Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 392723, 11 pages http://dx.doi.org/10.1155/2015/392723
Research Article Links between (𝛾𝑛, 𝜎𝑘)-KP Hierarchy, (𝛾𝑛, 𝜎𝑘)-mKP Hierarchy, and (2+1)-(𝛾𝑛, 𝜎𝑘)-Harry Dym Hierarchy Yehui Huang,1 Yuqin Yao,2 and Yunbo Zeng3 1
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China Department of Applied Mathematics, China Agricultural University, Beijing 100083, China 3 Department of Mathematical Science, Tsinghua University, Beijing 100084, China 2
Correspondence should be addressed to Yuqin Yao;
[email protected] Received 12 July 2015; Accepted 16 November 2015 Academic Editor: Boris G. Konopelchenko Copyright © 2015 Yehui Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The new (2+1)-(𝛾𝑛 , 𝜎𝑘 )-Harry Dym hierarchy and (𝛾𝑛 , 𝜎𝑘 )-mKP hierarchy with two new time series 𝛾𝑛 and 𝜎𝑘 , which consist of 𝛾𝑛 -flow, 𝜎𝑘 -flow, and mixed 𝛾𝑛 and 𝜎𝑘 evolution equations of eigenfunctions, are proposed. Gauge transformations and reciprocal transformations between (𝛾𝑛 , 𝜎𝑘 )-KP hierarchy, (𝛾𝑛 , 𝜎𝑘 )-mKP hierarchy, and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-Harry Dym hierarchy are studied. Their soliton solutions are presented.
1. Introduction Generalizations of soliton hierarchies are important topics since last century. In 2008, KdV6 equation is studied as a nonholonomic deformation of KdV equation. Kupershmidt developed a deformation from the bi-Hamiltonian structure of the soliton equations and KdV6 equation could be seen as the deformation of the KdV equation [1]. We generalized this Kupershmidt deformed system and find that KdV6 equation could be seen as the Rosochatius deformation of the KdV equation with self-consistent sources and firstly found the bi-Hamiltonian structure for the KdV6 equation [2]. The Kadomtsev-Petviashvili (KP) hierarchy is an important 2+1 dimensional integrable system and the generalizations of KP hierarchy (KPH) attract a lot of interests from both physical and mathematical points of view [3–11]. One kind of generalization is the multicomponent KP hierarchy [3], which contains many physical relevant nonlinear integrable systems such as Davey-Stewartson equation, two-dimensional Toda lattice, and three-wave resonant integrable equations. Another kind of generalization of KP equation is the so called KP equation with self-consistent sources (KPESCS) [10, 11]. The third kind of generalization is the extended KP hierarchy (exKPH), which is constructed by introducing another time
