Additional reductions in the K-constrained modified KP hierarchy

Запропоновано додатковi редукцiї в k-редукованiй модифiкованiй КП-iєрархiї. Як наслiдок, отримано узагальнення системи Каупа – Броера, рiвняння Кортевега – де Фрiза та модифiкованого рiвняння Кортевега – де Фрiза, якi належать до модифiкованої k-редукованої iєрархiї. Також запропоновано метод побудо...

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Datum:	2014
Hauptverfasser:	Chvartatskyi, O.I., Sydorenko, Y.M.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2014
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spelling	irk-123456789-1771012021-02-11T01:27:52Z Additional reductions in the K-constrained modified KP hierarchy Chvartatskyi, O.I. Sydorenko, Y.M. Запропоновано додатковi редукцiї в k-редукованiй модифiкованiй КП-iєрархiї. Як наслiдок, отримано узагальнення системи Каупа – Броера, рiвняння Кортевега – де Фрiза та модифiкованого рiвняння Кортевега – де Фрiза, якi належать до модифiкованої k-редукованої iєрархiї. Також запропоновано метод побудови розв’язкiв для отриманих рiвнянь, який базується на бiнарних перетвореннях Дарбу. Additional reductions in the modified k-constrained KP hierarchy are proposed. As a result we obtain generalizations of Kaup – Broer system, Korteweg – de Vries equation and a modification of Korteweg – de Vries equation that belongs to modified k-constrained KP hierarchy. We also propose solution generating technique based on binary Darboux transformations for the obtained equations. 2014 Article Additional reductions in the K-constrained modified KP hierarchy / O.I. Chvartatskyi, Y.M. Sydorenko // Нелінійні коливання. — 2014. — Т. 17, № 3. — С. 419-436 — Бібліогр.: 39 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177101 517.9 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Запропоновано додатковi редукцiї в k-редукованiй модифiкованiй КП-iєрархiї. Як наслiдок, отримано узагальнення системи Каупа – Броера, рiвняння Кортевега – де Фрiза та модифiкованого рiвняння Кортевега – де Фрiза, якi належать до модифiкованої k-редукованої iєрархiї. Також запропоновано метод побудови розв’язкiв для отриманих рiвнянь, який базується на бiнарних перетвореннях Дарбу.
format	Article
author	Chvartatskyi, O.I. Sydorenko, Y.M.
spellingShingle	Chvartatskyi, O.I. Sydorenko, Y.M. Additional reductions in the K-constrained modified KP hierarchy Нелінійні коливання
author_facet	Chvartatskyi, O.I. Sydorenko, Y.M.
author_sort	Chvartatskyi, O.I.
title	Additional reductions in the K-constrained modified KP hierarchy
title_short	Additional reductions in the K-constrained modified KP hierarchy
title_full	Additional reductions in the K-constrained modified KP hierarchy
title_fullStr	Additional reductions in the K-constrained modified KP hierarchy
title_full_unstemmed	Additional reductions in the K-constrained modified KP hierarchy
title_sort	additional reductions in the k-constrained modified kp hierarchy
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/177101
citation_txt	Additional reductions in the K-constrained modified KP hierarchy / O.I. Chvartatskyi, Y.M. Sydorenko // Нелінійні коливання. — 2014. — Т. 17, № 3. — С. 419-436 — Бібліогр.: 39 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT chvartatskyioi additionalreductionsinthekconstrainedmodifiedkphierarchy AT sydorenkoym additionalreductionsinthekconstrainedmodifiedkphierarchy
first_indexed	2025-07-15T15:03:55Z
last_indexed	2025-07-15T15:03:55Z
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fulltext	UDC 517.9 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED MODIFIED KP HIERARCHY ДОДАТКОВI РЕДУКЦIЇ В K-РЕДУКОВАНIЙ МОДИФIКОВАНIЙ КП-IЄРАРХIЇ O. Chvartatskyi, Yu. Sydorenko Lviv Nat. Ivan Franko Univ. Ukraine, 79000, Lviv, Universytets’ka st., 1 e-mail: alex.chvartatskyy@gmail.com y_sydorenko@franko.lviv.ua Additional reductions in the modified k-constrained KP hierarchy are proposed. As a result we obtain generalizations of Kaup – Broer system, Korteweg – de Vries equation and a modification of Korteweg – de Vries equation that belongs to modified k-constrained KP hierarchy. We also propose solution generating technique based on binary Darboux transformations for the obtained equations. Запропоновано додатковi редукцiї в k-редукованiй модифiкованiй КП-iєрархiї. Як наслiдок, от- римано узагальнення системи Каупа – Броера, рiвняння Кортевега – де Фрiза та модифiковано- го рiвняння Кортевега – де Фрiза, якi належать до модифiкованої k-редукованої iєрархiї. Також запропоновано метод побудови розв’язкiв для отриманих рiвнянь, який базується на бiнарних перетвореннях Дарбу. 