3 = β3∂τ3 − c1 ( D3 + wD2 + vD ) − qM0D −1r>D, M2 = α2∂t2 − c2D2 − uD − q̃M̃0D −1r̃>D. Using (4.3) we get that Lax representation [L3,M2] = 0 is equivalent to the following system 3ux − 2c2wx − [u,w] = 0, α2c1wt2 − c1c2wxx + 2c1wux − c1uwx + 3c1uxx + 3c1 ( q̃M̃0r̃ >) x − 2c2c1vx − c2 [ q̃M̃0r̃ >, w ] + c1[v, u] = 0, −β3uτ3 + c1uxxx + c1wuxx + 3c1 ( q̃xM̃0r̃ >) x + 2c1wq̃xM̃0r̃ > 14 O. Chvartatskyi and Yu. Sydorenko + c1q̃M̃0r̃ > xw + α2c1vt2 − c1c2vxx + c1vux − c1uvx + c1 [ v, q̃M0r̃ >] − 2c2 ( qM0r >) x + c1wq̃M̃0r̃ > x + [ qM0r >, u ] = 0, β3q̃τ3 − c1q̃xxx − c1wq̃xx − c3vq̃x − qM0D −1{r>q̃x} = q̃Λq̃, α2qt2 − c2qxx − uqx − q̃M̃0D −1{r̃>qx} = qΛq, −β3r̃τ3 + c1r̃xxx − c1 ( w>r̃x ) x + c1v >r̃x − rM>0 D−1 { q>r̃x } = r̃Λr̃, −α2rt2 − c2rxx + u>rx − r̃M̃>0 D−1 { q̃>rx } = rΛr. Set c1 = c2 = 1 in the scalar case (N = 1). Eliminating variables w and v from the first and second equation respectively, we get −β3uτ3 − 1 4 uxxx − 3 8 u2ux + 3 4 α2uxD −1{ut2}+ 3 4 α2 2D −1{ut2t2} + 3 2 ( uq̃M̃0r̃ >) x + 3 2 α2 ( q̃M0r̃ >) t2 − 3 2 ( q̃M̃0r̃ >) xx − 2(qM0r >)x = 0, β3q̃τ3 − q̃xxx − 3 2 uq̃xx − vq̃x − qM0D −1{r>q̃x} = q̃Λq̃, α2qt2 − qxx − uqx − q̃M̃0D −1{r̃>qx} = qΛq, −β3r̃τ3 + r̃xxx − 3 2 (ur̃x)x + vr̃x − rM>0 D−1 { q>r̃x } = r̃Λr̃, −α2rt2 − rxx + urx − r̃M̃>0 D−1 { q̃>rx } = rΛr, v = 3 4 ux + 3 8 u2 + 3 2 ( q̃M̃0r̃ >)+ 3 4 α2D −1{ut2}. The latter under the Hermitian conjugation reduction α2 ∈ iR, β3 ∈ R,M∗0 = −M0, M̃0 = M̃∗0, ū = −u, ¯̃r = q̃, r̄ = q (L∗3 = −DL3D −1, M∗2 = DM2D −1) reads: −β3uτ3 − 1 4 uxxx − 3 8 u2ux + 3 4 α2uxD −1{ut2}+ 3 4 α2 2D −1{ut2t2} + 3 2 (uq̃M̃0q̃ ∗)x + 3 2 α2(q̃M̃0q̃ ∗)t2 − 3 2 (q̃M̃0q̃ ∗)xx − 2 ( qM0q ∗) x = 0, β3q̃τ3 − q̃xxx − 3 2 uq̃xx − ( 3 4 ux + 3 8 u2 + 3 2 (q̃M̃0q̃ ∗) + 3 4 α2D −1{ut2} ) q̃x − qM0D −1{q∗q̃x} = q̃Λq̃, α2qt2 − qxx − uqx − q̃M̃0D −1{r̃>qx} = qΛq. (4.5) q̃ = 0, Λq = 0 lead to the modified KPSCS of the first type −β3uτ3 − 1 4 uxxx − 3 8 u2ux + 3 4 α2uxD −1{ut2}+ 3 4 α2 2D −1{ut2t2} = 2 ( qM0q ∗) x , α2qt2 − qxx − uqx = 0. If q = 0, Λ̃q̃ = 0 in (4.5) we recover the second type of the modified KPSCS −β3uτ3 − 1 4 uxxx − 3 8 u2ux + 3 4 α2uxD −1{ut2}+ 3 4 α2 2D −1{ut2t2} + 3 2 (uq̃M̃0q̃ ∗)x + 3 2 α2(q̃M̃0q̃ ∗)t2 − 3 2 (q̃M̃0q̃ ∗)xx = 0, β3q̃τ3 − q̃xxx − 3 2 uq̃xx − ( 3 4 ux + 3 8 u2 + 3 2 (q̃M̃0q̃ ∗) + 3 4 α2D −1{ut2} ) q̃x = 0, Both types were investigated in [47] within (2 + 1)-dimensional extensions of the k-cmKP hier- archy. Darboux Transformations for (2 + 1)-Dimensional Extensions of the KP Hierarchy 15 4.1 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 (4.1) and (4.3). First of all, we start with the matrix version of the theorem that was proven in [12]. Theorem 4.2. Let (N ×K)-matrix functions ϕ and ψ satisfy linear