series {𝜏𝑘 } [12–14]. The exKPH consists of 𝑡𝑛 -flow of KP hierarchy, 𝜏𝑘 -flow, and 𝑡𝑛 -evolution equations of eigenfunctions. To make difference, we may call the exKPH as (𝑡𝑛 , 𝜏𝑘 )-KPH. Recently, we generalize the (𝑡𝑛 , 𝜏𝑘 )-KPH to (𝛾𝑛 , 𝜎𝑘 )-KPH by introducing two new time series 𝛾𝑛 and 𝜎𝑘 with two parameters 𝛼𝑛 and 𝛽𝑘 [15]: 𝑁
𝐿 𝛾𝑛 = [𝐵𝑛 + 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 , 𝐿] , 𝐵𝑛 = 𝐿𝑛≥0 ,
(1a)
𝑖=1 𝑁
𝐿 𝜎𝑘 = [𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 , 𝐿] ,
(1b)
𝑖=1
𝛼𝑛 (𝑞𝑖,𝜎𝑘 − 𝐵𝑘 (𝑞𝑖 )) − 𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐵𝑛 (𝑞𝑖 )) = 0, 𝛼𝑛 (𝑟𝑖,𝜎𝑘 + 𝐵𝑘∗ (𝑟𝑖 )) − 𝛽𝑘 (𝑟𝑖,𝛾𝑛 + 𝐵𝑛∗ (𝑟𝑖 )) = 0,
(1c)
𝑖 = 1, 2, . . . , 𝑁, where the pseudodifferential operator 𝐿 with potential functions 𝑢𝑖 is defined as 𝐿 = 𝜕 + 𝑢1 𝜕−1 + 𝑢2 𝜕−2 + ⋅ ⋅ ⋅
(2)
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and ∗ denotes the adjoint operator. The compatibility of 𝛾𝑛 flow (1a) and 𝜎𝑘 -flow (1b) under (1c) gives rise to the zerocurvature representation for (1a), (1b), and (1c), 𝑁
2. A New (2+1)-(𝛾𝑛 , 𝜎𝑘 )-Harry Dym Hierarchy 2.1. A New (2+1)-(𝛾𝑛 , 𝜎𝑘 )-Harry Dym Hierarchy. It is well known that the pseudodifferential operator 𝐿 for (2+1)-HDH with potential functions 𝑤𝑖 is defined as 𝐿 = 𝑤𝜕 + 𝑤0 + 𝑤1 𝜕−1 + 𝑤2 𝜕−2 + ⋅ ⋅ ⋅ .
𝐵𝑛,𝜎𝑘 − 𝐵𝑘,𝛾𝑛 + [𝐵𝑛 , 𝐵𝑘 ] + 𝛽𝑘 [𝐵𝑛 , ∑𝑞𝑖 𝜕−1 𝑟𝑖 ] 𝑖=1
𝑁
+
(3a)
The (2+1)-HDH is given by [17]
−1
+ 𝛼𝑛 [∑𝑞𝑖 𝜕 𝑟𝑖 , 𝐵𝑘 ] = 0, 𝑖=1
𝐿𝑡𝑛 = [𝐵𝑛 , 𝐿] ,
+
𝑛
𝛼𝑛 (𝑞𝑖,𝜎𝑘 − 𝐵𝑘 (𝑞𝑖 )) − 𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐵𝑛 (𝑞𝑖 )) = 0, 𝛼𝑛 (𝑟𝑖,𝜎𝑘 + 𝐵𝑘∗ (𝑟𝑖 )) − 𝛽𝑘 (𝑟𝑖,𝛾𝑛 + 𝐵𝑛∗ (𝑟𝑖 )) = 0,
(3b)
(5)
𝑛 ≥ 1,
(6)
𝑛
where 𝐵𝑛 = 𝐿+ = (𝐿 )≥2 , 𝑛 ≥ 1. The compatibility of the 𝑡𝑛 -flow and 𝑡𝑘 -flow of (6) leads to the zero-curvature representation of (2+1)-HDH: 𝐵𝑛,𝑡𝑘 − 𝐵𝑘,𝑡𝑛 + [𝐵𝑛 , 𝐵𝑘 ] = 0.
𝑖 = 1, 2, . . . , 𝑁,
(7)
In particular, when taking 𝐵2 = 𝑤2 𝜕2 and 𝐵3 = 𝑤3 𝜕3 + 3𝑤2 (𝑤𝑥 + 𝑤0 )𝜕2 and setting 𝑡2 = 𝑦, 𝑡3 = 𝑡, and 𝑤0 = (1/2)𝜕−1 (𝑤𝑦 /𝑤2 ) − (1/2)𝑤𝑥 , (7) yields the (2+1)-HD equation
with the Lax representation 𝑁
4𝑤𝑡 = 𝑤3 𝑤𝑥𝑥𝑥 − 3
𝜓𝛾𝑛 = (𝐵𝑛 + 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝜓) , 𝑖=1 𝑁
(4)
(8)
which is reduced to the Harry Dym equation. Consider
−1
𝜓𝜎𝑘 = (𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕 𝑟𝑖 ) (𝜓) .