1. Introduction. The algebraic constructions of the well-known Kyoto group [1], which are called the Sato theory, play an important role in the contemporary theory of nonlinear integrable systems of mathematical and theoretical physics. The leading place in these investigations is occupied by the theory of equations of Kadomtsev – Petviashvili type (KP hierarchy) and their generalizations and applications [1 – 3]. One of known generalizations of the KP hierarchy arises as a result of k-symmetry constra- ints (so-called k-cKP hierarchy) that were investigated in [4 – 8]. k-cKP hierarchy are closely connected with so-called KP equation with self-consistent sources (KPSCS) [9 – 12]. Multi- component k-constraints of the KP hierarchy were introduced in [13] and investigated in [14 – 18]. This extension of k-cKP hierarchy contains vector (multicomponent) generalizations of physically relevant systems like the nonlinear Schrödinger equation, the Yajima – Oikawa sys- tem, a generalization of the Boussinesq equation, and the Melnikov system. The modified k-constrained KP (k-cmKP) hierarchy was proposed in [19, 20]. It contains, for example, the vector Chen – Lee – Liu, the modified Korteweg – de Vries (mKdV) equation and their multicomponent extensions. The k-cmKP hierarchy and dressing methods for it via integral transformations were investigated in [21 – 23]. In [24, 25] (2+1)-dimensional extensions of the k-cKP hierarchy ((2+1)-dimensional k-cKP hierarchy) were introduced and dressing methods via differential transformations were investi- gated. Some systems of this hierarchy were investigated via binary Darboux transformations in [22, 23]. This hierarchy was also rediscovered recently in [26, 27]. Matrix generalizations of (2 + 1)-dimensional k-cKP hierarchy were considered in [28, 29]. c© O. Chvartatskyi, Yu. Sydorenko, 2014 ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 419 420 O. CHVARTATSKYI, YU. SYDORENKO In this paper our aim is to consider additional reductions of the k-cmKP hierarchy that lead to new generalizations of well-known integrable systems. We also investigated dressing methods for the obtained systems via integral transformations that arise from Binary Darboux Transformations (BDT). This work is organized as follows. In Section 2 we present a short survey of results on constraints for the KP hierarchies including the k-cmKP hierarchy. In Section 3 we investigate Lax representations obtained as a result of additional reductions in the k-cmKP hierarchy and corresponding nonlinear systems. Section 4 presents results on dressing methods for Lax pairs obtained in Section 3. In the final section, we discuss the obtained results and mention problems for further investigations. 2. Symmetry constraints of the KP hierarchy. Let us recall some basic objects and notations concerning KP hierarchy, modified KP hierarchy, their multicomponent k-constraints and their (2 + 1)-extensions. A Lax representation of the KP hierarchy is given by Ltn = [Bn, L], n ≥ 1, (1) where L = D+U1D −1+U2D −2+ . . . is a scalar pseudodifferential operator, t1 := x, D := ∂ ∂x , and Bn := (Ln)+ := (Ln)≥0 = Dn + ∑n−2 i=0 uiD i is the differential operator part of Ln. The consistency condition (zero-curvature equations), arising from the commutativity of flows (1), is Bn,tk −Bk,tn + [Bn, Bk] = 0. (2) Let Bτ n denote the formal transpose of Bn, i. e., Bτ n := (−1)nDn + ∑n−2 i=0 (−1)iDiu>i , where > denotes the matrix transpose. We will use curly brackets to denote the action of an operator on a function whereas, for example, Bnq means the composition of the operator Bn and the operator of multiplication by the function q. The following formula holds forBnq andBn{q} := := (Bnq)=0 = Bnq− (Bnq)>0. In the case k = 2, n = 3 formula (2) presents a Lax pair for the Kadomtsev – Petviashvili equation. Its Lax pair was obtained in [33] (see also [34]). The multicomponent k-constraints of the KP hierarchy is given by [13] Ltn = [Bn, L], (3) with the k-symmetry reduction Lk := Lk = Bk + m∑ i=1 m∑ j=1 qimijD −1rj = Bk + qM0D −1r>, (4) where q = (q1, . . . , qm) and r = (r1, . . . , rm) are vector functions, M0 = (mij) m i,j=1 is a constant (m ×m)-matrix. In the scalar case (m = 1) we obtain a k-constrained KP hierarchy [4 – 8]. The hierarchy given by (3), (4) admits the Lax representation (here k ∈ N is fixed): [Lk,Mn] = 0, Lk = Bk + qM0D −1r>, Mn = ∂tn −Bn. (5) ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 421 Lax equation (5) is equivalent to the following system: [Lk,Mn]≥0 = 0, Mn{q} = 0, M τ n{r} = 0. (6) Below we will also use the formal adjoint B∗n := B̄τ n = (−1)nDn + ∑n−2 i=0 (−1)iDiu∗i of Bn, where ∗ denotes the Hermitian conjugation (complex conjugation and transpose). For k = 1, the hierarchy given by (6) is a multicomponent generalization of the AKNS hierarchy. For k = 2 and k = 3, one obtains vector generalizations of the Yajima – Oikawa and Melnikov [9 – 11] hierarchies, respectively. An essential extension of the k-cKP hierarchy is its (2 + 1)-dimensional generalization introduced in [24, 25] and rediscovered in [26, 27]. In [19, 20], a k-constrained modified KP (k-cmKP) hierarchy was introduced and investi- gated. Dressing methods for k-cmKP hierarchy under additional D-Hermitian reductions were also investigated in [21, 22]. At first we recall the definition of the modified KP hierarchy. A Lax representation of this hierarchy is given by Ltn = [Bn, L], n ≥ 1, (7) where L = D+U0 +U1D −1 +U2D −2 + . . . andBn := (Ln)>0 := Dn+ ∑n−1 i=1 uiD i is the