problems: Lk{ϕ} = ϕΛk, L τ k{ψ} = ψΛ̃k, Λk, Λ̃k ∈ MatK×K(C), Lk = βk∂τk −Bk − qM0D −1r>D, Bk = k∑ i=1 uiD i. Then the operator Lk transformed via Wm := w−10 W = w−10 ( I − ϕ∆−1D−1ψ> ) = I − ϕ∆̃−1D−1(D−1{ψ})>D, (4.6) where w0 = IN − ϕ∆−1D−1 { ψ> } , ∆̃ = −C +D−1 { D−1 { ψ> } ϕx } , ∆ = C +D−1 { ψ>ϕ } , (4.7) has the form: L̂k := WmLkW −1 m = βk∂τk − B̂k − q̂M0D −1r̂>D + ΦMkD −1Ψ>D, B̂k = k∑ j=1 ûjD j , ûk = uk, ûk−1 = uk−1 + kukw −1 0 w0,x, . . . , where Mk = CΛk − Λ̃>k C, Φ̃ = −Wm{ϕ}C−1 = ϕ∆̃−1, Ψ̃ = D−1 { W τ,−1 m {ψ} } C−1,> = D−1{ψ}∆−1,>, q̂ = Wm{q}, r̂ = D−1W−1,τm D{r}. Proof. The proof is analogous to the proof of Theorem 2 in [12]. � The following consequence of the latter theorem provides a solution generating method for the hierarchy given by (4.1) and (4.3): Corollary 4.3. Let (N ×K)-matrix functions ϕ and ψ satisfy linear problems: Lk{ϕ} = ϕΛk,1, Lτk{ψ} = ψΛ̃k,1, Λk,1, Λ̃k,1 ∈ MatK×K(C), Mn{ϕ} = ϕΛn,2, M τ n{ψ} = ψΛ̃n,2, Λn,2, Λ̃n,2 ∈ MatK×K(C) with operators Lk and Mn given by (4.1). The operators L̂k = WmLkW −1 m and M̂n = WmMnW −1 m transformed via Wm (4.6), (4.7) have the form: L̂k := βk∂τk − B̂k − q̂M0D −1r̂>D + Φ̃Mk,1D −1Ψ̃>D, B̂k = k∑ j=1 ûjD j , M̂n = αn∂tn − Ân − q̂M̃0D −1r̂>D + Φ̃Mn,2D −1Ψ̃>D, Ân = n∑ i=1 v̂iD i, where Mk,1 = CΛk,1 − Λ̃>k,1C, Mn,2 = CΛn,2 − Λ̃>n,2C, Φ̃ = −Wm{ϕ}C−1 = ϕ∆̃−1, Ψ̃ = D−1 { W τ,−1 m {ψ} } C−1,> = D−1{ψ}∆−1,>, q̂ = Wm{q}, r̂ = D−1W−1,τm D{r}, ∆̃ = −C +D−1 { D−1 { ψ> } ϕx } . 16 O. Chvartatskyi and Yu. Sydorenko 5 Conclusion In this work we proposed new integrable generalizations of the KP and modified KP hierarchy with self-consistent sources. The obtained hierarchies of nonlinear equations include, in particu- lar, 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 generaliza- tions of the N -wave problem and the DS-III system. Under reductions (2.11) imposed on the obtained hierarchies one recovers (2 + 1)-BDk-cKP hierarchy. The latter contains (tA, τB)- and (γA, σB)-matrix KP hierarchies [36, 37] (see [11] for details). Remark 5.1. It should be pointed out that in the scalar case (N = 1) Lax pairs (4.1) admit the following reduction: q = ( q1,q2,−q2M0r > 2 −D−1{u}, 1 ) , r = ( r1, D −1{r2}, 1, D−1{u} ) , q̃ = ( q̃1, q̃2,−q̃2M̃1r̃ > 2 −D−1{ũ}, 1,q1[0],q1[1], . . . ,q1[l] ) , q1[j] := Lj{q1}, r̃ = ( r̃1, r̃2, 1, D −1{ũ}, r1[l], r1[l − 1], . . . , r1[0] ) , r1[j] := (Lτ )j{r1}, M0 = diag(M1,M1, 1, 1), M̃0 = diag(M̃1,M̃1, 1, 1, Il+1 ⊗M1), where qj , rj and q̃j , r̃j , j = 1, 2, are vectors of functions of dimensions (1×m0) and (1× m̃0) respectively. M1 and M̃1 are square matrices of dimensions m0 and m̃0. Il+1 ⊗M1 denotes the tensor product of the identity matrix Il+1 and M1. It leads to the following family