4𝑤𝑡 = 𝑤3 𝑤𝑥𝑥𝑥
𝑖=1
The (𝛾𝑛 , 𝜎𝑘 )-KPH consists of 𝛾𝑛 -flow, 𝜎𝑘 -flow, and one mixed 𝛾𝑛 and 𝜎𝑘 evolution equation of eigenfunctions. The (𝛾𝑛 , 𝜎𝑘 )KPH can be reduced to the KPH and (𝑡𝑛 , 𝜏𝑘 )-KPH and contains first type and second type as well as mixed type of KPESCS as special cases. We also develop the dressing method to solve the (𝛾𝑛 , 𝜎𝑘 )-KPH [15]. The (2+1)-Harry Dym equation has been firstly defined by Konopelchenko and Dubrovsky in 1984 [16]. The Harry Dym hierarchy is the third hierarchy of soliton hierarchies presented by the pseudodifferential operator technique after the KP hierarchy and mKP hierarchy [17]. In this paper, we first constructed the (2+1)-(𝛾𝑛 , 𝜎𝑘 )-Harry Dym hierarchy ((2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH), which consists of 𝛾𝑛 -flow, 𝜎𝑘 -flow, and one mixed 𝛾𝑛 and 𝜎𝑘 evolution equation of eigenfunctions. The (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH can be reduced to the (2+1)-Harry Dym hierarchy ((2+1)-HDH) and (𝑡𝑛 , 𝜏𝑘 )-HDH [18] and contains first type and second type as well as mixed type of Harry Dym equation with self-consistent sources (HDESCS) as special cases. Then the (𝛾𝑛 , 𝜎𝑘 )-mKPH is proposed. The generalized gauge transformations and generalized reciprocal links between these three kinds of (𝛾𝑛 , 𝜎𝑘 ) hierarchies are studied. Further, based on these transformations and the solutions of the (𝛾𝑛 , 𝜎𝑘 )-KPH [15], the solutions of (𝛾𝑛 , 𝜎𝑘 )mKPH and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH are obtained, respectively. The paper is organized as follows. In Section 2, we propose a new (𝛾𝑛 , 𝜎𝑘 )-HDH and present its reduction. A new (𝛾𝑛 , 𝜎𝑘 )-mKPH is proposed in Section 3. Section 4 is devoted to studying the links between the three kinds of (𝛾𝑛 , 𝜎𝑘 ) hierarchies. Section 5 presents the soliton solutions and a conclusion is given in the last section.
𝑤𝑦 1 [𝑤2 𝜕−1 ( 2 )] , 𝑤 𝑤 𝑦
(9)
by dropping the 𝑦-dependence. Based on the squared eigenfunction symmetry constraint 𝑘
𝑁
𝐿 = 𝐵𝑘 + ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 , 𝑖=1
(10)
𝑞𝑖,𝑡𝑛 = 𝐵𝑛 (𝑞𝑖 ) , ∗
𝑟𝑖,𝑡𝑛 = −𝜕−2 𝐵𝑛 𝜕2 (𝑟𝑖 ) , which is compatible with (2+1)-HDH [17], we propose the following generalized (2+1)-HDH with two generalized time series 𝛾𝑛 and 𝜎𝑘 : 𝑁
𝐿𝛾𝑛 = [𝐵𝑛 + 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 , 𝐿] ,
(11a)
𝑖=1 𝑁
𝐿𝜎𝑘 = [𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 , 𝐿] ,
(11b)
𝑖=1
𝛼𝑛 (𝑞𝑖,𝜎𝑘 − 𝐵𝑘 (𝑞𝑖 )) − 𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐵𝑛 (𝑞𝑖 )) = 0, ∗
∗
𝛼𝑛 (𝑟𝑖,𝜎𝑘 + 𝜕−2 𝐵𝑘 𝜕2 (𝑟𝑖 )) − 𝛽𝑘 (𝑟𝑖,𝛾𝑛 + 𝜕−2 𝐵𝑛 𝜕2 (𝑟𝑖 )) = 0,
(11c)
𝑖 = 1, 2, . . . , 𝑁.