purely differential operator part of Ln. The consistency condition arising from the commutativity of flows (7) is Bn,tk −Bk,tn + [Bn, Bk] = 0. The multicomponent k-constraints of the modified KP hierarchy are given by the operator equation Ltn = [Bn, L], (8) with the k-symmetry reduction Lk := Lk = Bk − m∑ i=1 m∑ j=1 qimijD −1rjD = Bk − qM0D −1r>D, (9) where q = (q1, . . . , qm) and r = (r1, . . . , rm) are vector functions, M0 = (mij) m i,j=1 is a constant (m × m)-matrix. The hierarchy (8), (9) admits the Lax representation (here k ∈ N is fixed) [Lk,Mn] = 0, Lk = Bk − qM0D −1r>D, (10) Mn = αn∂tn −Bn, Bn = Dn + n−1∑ i=1 uiD i. We can rewrite the Lax pair (10) in the following way: [Lk,Mn] = 0, Lk = Bk − qM0r > + qM0D −1r>x , (11) Mn = αn∂tn −Bn, Bn = Dn + n−1∑ i=1 uiD i. ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 422 O. CHVARTATSKYI, YU. SYDORENKO From Lax representation for k-cKP hierarchy (5), (6) and representation (11) we come to conclusion that the equation [Lk,Mn] = 0 in (10) is equivalent to the following system: [Lk,Mn]>0 = 0, Mn{q} = 0, (M τ n){rx} = 0 ([Lk,Mn]=0 = 0 since [Lk,Mn]{1} = 0). We can rewrite the last equation in the following form: (D−1M τ nD){r} = 0 to keep the order of differentiation equal to n. As a result we obtain [Lk,Mn]>0 = 0, Mn{q} = 0, (D−1M τ nD){r} = 0. (12) The hierarchy (10) contains vector generalizations of the Chen – Lee – Liu (k = 1), the modified multicomponent Yajima – Oikawa (k = 2) and Melnikov (k = 3) hierarchies. Consider some equations that can be obtained from (10) under certain choice of k and n (see [23]). (1) k = 1, n = 2. Then (10) becomes L1 = D − qM0D −1r>D, M2 = α2∂t2 −D2 + 2qM0r >D. (13) In this case equation (12) becomes the following system: α2qt2 − qxx + 2qM0r >qx = 0, α2r > t2 + r>xx + 2r>x qM0r > = 0. (14) Under the additional Hermitian conjugation reduction α2 = i,M0 = −M∗0, r> = q∗ (L∗1 = = −D−1L1D, M ∗ 2 = D−1M2D) in (14), we obtain the Chen – Lee – Liu equation iqt2 − qxx + 2qM0q ∗qx = 0. (15) (2) k = 1, n = 3. In this case (10) takes the form L1 = D − qM0D −1r>D, (16) M3 = α3∂t3 −D3 + 3qM0r >D2 + 3[qxM0r > − (qM0r >)2]D, and equations (12) read α3qt3 = qxxx − 3(qM0r >)qxx − 3(qxM0r > − (qM0r >)2)qx, (17) α3r > t3 = r>xxx + 3r>xx(qM0r >) + 3r>x (qM0r > x + (qM0r >)2). After reduction of Hermitian conjugation: α3 = 1, r> = q∗,M0 = −M∗0 (L∗1 = −D−1L1D, M∗3 = −D−1M3D), (17) becomes: qt3 = qxxx − 3(qM0q ∗)qxx − 3(qxM0q ∗ − (qM0q ∗)2)qx. (18) (3) k = 2, n = 2. After additional reduction in (10): α2 = i, u1 := iu, u = u(x, t2) ∈ R, M0 = M∗0, the Lax pair in (12) reads [L2,M2] = 0, L2 = D2 + iuD − qM0D −1q∗D, M2 = i∂t2 −D2 − iuD, (19) ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 423 and equation (12) becomes the modified Yajima – Oikawa equation iqt2 = qxx + iuqx, ut2 = 2(qM0q ∗)x. In the next section we will introduce additional reductions in Chen – Lee – Liu hierarchy. As a result we will obtain generalizations of the Kaup – Broer system, KdV equation, modified KdV equation and their scalar coupled versions. 3. Additional reductions in the modified k-constrained KP hierarchy. For further conveni- ence let us make a change in formulae (10), q → q̃, r → r̃, M0 → M̃0. (20) After the change (20) the hierarchy (10) reads [Lk,Mn] = 0, Lk = Bk − q̃M̃0D −1r̃>D, (21) Mn = αn∂tn −Bn, Bn = Dn + n−1∑ i=1 uiD i. Let us make the additional reduction in (21), q̃ := (q1, . . . , qm,−v − βD−1{u}, 1) = (q,−v − βD−1{u}, 1), (22) M̃0 =  M0 0 0 0 1 0 0 0 1  , r̃ := (r1, . . . , rm, 1, βD −1{u}) = (r, 1, βD−1{u}), whereM0 is an (m×m)-constant matrix, q and r are m-component vectors, u and v are scalar functions, β ∈ R, D−1{u} denotes indefinite integral of the function u with respect to x. After reduction (22) k-cmKP hierarchy (21) takes the form [Lk,Mn] = 0, Lk = Bk − qM0D −1r>D + v + βD−1u, (23) Mn = αn∂tn −Bn, Bn = Dn + n−1∑ i=1 uiD i. In the following subsections we will investigate hierarchy (23) in case k = 1. 3.1. Reductions of the Chen – Lee – Liu system. Let us put k = 1, n = 2. Then Lax pair (23) becomes [L1,M2] = 0, L1 = D − qM0D −1r>D + βD−1u+ v, (24) M2 = α2∂t2 −D2 + 2(qM0r > − v)D. ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 424 O. CHVARTATSKYI, YU. SYDORENKO A system that corresponds to equation (24) has the form α2qt2 = qxx − 2(qM0r > − v)qx, α2r > t2 = −r>xx − 2r>x (qM0r > − v), α2ut2 + uxx + 2 ( u(qM0r > − v) ) x = 0, (25) −α2vt2 + 2βux + vxx − 2 ( qM0r > − v ) vx = 0. Consider additional reductions of Lax pair (24) and system (25). (1) Assume thatM0 = −M∗0, r> = q∗, v = −2iIm (βD−1{u}) (L∗1 = −DL1D −1, M∗2 = = DM2D −1). Then equation (25) takes the form α2qt2 = qxx − 2(2iIm (βD−1{u}) + qM0q ∗)qx, α2ut2 + uxx + 2 ( u(2iIm (βD−1{u}) + qM0q ∗) ) x = 0. (2) Let us putM0 = 0 in the operators L1 and M2, L1 = D + βD−1u + v, M2 = α2∂t2 − −D2 − 2vD. Then equation (25) becomes the Kaup – Broer system α2ut2 + uxx − 2(uv)x = 0, −α2vt2 + 2βux + vxx + 2vvx = 0. (26) In case u = 0 in (26) we obtain the Burgers equation −α2vt2 + vxx − vvx = 0. (3) Consider the case u = 0 for the operators L1 and M2 (25), L1 = D−qM0D −1r>D+v, M2 = α2∂t2 −D2 + 2(v + qM0r >)D. Then (25) reads α2qt2 = qxx − 2(qM0r > − v)qx, α2r > t2 = −r>xx − 2r>x (qM0r > − v), −α2vt2 + vxx − ( qM0r > − v ) vx = 0. 