of integro-differential operators in (4.1) Lk = βk∂τk −Bk − q1M1D −1r>1 D + q2M1D −1r>2 +D−1u, Mn = αn∂tn −An − q̃1M̃1D −1r̃>1 D + q̃2M̃1D −1r̃>2 +D−1ũ − cl l∑ j=0 q1[j]M0D −1r>1 [l − j]D. Lax equation [Lk,Mn] = 0 involving the latter operators should lead (under additional reduc- tions) to (2 + 1)-dimensional generalizations of the corresponding integrable systems that were obtained in [12]. In particular it concerns systems that extend KdV, mKdV and Kaup–Broer equations. In this paper we also elaborated solution generating methods for the proposed hierarchies (2.2), (2.6) and (4.1), (4.3) respectively via DTs and BDTs. 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 [36, 49, 85]. 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 [3, 20, 21, 22, 23, 24, 62, 81] It is also known that inverse scattering and spectral methods [1, 7, 41, 57] were applied to generate solutions of equations with self-consistent sources [6, 27, 46]. An extension of these methods to the obtained hierarchies and comparison with results that can be provided by BDTs (e.g., following [67]) presents an interest for us. The search for the corresponding discrete counterparts of the constructed hierarchies is an- other problem for future investigation. The latter is expected to contain the discrete KP equa- tion with self-consistent sources [19, 35]. 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. 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Gen. 36 (2003), 5035–5043, nlin.SI/0304030. http://dx.doi.org/10.1088/0266-5611/10/3/013 http://dx.doi.org/10.1088/0305-4470/36/18/307 http://dx.doi.org/10.1007/BF02591917 http://dx.doi.org/10.1007/BF02591917 http://dx.doi.org/10.4303/jglta/S080326 http://dx.doi.org/10.4303/jglta/S080326 http://dx.doi.org/10.1007/BF01075696 http://dx.doi.org/10.1007/BF01077483 http://dx.doi.org/10.1016/0375-9601(96)00090-4 http://dx.doi.org/10.1103/PhysRevE.85.026601 http://dx.doi.org/10.1088/0266-5611/13/3/019 http://dx.doi.org/10.1088/0266-5611/13/3/019 http://dx.doi.org/10.1088/0305-4470/37/28/006 http://arxiv.org/abs/nlin.SI/0412070 http://dx.doi.org/10.1088/0305-4470/36/18/308 http://arxiv.org/abs/nlin.SI/0304030 1 Introduction 2 New (2+1)-dimensional generalizations of the k-constrained KP hierarchy 3 Dressing methods for the new (2+1)-dimensional generalizations of k-constrained KP hierarchy 3.1 Dressing via Darboux transformations 3.2 Dressing via binary Darboux transformations 4 New (2+1)-dimensional generalizations of the modified k-constrained KP hierarchy 4.1 Dressing via binary Darboux transformations 5 Conclusion References

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

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