We will prove the compatibility of (11a) and (11b) under (11c) in the following theorem. Theorem 1. The 𝛾𝑛 -flow (11a) and 𝜎𝑘 -flow (11b) under (11c) are compatible.
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Proof. Denote
Then (15), (16), and (18) under (11c) yield 𝑁
𝐵̂𝑛,𝜎𝑘 − 𝐵̂𝑘,𝛾𝑛 + [𝐵̂𝑛 , 𝐵̂𝑘 ] = [𝛼𝑛 (𝑞𝜎𝑘 − 𝐵𝑘 (𝑞))
𝐵̂𝑛 = 𝐵𝑛 + 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 , 𝑖=1
− 𝛽𝑘 (𝑞𝛾𝑛 − 𝐵𝑛 (𝑞))] 𝜕−1 𝑟𝜕2
(12)
𝑁
𝐵̂𝑘 = 𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 . 𝑖=1
∗
− 𝛽𝑘 (𝑟𝛾𝑛 + 𝜕−2 𝐵𝑛 𝜕2 (𝑟))] 𝜕2 = 0.
In order to prove 𝐿𝛾𝑛 ,𝜎𝑘 = 𝐿𝜎𝑘 ,𝛾𝑛 , that is, [𝐵̂𝑛,𝜎𝑘 − 𝐵̂𝑘,𝛾𝑛 + [𝐵̂𝑛 , 𝐵̂𝑘 ] , 𝐿] = 0,
(13)
we only need to prove 𝐵̂𝑛,𝜎𝑘 − 𝐵̂𝑘,𝛾𝑛 + [𝐵̂𝑛 , 𝐵̂𝑘 ] = 0.
= [𝐵𝑘 + 𝛽𝑘 (𝑞𝜕−1 𝑟𝜕2 ) , 𝐿 ]+ + 𝛼𝑛 (𝑞𝜕−1 𝑟𝜕2 )𝜎
𝑘
𝑛
= [𝐵𝑘 , 𝐿 ]+ + 𝛽𝑘 [𝑞𝜕−1 𝑟𝜕2 , 𝐿 ]+ + 𝛼𝑛 𝑞𝜎𝑘 𝜕−1 𝑟𝜕2
(15)
and, similarly, 𝑘
𝑘
𝐵̂𝑘,𝛾𝑛 = [𝐵𝑛 , 𝐿 ] + 𝛼𝑛 [𝑞𝜕 𝑟𝜕 , 𝐿 ] + 𝛽𝑘 𝑞𝛾𝑛 𝜕 𝑟𝜕 −1
2
+
−1
2
+
2
(16)
+ 𝛽𝑘 𝑞𝜕 𝑟𝛾𝑛 𝜕 . 𝑁
−1
[𝐵𝑛 , ∑𝑞𝑖 𝜕 𝑟𝑖 𝜕 ] = ∑𝐵𝑛 (𝑞𝑖 ) 𝜕 𝑟𝑖 𝜕
𝑖=1
=0
Theorem 2. The commutativity of (11a) and (11b) under (11c) gives rise to the zero-curvature equation for the generalized (2+1)-HDH with two generalized time series:
𝑖=1
𝑁
2
(17)
𝑁
𝑖=1
𝑖=1
+
∗
𝛼𝑛 (𝑟𝑖,𝜎𝑘 + 𝜕−2 𝐵𝑘 𝜕2 (𝑟𝑖 ))
we have
(21b)
∗
− 𝛽𝑘 (𝑟𝑖,𝛾𝑛 + 𝜕−2 𝐵𝑛 𝜕2 (𝑟𝑖 )) = 0,
[𝐵̂𝑛 , 𝐵̂𝑘 ] = [𝐵𝑛 , 𝐵𝑘 ] + [𝐵𝑛 , 𝛽𝑘 𝑞𝜕−1 𝑟𝜕2 ] + [𝛼𝑛 𝑞𝜕−1 𝑟𝜕2 , 𝐵𝑘 ]
𝑖 = 1, 2, . . . , 𝑁,
with the Lax representation 𝑁
𝑘