3.2. Reductions of the modification of KdV system (18). Now let us consider the hierarchy (23) in case k = 1, n = 3. Then its Lax pair L1, M3 in (23) reads [L1,M3] = 0, L1 = D − qM0D −1r>D + βD−1u+ v, (27) M3 = α3∂t3−D3−3(v − qM0r >)D2−3 ( (qM0r > − v)2 − qxM0r > + βu+ vx ) D. Commutator equation in (27) is equivalent to the system −α3vt3 + vxxx + 3vvxx + 3v2vx + 3v2x + 6β(uv)x+ + 3 { (qM0r >)2 − qxM0r > } vx − 3qM0r >vxx− − 6qM0r >vvx − 3β(qM0r >u)x − 3βqM0r >ux = 0, ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 425 α3qt3 = qxxx+3(v − qM0r >)qxx+3 { (qM0r > − v)2 − qxM0r > + vx + βu } qx, (28) α3r > t3 = r>xxx−3 ( r>x ( v − qM0r > )) x +3r>x { (qM0r > − v)2 − qxM0r > + vx + βu } , α3ut3 = uxxx−3 ( u(v − qM0r >) ) xx +3 ( u ( (qM0r > − v)2 − qxM0r > + vx + βu )) x . Consider additional reductions in Lax pair (27) and corresponding system (28). (1) Assume that v = −2iIm (βD−1{u}), q∗ = r>, u ∈ R,M0 = −M∗0 (L∗1 = −DL1D −1, M∗3 = −DM3D −1). Then system (28) takes the form α3qt3 = qxxx − 3(2iIm (βu) + qM0q ∗)qxx+ + 3 ( (qM0q ∗ + 2iIm (βu))2 − qxM0q ∗ + βu− 2iIm (βu) ) qx, (29) α3ut3 = uxxx + 3 {u (2iIm (βu) + qM0q ∗)}xx + + 3 ( u { (qM0q ∗ + 2iIm (βu))2 − qxM0q ∗ + βu− 2iIm (βu) }) x . (a) Let us assume that in addition to the reductions described in item 1, the functions q and u with the matrix M0 are real-valued (i.e., the matrix M0 is skew-symmetric, M>0 = −M0) and v = 0. Then the scalar qM0q > = 0 since qM0q > = −(qM0q >)>, and equation (29) reads α3qt3 = qxxx − 3qxM0q >qx + 3βuqx, (30) α3ut3 = uxxx − 3(uqxM0q >)x + 6βuux. (2) Let us putM0 = 0 in the operators L1, M3 (27), L1 = D + βD−1u+ v, M3 = α3∂t3 −D3 − 3vD2 − 3(v2 + vx + βu)D. Then equation (28) takes the form −α3vt3 + vxxx + 3vvxx + 3v2vx + 3v2x + 6β(uv)x = 0, (31) α3ut3 = uxxx − 3(uv)xx + 3(u(v2 + vx + βu))x. (a) Under the additional restrictions v = −2iIm (D−1{βu}) (L∗1 = −DL1D −1, M∗3 = = −DM3 D −1) in item 2 we obtain a complex generalization of the modified KdV equation, α3ut3 = uxxx + 6i(uIm (D−1{βu}))xx + 3(u(−4Im(D−1{βu})2 − 2iIm (αu) + βu))x. (32) In the real case (β ∈ R, u is a real-valued function, v = 0) the operators L1 and M3 take the form L1 = D + βD−1u, M3 = β∂t −D3 − 3βuD, and we obtain the KdV equation in (32), α3ut3 = uxxx + 6βuux. (33) ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 426 O. CHVARTATSKYI, YU. SYDORENKO (3) Let us put u = 0 in Lax pair (27), L1 = D − qM0D −1r>D + v, M3 = α3∂t3 − D3 − −3(v − qM0r >)D2 − 3 ( (qM0r > − v)2 − qM0r > + vx ) D. Equation (28) becomes −α3vt3 + vxxx + 3vvxx + 3v2vx + 3v2x + 3 { (qM0r >)2 − qxM0r > } vx− − 3qM0r >vxx − 6qM0r >vvx = 0, α3qt3 = qxxx+3(v − qM0r >)qxx+3 { (qM0r > − v)2 − qxM0r > + vx } qx, (34) α3r > t3 = r>xxx−3 ( r>x ( v − qM0r > )) x +3r>x { (qM0r > − v)2−qxM0r > + vx } . 4. Dressing methods for k-cmKP hierarchy. In this section our aim is to elaborate dres- sing methods for the k-cmKP hierarchy (10). At first we recall a main result from paper [35]. Let (1 ×K)-matrix functions ϕ and ψ be solutions of linear problems with (2+1)-dimensional generalization of the operator Lk (4) with a more general differential part Bk, Lk{ϕ} = ϕΛ, Lτk{ψ} = ψΛ̃, Λ, Λ̃ ∈ MatK×K(C), (35) Lk = βk∂τk +Bk + qM0D −1r>, Bk = k∑ j=0 ujD j . Introduce a binary Darboux transformation (BDT) in the following way: W = I − ϕ ( C +D−1{ψ>ϕ} )−1 D−1ψ> := I − ϕ∆−1D−1ψ>, (36) 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> = I + ϕD−1∆−1ψ>. (37) The following theorem is proved in [35]. Theorem 1 [35]. The operator L̂k := WLkW −1 obtained from Lk in (35) via BDT (36) has the form L̂k := WLkW −1 = βk∂τk + B̂k + q̂M0D −1r̂> + ΦMD−1Ψ>, (38) B̂k = k∑ j=0 ûjD j , where M = CΛ− Λ̃>C, Φ = ϕ∆−1, Ψ = ψ∆−1,>, (39) ∆ = C +D−1{ψ>ϕ}, q̂ = W{q}, r̂ = W−1,τ{r}. ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 427 The coefficients ûk and ûk−1 of the operator L̂k remain the same, i.e., ûk = uk, ûk−1 = uk−1. All other coefficients ûj , j = 0, k − 2, depend on the functions ϕ, ψ and ui, i = 0, j. Exact forms of all coefficients ûj can be found in [35]. Using the previous theorem we obtain the following result for the (2 + 1)-generalization of operator Lk from the k-cmKP hierarchy (10): Theorem 2. Let (1×K)-vector functions ϕ and ψ satisfy linear problems: Lk{ϕ} = ϕΛ, Lτk{ψ} = ψΛ̃, Λ, Λ̃ ∈ MatK×K(C), (40) Lk = βk∂τk +Bk − qM0D −1r>D, Bk = k∑ i=1 uiD i. Then the operator L̂k transformed via the operator Wm := w−10 W = w−10 ( I − ϕ∆−1D−1ψ> ) = I − ϕ∆̃−1D−1(D−1{ψ})>D, (41) where w0 = I − ϕ∆−1D−1{ψ>}, ∆̃ = −C +D−1{D−1{ψ>}ϕx}, ∆ = C +D−1{ψ>ϕ}, has the form L̃k := WmLkW −1 m = βk∂τk + B̃k − q̃M0D −1r̃>D + Φ̃MD−1Ψ̃>D, (42) B̃k = k∑ j=1 ũjD j , ũk = uk, ũk−1 = uk−1 + kukw −1 0 w0,x, . . . , where M = CΛ− Λ̃>C, Φ̃ = −Wm{ϕ}C−1 = ϕ∆̃−1, Ψ̃ = D−1{W τ,−1 m {ψ}}C−1,> = D−1{ψ}∆−1,>, q̃ = Wm{q}, (43) r̃ = D−1W−1,τm D{r}, ∆̃ = C +D−1{D−1{ψ>}ϕx}. Proof. Let us check that w−10 = I − ϕ∆̃D−1{ψ>}, ∆̃ = −C +D−1{D−1{ψ>}ϕx}. In order to do that we have to verify the equality w0w −1 0 = I, w0w −1 0 = I − ϕ∆−1D−1{ψ>} − ϕ∆̃−1D−1{ψ>}+ + ϕ∆̃−1 ( C +D−1{ψ>ϕ} − C +D−1{D−1{ψ>}ϕx} ) ϕ∆−1D−1{ψ>} = I. ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 428 O. CHVARTATSKYI, YU. SYDORENKO Analogously it can be verified that w−10 w0 = I. By Theorem 1 we obtain WmLkW −1 m = w−10 W ( βk∂τk +Bk − qM0r > + qM0D −1r>x ) W−1w0 = = βk∂τk + (WmLkW −1 m )≥0 − w−10 W{q}M0D −1 (W−1,τ{rx})>w0+ + w−10 ΦMD−1Ψ>w0. (44) We shall point out that Ψ>w0 = ∆−1ψ>(I − ϕ∆−1D−1{ψ>}) = ( ∆−1D−1{ψ>} ) x = Ψ̃>x . We shall also observe that (W−1,τ{rx})>w0 = ( r>x −D−1{r>x ϕ}∆−1ψ> )( I − ϕ∆−1D−1{ψ>} ) = = ( r> −D−1{r>x ϕ}∆−1D−1{ψ>} ) x = (D−1W−1,τm D{r})>x = r̃>x . Thus (44) can be rewritten as L̃k = WmLkW −1 m = w−10 W ( βk∂τk +Bk − qM0r > + qMD−1r>x ) W−1w0 = = βk∂τk + (WmLkW −1 m )≥0 + q̃M0D −1r̃>x − Φ̃MD−1Ψ̃>x = = βk∂τk + (WmLkW −1 m )≥0 + q̃M0r̃ > − Φ̃MΨ̃> − q̃M0D −1r̃>D + Φ̃MD−1Ψ̃>D. (45) Using that L̃k{1} = ũ0 = 0 we obtain the form of B̃k, i.e., B̃k := (WmLkW −1 m )≥0 + q̃M0r̃ > − −Φ̃MΨ̃> = ∑k j=1 ũjD j . Theorem 2 is proved. Theorem 2 provides us with a dressing method for k-cmKP hierarchy (10), i.e., the following corollary directly follows from the previous theorem: Corollary 1. Assume that the operatorsLk andMn in (10) satisfy the Lax equation [Lk,Mn] = = 0. Let functions ϕ and ψ satisfy the equations Lk{ϕ} = ϕΛ, Lτk{ψ} = ψΛ̃, Λ, Λ̃ ∈ Mat(K×K)(C), Mn{ϕ} = 0, M τ n{ψ} = 0. (46) Then the transformed operators L̃k = WmLkW −1 m (see (42) with βk = 0) and M̃n = WmMnW −1 m = αn∂tn −Dn − n−1∑ i=1 ũiD i (47) via the transformation Wm (41) also satisfy the Lax equation [L̃k, M̃n] = 0. Proof. It can be checked directly that [L̃k, M̃n] = [WmLkW −1 m ,WmMnW −1 m ] = Wm[Lk,Mn]W−1m = 0. The exact form of the operators L̃k and M̃n follows from Theorem 2. ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 429 The following corollary follows from Corollary 1 and Theorem 2: Corollary 2. Suppose that the functions ϕ and ψ satisfy equations (46) with operators Lk and Mn defined by (23). Then the transformed operators have the form L̃k = Bk − q̃M0D −1r̃>D + Φ̃MD−1Ψ̃>D + ṽ + βD−1ũ, (48) M̃n = αn∂tn − B̃n, B̃n = Dn + n−1∑ i=1 ũiD i, where M = CΛ− Λ̃>C, Φ̃ = −Wm{ϕ}C−1 = ϕ∆̃−1, Ψ̃ = D−1{W τ,−1 m {ψ}}C−1,> = D−1{ψ}∆−1,>, q̃ = Wm{q}, (49) r̃ = W−1,τm {r} , ∆̃ = C +D−1{D−1{ψ>}ϕx}, ∆ = C +D−1{ψ>ϕ}, ũ = W−1,τm {D−1{u}}, ṽ = Wm{v}+ βD−1W−1,τm {u} − βWm{D−1{u}}. As it was shown in previous sections, the most interesting systems arise from the k-cmKP hierarchy (10) and its reduction (23) after a Hermitian conjugation reduction. Our aim is to show that under additional restrictions, the BDT Wm (41) preserves this reduction. Proposition 1. (1) Let ψ = ϕ̄x and C = −C∗ in the dressing operator Wm (41). Then the operator Wm is D-unitary (W−1m = D−1W ∗mD). (2) Let the operator Lk (10) be D-Hermitian, L∗k = DLkD −1 (D-skew-Hermitian, L∗k = = −DLkD−1) and Mn (10) be D-Hermitian (D-skew-Hermitian). Then the operator L̂k = = WmLkW −1 m (see (42)) transformed via the D-unitary operator Wm is D-Hermitian (D-skew- Hermitian) and M̂n := WmMnW −1 m (47) is D-Hermitian (D-skew-Hermitian). (3) Assume that the conditions of items 1 and 2 hold. Let Λ̃ = Λ̄ in the case of D-Hermitian Lk (Λ̃ = −Λ̄ in D-skew-Hermitian case). We shall also assume that the function ϕ satisfies the corresponding equations in formulae (46). Then M = M∗ (M = −M∗) and Ψ̃ = ¯̃Φ (see formulae (39)). In Subsection 4.1 we will show how one can use the methods described in Theorem 2 and its corollaries in order to obtain solutions of KdV equation (33) and its generalization (30). 4.1. Solution generating technique for system (30) and KdV equation (33). We shall consider equation (30) in the case where the dimension of the vector q and the matrixM0 is even, i.e., m = 2m̃, m̃ ∈ N (in this situation, the skew-symmetric matrix M0 can be nondegenerate). Assume that the skew-symmetric matrixM0 in (30) and the vector-function q has the form M0 = ( 0m̃ Im̃ −Im̃ 0m̃ ) , q = (q1,q2) = (q11, q12, . . . , q1m̃, q21, q22, . . . , q2m̃) , (50) where 0m̃ is a (m̃× m̃)-dimensional matrix consisting of zeros, Im̃ is an identity matrix with the dimension m̃ × m̃. Equation (30) with the notation ũ := u can be rewritten in the following form: ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 430 O. CHVARTATSKYI, YU. SYDORENKO α3q1,t3 = q1,xxx − 3(q1,xq > 2 − q2,xq > 1 )q1,x + 3βũq1,x, α3q2,t3 = q2,xxx − 3(q1,xq > 2 − q2,xq > 1 )q2,x + 3βũq2,x, (51) α3ũt3 = ũxxx − 3(ũ(q1,xq > 2 − q2,xq > 1 ))x + 6βũũx. In this subsection our aim is to consider the case m̃ = 1 (although the corresponding solution generating technique can be generalized to the case of an arbitrary natural m̃). In this situation q1 = q1 and q2 = q2 are scalars. We shall suppose that K = 2K̃ is an even natural number. Assume that the function ϕ is a (1×K)-vector solution of the system L10{ϕ} = ϕx + βD−1{uϕ} = ϕΛ, Λ ∈ MatK×K(C), β ∈ R, (52) M30{ϕ} = α3ϕt3 − ϕxxx − 3βuϕx = 0, with a number u ∈ R. Using Theorem 2 and Proposition 1 we obtain that the dressed operators L̃10 and M̃30 via the operator Wm (41) with the skew-Hermitian matrix C and ψ = ϕ̄x has the form L̃10 = WmL10W −1 m = D + Φ̃MD−1Φ̃∗D + βD−1ũ+ ṽ, (53) M̃30 = WmM30W −1 m = α3∂t3 −D3 − (ṽ + Φ̃MΦ̃∗)D2− − 3 ( (Φ̃MΦ̃∗ + ṽ)2 + Φ̃xMΦ̃∗ + ṽx + βũ ) D, where M = CΛ − Λ∗C∗, Φ̃ = ϕ∆̃−1, ũ = uD{ϕ̄ ¯̃∆−1D−1{ϕ>u}}, ṽ = β(Φ̃D−1{ϕ∗u} − −D−1{uϕ}Φ̃∗), ∆̃ = −C + D−1{ϕ∗ϕx}. It has to be pointed out that the function Φ̃ = = −Wm{ϕ}C−1 = ϕ∆̃−1 satisfies the equation M̃30{Φ̃} = 0 because M̃30{Φ̃} = WmM30W −1 m {Wm{ϕ}C−1} = 0. Now we assume that the functionϕ and the matricesC and Λ are real. In this case, ṽ = ṽ> = = β(Φ̃D−1{ϕ>u} −D−1{uϕ}Φ̃>)> = −ṽ = 0. Let us put Λ = diag (λ11, λ12, λ21, λ22, . . . , λK̃1, λK̃2), λij ∈ R, C =  C11 C12 . . . C1K̃ C21 C22 . . . C2K̃ ... ... . . . ... CK̃1 CK̃2 . . . CK̃K̃  , (54) ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 431 where the elements Cij are (2× 2)-matrices of the form Cij =  0 − 1 λj2 + λi1 1 λj1 + λi2 0  . (55) Under such a choice of C (55) and Λ (54) we obtain that the (2K̃×2K̃)-dimensional matrix M = CΛ−Λ>C> has the block form,M = (Mij) K̃ i,j=1,whereMij = M0 (see formula (50) in case m̃ = 1). Let us denote by 1K̃ = (I2, . . . , I2) a matrix that consists of K̃ (2×2)-dimensional identity matrices I2. ThenM = −1> K̃ M01K̃ . Let us put u = const and choose a solution of system (52) in the form ϕ = ( ϕ11, ϕ12ϕ21, ϕ22, . . . , ϕK̃1, ϕK̃2 ) , ϕij = exp {( 1 2 λij + γij ) x+ aijt } , where γij = √ 1 4 λ2ij − βu, aij = {( 1 2 λij + γij )3 + 3βu ( 1 2 λij + γij )} /α3. The (2K̃ × 2K̃)-matrix ∆̃ then takes the block form ∆̃ = −C +D−1{ϕ>ϕx} = ( ∆̃ij )K̃ i,j=1 = =  αi1 αi1+αj1 e(αi1+αj1)x+(ai1+aj1)t αi2 αi2+αj1 e(αi2+αj1)x+(ai2+aj1)t + 1 λj2+λi1 αi1 αi1+αj2 e(αi1+αj2)x+(ai1+aj2)t − 1 λj1+λi2 αi2 αi2+αj2 e(αi2+αj2)x+(ai2+aj2)t  K̃ i,j=1 , (56) where αij = 1 2 λij+γij . The functions q = (q1, q2) =ϕ∆̃−11> K̃ and ũ= u−D { ϕ∆̃−1D−1{ϕ>u} } will be solutions of system (51). We shall point out that in case β = 0, K̃ = 1, α3 = 1, we obtain the following solution of the real version of the mKdV-type equation (equation (51) with ũ = 0): q = (q1, q2), q1 = − 2(λ11 + λ12)ϕ12 (λ11 − λ12)ϕ11ϕ12 − 2 , q2 = 2(λ11 + λ12)ϕ11 (λ11 − λ12)ϕ11ϕ12 − 2 , ϕ1j = eλ1jx+λ 3 1jt3 , λ1j > 0, j = 1, 2. It is also possible to choose other types of matrices C and Λ in (54) and (55). In particular the following remark holds. Remark 1. In case K̃ = 1, the vector of the functions ϕ = (ϕ1, ϕ2), ϕ1 = cos ( xλ12 + (3λ211λ12 − λ312)t+ π 4 ) exλ11+(λ311−3λ11λ212)t, ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 432 O. CHVARTATSKYI, YU. SYDORENKO ϕ2 = sin ( xλ12 + (3λ211λ12 − λ312)t+ π 4 ) exλ11+(λ311−3λ11λ212)t will be a solution of system (52) with u = 0 and Λ = ( λ11 λ12 −λ12 λ11 ) . The corresponding solution generating technique given by (54) – (56) in case K̃ = 1, CK̃ = C1 =  0 1 2λ11 − 1 2λ11 0  gives us a solution of mKdV-type equation (51) with ũ = 0 that coincides with a solution obtai- ned in [36]. Now we will consider solution generating technique for KdV (33). For this purpose we assume that the function ϕ, the matrices Λ = diag (Λ1, . . . ,ΛK̃) and C = diag (C1 . . . , CK̃) are real and have the form Λj = ( 0 λj λj 0 ) , Cj = ( 0 −cj cj 0 ) . (57) In this case we obtain that the matrixM = CΛ−Λ>C> consists of zeros in (53). Consider the following solution of system (52): ϕ = ( ϕ11, ϕ12, ϕ21, ϕ22, . . . , ϕK̃1, ϕK̃2 ) , ϕj1 = eγjx+ajt cosh ( λj 2 x+ bjt ) , ϕj2 = eγjx+ajt sinh ( λj 2 x+ bjt ) , where γj = √ 1 4 λ2j − βu, aj = ( γ3j + 3 4 γjλ 2 j + 3βuγj ) /α3, bj = ( 3γ2j λj 2 + λ3j 8 + 3 2 βuλj ) /α3 and λj , α3, β, u ∈ R. Thus, ṽ = 0 and we obtain a Lax pair for the KdV equation in (53), L̃10 = D + βD−1ũ, M̃30 = α3∂t3 −D3 − 3βũD. The formula ũ = u−D { ϕ∆̃−1D−1{ϕ>u} } := u+ û (58) gives us a finite density solution of equation (33). In particular, if K̃ = 1 and c1 = 1 8 λ1 γ1 we obtain the following solution: ũ = u+ 2γ21 β cosh2 (γ1x+ a1t) . (59) Now we shall substitute ũ (58) in KdV equation (33), α3ût3 = ûxxx + 6βûûx + 6βuûx. (60) ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 433 The corresponding pair of operators has the form L1 = D+ βD−1(û+ u), M3 = α3∂t3 −D3− −3βûD − 3βuD. We have two ways to obtain soliton solutions (that are rapidly decreasing at both infinities in contrast to finite density solutions (58) that tend to an arbitrary real number u) for KdV from formula (58): (1) By taking the limit u → 0 in (58) – (60). (2) By making a change of the independent variables: x̃ := x + 6α−13 βut3, t̃3 := t3 and v̂(x̃, t̃3) := û(x, t3) in equation (60) and solutions (58), (59). This change corresponds to the change of differential operators in the Lax pair for equation (60) consisting of the operators L1 and M3, α3∂t̃3 = α3∂t3 − 3βuD. 5. Conclusions. In this paper we obtain new generalizations (23) of the modified k-cKP (k-cmKP) hierarchy (10). The obtained hierarchy also generalizes the BKP hierarchy [36 – 38] which is a special case of the k-cmKP hierarchy. Dressing methods elaborated via BDT-type operators (Section 4) give rise to exact solutions of the integrable systems that hierarchy (23) contains. In particular, soliton solutions for generalization of mKdV-type equation (51) and fi- nite density solutions as well as regular soliton solutions were constructed for the KdV equation using the proposed dressing methods. These methods also allow to obtain rational and singular multisoliton solutions of the corresponding nonlinear systems under a special choice of spectral matrix Λ in the linear system (52). In order to minimize the size of this article we do not include those results here. We shall point out that the special case of equation (51) (ũ = 0) and its solutions were considered in [36]. Generalizations (23) of the k-cmKP hierarchy (10) together with different extensions of k-cKP hierarchy is a good basis for construction of other hierarchi- es of nonlinear integrable equations with corresponding dressing methods. In particular in our forthcoming papers we plan to introduce a (2+1)-BDk-cmKP hierarchy and investigate soluti- on generating technique for the corresponding integrable systems. Consider as an example the Lax pair from the (1+1)-BDk-cKP hierarchy that was investigated in [31], P1,1 = D + c1M2{q}M0D −1r> + c1qM0D −1(M τ 2 {r})> + c0qM0D −1r> = = D + c1 ( α2qt2M0D −1r> − α2qM0D −1r>t2 − qxxM0D −1r> − − qM0D −1r>xx − uqM0D −1r> − qM0D −1r>u ) + c0qM0D −1r>, (61) M2 = α2∂t2 −D2 − u with vector-functions q and r satisfying c1M2 2 {q}+ c0M2{q} = 0, c1(M τ 2 )2{r}+ c0M τ 2 {r} = 0. It was shown in [31] that the Lax equation [P1,1,M2] = 0 in (61) is equivalent to the system [P1,1,M2]≥0 = 0, c1M 2 2 {q}+ c0M2{q} = 0, c1(M τ 2 )2{r}+ c0M τ 2 {r} = 0 (62) that is equivalent to a generalization of the AKNS system. In case c0 = 1, c1 = 0 we obtain the AKNS system in (62), α2qt2 − qxx − uq = 0, −α2rt2 − rxx − ur = 0, u = 2qM0r >. Assume that the scalar function f satisfies the equations P1,1{f} = fλ, M2{f} = 0. We shall introduce the notations M̃2 := f−1M2f, M̂2 := DM̃2D −1, P̃1,1 := f−1P1,1f, q̃ := f−1q, ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 434 O. CHVARTATSKYI, YU. SYDORENKO r̃> := D−1{r>f} and consider the following gauge transformations: M̃2 = f−1M2f = α2∂t2 −D2 − 2ũD, ũ = f−1fx, P̃1,1, = f−1P1,1f = D + f−1fx + c1f −1M2{q}M0D −1r>f+ + c1f −1qM0D −1(M τ 2 {r})>f + c0f −1qM0D −1r>f = = D − c1M̃2{q̃}M0D −1r̃>D − c1q̃M0D −1(M̂ τ 2 {r̃})>D − c0q̃M0D −1r̃>D. The equation [M̃2, P̃1,1] = 0 is equivalent to the following system: [P̃1,1, M̃2]>0 = 0, c1M̃ 2 2 {q̃}+ c0M̃2{q̃} = 0, c1(M̂ τ 2 )2{r̃}+ c0M̂ τ 2 {r̃} = 0 or in an equivalent form (after notation q0 := q̃, r0 := r̃), [P̃1,1, M̃2]>0 = 0, q1 = M̃2{q0}, r1 = M̂ τ 2 {r0}, (63) c1M̃2{q1}+ c0M̃2{q0} = 0, c1M̂ τ 2 {r1}+ c0M̂ τ 2 {r0} = 0. System (63) is a generalization of the Chen – Lee – Liu system (case c1 = 0, c0 = 1). In case of the additional reduction α2 ∈ iR, c0 = 0, c1 ∈ R,M∗0 = −M0, r = q̄, (63) reads as follows: α2q0,t2 − q0,xx + 2c1(q1M0q ∗ 0 + q0M0q ∗ 1)q0,x − q1 = 0, α2q1,t2 − q1,xx + 2c1(q1M0q ∗ 0 + q0M0q ∗ 1)q1,x = 0. We shall also point out that the extension of the k-cmKP hierarchy (23) can also be generali- zed to the matrix case. It leads to matrix generalizations of integrable systems that hierarchy (23) contains (including Chen – Lee – Liu (15) and modified-type KdV equation (18)). In parti- cular, the matrix generalization of the modified KdV-type equation (18) differs from the well- known matrix mKdV equation that was investigated by the inverse scattering method in [39]. 6. Acknowledgement. The second-named author Yu. M. Sydorenko (J. Sidorenko till 1998 in earlier transliteration) thanks the Ministry of Education, Science, Youth and Sports of Ukraine for partial financial support (Research Grant MA-107F). 1. Ohta Y., Satsuma J., Takahashi D., Tokihiro T. An elementary introduction to Sato theory // Prog. Theor. Phys. Suppl. — 1988. — 94. — P. 210 – 241. 2. Dickey L. A. Soliton equations and Hamiltonian systems // Adv. Ser. Math. Phys. — 2nd ed. — 2003. — 26. 3. Blaszak M. Multi-Hamiltonian theory of dynamical systems. — Springer-Verlag, 1998. 4. Sidorenko J., Strampp W. Symmetry constraints of the KP-hierarchy // Inverse Problems. — 1991. — 7. — P. L37 – L43. 5. Konopelchenko B. G., Sidorenko J., Strampp W. (1+1)-Dimensional integrable systems as symmetry constra- ints of (2+1)-dimensional systems // Phys. Lett. A. — 1991. — 157. — P. 17 – 21. ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 ADDITIONAL REDUCTIONS IN THE K-CONSTRAINED . . . 435 6. Cheng Y., Li Y. S. The constraint of the Kadomtsev – Petviashvili equation and its special solutions // Phys. Lett. A. — 1991. — 157. — P. 22 – 26. 7. Cheng Y. Constrained of the Kadomtsev – Petviashvili hierarchy // J. Math. Phys. — 1992. — 33. — P. 3774 – 3787. 8. Cheng Y., Li Y. S. Constraints of the 2+1 dimensional integrable soliton systems // J. Phys. A. — 1992. — 25, № 2. — P. 419 – 431. 9. Melnikov V. K. On equations for wave interactions // Lett. Math. Phys. — 1983. — 7, № 2. — P. 129 – 136. 10. Melnikov V. K. A direct method for deriving a multi-soliton solution for the problem of interaction of waves on thex, y plane // Commun. Math. Phys. — 1987. — 112, № 4. — P. 639 – 652. 11. Melnikov V. K. Exact solutions of the Korteweg – de Vries equation with a self-consistent source // Phys. Lett. A. — 1988. — 128. — P. 488 – 492. 12. Samoilenko V. G. Differential-geometric structure and spectral properties of nonlinear completely integrable dynamical systems of the Mel’nikov type // Ukr. Math. J. — 1990. — 42, № 5. — P. 579 – 583. 13. Sidorenko J., Strampp W. Multicomponent integrable reductions in Kadomtsev – Petviashvilli hierarchy // J. Math. Phys. — 1993. — 34, № 4. — P. 1429 – 1446. 14. Oevel W. Darboux theorems and Wronskian formulas for integrable systems I: Constrained KP flows // Physi- ca A. — 1993. — 195. — P. 533 – 576. 15. Zhang Y.-J., Cheng Y. Solutions for the vector k-constrained KP-hierarchy // J. Math. Phys. — 1994. — 35. — P. 5869 – 5884. 16. Oevel W., Strampp W. Wronskian solutions of the constrained KP hierarchy // J. Math. Phys. — 1996. — 37. — P. 6213 – 6219. 17. Aratyn H., Nissimov E., Pacheva S. Constrained KP hierarchy: additional symmetries, Darboux – Backlund solutions and relations to multi-matrix models // Int. J. Mod. Phys. A. — 1997. — 12. — P. 1265 – 1340. 18. Willox R., Loris I., Gilson C. R. Binary Darboux transformations for constrained KP hierarchies // Inverse Problems. — 1997. — 13. — P. 849 – 865. 19. Kundu A., Strampp W., Oevel W. Gauge transformations of constrained KP flows: new integrable hierarchies // J. Math. Phys. — 1995. — 36. — P. 2972 – 2984. 20. Oevel W., Carillo S. Squared eigenfunction symmetries for soliton equations // J. Math. Anal. and Appl. — 1998. — 217. — P. 161 – 199. 21. Sidorenko Yu. Transformation operators for integrable hierarchies with additional reductions // Proc. Inst. Math. Nat. Acad. Sci. Ukraine. — 2002. — 43, Pt 1. — P. 352 – 357. 22. Berkela Yu. Yu., Sydorenko Yu. M. Darboux type theorems and transformation operators for nonlocal reduced Hermitian Kadomtsev – Petviashvili hierarchy (Hk-cKP) // Mat. Stud. — 2006. — 25, № 1. — P. 38 – 64 (in Ukrainian). 23. Berkela Yu. Yu. Integration of nonlinear evolution systems with nonlocal constraints: Ph. D. thesis. — Lviv, 2005. 24. Mytropolsky Yu. O., Samoilenko V. H., Sidorenko Yu. M. Spatially two-dimensional generalization of the KP hierarchy with nonlocal constraints // Dop. Nat. Acad. Sci. Ukraine. — 1999. — 9. — P. 19 – 23. 25. Samoilenko A. M., Samoilenko V. G., Sidorenko Yu. M. Hierarchy of the Kadomtsev – Petviashvili equations under nonlocal constraints: Manydimensional generalizations and exact solutions of reduced systems // Ukr. Math. J. — 1999. — 51, № 1. — P. 86 – 106. 26. Liu X. J., Zeng Y. B., Lin R. A new extended KP hierarchy // Phys. Lett. A. — 2008. — 372. — P. 3819 – 3823. 27. Liu X. J., Lin R., Jin B., Zeng Y.-B. A generalized dressing approach for solving the extended KP and the extended mKP hierarchy // J. Math. Phys. — 2009. — 50. — P. 053506-1 – 053506-14. 28. Huang Y., Liu X., Yao Y., Zeng Y. A new extended matrix KP hierarchy and its solutions // Theor. Math. Phys. — 2011. — 167, № 2. — P. 590 – 605. 29. Huang Y.-H., Yao Y. Q., Zeng Y. B. A new (γA, σB)-matrix KP hierarchy and its solutions // Commun. Theor. Phys. — 2012. — 57, № 4. — P. 515 – 522. ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3 436 O. CHVARTATSKYI, YU. SYDORENKO 30. Chvartatskyi O., Sydorenko Yu. Matrix generalizations of integrable systems with Lax integro-differential representations // J. Phys.: Conf. Ser. — 2013. — 411. — P. 012010-1 – 012010-11. e-print:http://arxiv.org/abs/1212.3444 31. Chvartatskyi O., Sydorenko Yu. A new (1+1)-dimensional matrix KP hierarchy // Mat. Stud. — 2013. — 39, № 2. — P. 164 – 178. e-print:http://arxiv.org/abs/1303.7064 32. Chvartatskyi O., Sydorenko Yu. A new bidirectional generalization of (2+1)-dimensional matrix k-constrained KP hierarchy // J. Math. Phys. — 2013. — 54. — P. 113508-1 – 113508-22. e-print:http://arxiv.org/abs/1303.6510 33. Dryuma V. S. On the analytic solution of the two-dimensional Korteweg – de Vries equation // JETP Lett. — 1974. — 19, № 12. — P. 753 – 755. 34. Zakharov V. E., Shabat A. B. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I // Funct. Anal. and Appl. — 1974. — 8, № 3. — P. 226 – 235. 35. Sydorenko Yu., Chvartatskyi O. Binary transformations of the spatially two-dimensional operators and Lax equations // Visn. Kyiv Univ. Mech. and Math. — 2009. — 22. — P. 32 – 35 (in Ukranian). 36. Loris I., Willox R. Symmetry reduction of the BKP hierarchy // J. Math. Phys. — 1999. — 40. — P. 1420 – 1431. 37. He J., Wu Zh., Cheng Yi. Gauge transformations for the constrained CKP and BKP hierarchies // J. Math. Phys. — 2007. — 48, № 11. — P. 113519-1 – 13519-16. 38. Wu H. -X., Liu X. -J., Zeng Y. -B. Two new multi-component BKP hierarchies // Commun. Theor. Phys. — 2009. — 51. — P. 193 – 204. 39. Khalilov F. A., Khruslov E. Ya. Matrix generalisation of the modified Korteweg – de Vries equation // Inverse Problems. — 1990. — 6. Received 28.02.13, after revision — 29.03.14 ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 3

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