= [𝐿 − (𝐿 )− , 𝐿 − (𝐿 ) ]
𝜓𝛾𝑛 = (𝐵𝑛 + 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ) (𝜓) ,
− +
𝑖=1
+ 𝛽𝑘 [𝐵𝑛 , 𝑞𝜕−1 𝑟𝜕2 ]+ + 𝛼𝑛 [𝑞𝜕−1 𝑟𝜕2 , 𝐵𝑘 ]+ + 𝛽𝑘 [𝐵𝑛 , 𝑞𝜕−1 𝑟𝜕2 ]− + 𝛼𝑛 [𝑞𝜕−1 𝑟𝜕2 , 𝐵𝑘 ]− 𝑛
(21a)
2
𝛼𝑛 (𝑞𝑖,𝜎𝑘 − 𝐵𝑘 (𝑞𝑖 )) − 𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐵𝑛 (𝑞𝑖 )) = 0,
∗
− ∑𝑞𝑖 𝜕−3 𝐵𝑛 (𝑟𝑖 ) 𝜕4 ,
𝑘
−1
+
+ 𝛼𝑛 [∑𝑞𝑖 𝜕 𝑟𝑖 𝜕 , 𝐵𝑘 ] = 0,
𝑖=1
−
𝑘
𝑖=1
(20)
𝑁
2
𝑛
𝑁
𝐵𝑛,𝜎𝑘 − 𝐵𝑘,𝛾𝑛 + [𝐵𝑛 , 𝐵𝑘 ] + 𝛽𝑘 [𝐵𝑛 , ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ]
Moreover, on the other hand, under the formula [18]
𝑛
𝑁
𝛾𝑛
which under (11c) can be simplified as follows. Then we have the following.
+ 𝛼𝑛 𝑞𝜕−1 𝑟𝜎𝑘 𝜕2 ,
𝑖=1
𝑖=1
𝜎𝑘
+ [𝐵𝑛 + 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 , 𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ]
𝑛
−1
𝑁
𝑖=1
𝑘
𝑁
𝑁
(𝐵𝑛 + 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ) − (𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 )
𝐵̂𝑛,𝜎𝑘 = 𝐵𝑛,𝜎𝑘 + 𝛼𝑛 (𝑞𝜕−1 𝑟𝜕2 )𝜎
−1
The compatibility of 𝛾𝑛 -flow (11a) and 𝜎𝑘 -flow (11b) under (11c) gives rise to the zero-curvature representation for (11a), (11b), and (11c):
(14)
For convenience, we omit ∑. We can find that
𝑛
(19)
∗
+ 𝑞𝜕−1 [𝛼𝑛 (𝑟𝜎𝑘 + 𝜕−2 𝐵𝑘 𝜕2 (𝑟))
𝑛
𝑘
= [𝐵𝑛 , 𝐿 ] + [𝐿 , 𝐵𝑘 ]+ − [(𝐿 )− , (𝐿 ) ] +
− +
+ 𝛽𝑘 [𝐵𝑛 , 𝑞𝜕−1 𝑟𝜕2 ]+ + 𝛼𝑛 [𝑞𝜕−1 𝑟𝜕2 , 𝐵𝑘 ]+ ∗
+ 𝛽𝑘 𝐵𝑛 (𝑞) 𝜕−1 𝑟𝜕2 − 𝛽𝑘 𝑞𝜕−3 𝐵𝑛 (𝑟) 𝜕4 ∗
− 𝛼𝑛 𝐵𝑘 (𝑞) 𝜕−1 𝑟𝜕2 + 𝛼𝑛 𝑞𝜕−3 𝐵𝑘 (𝑟) 𝜕4 .
𝑁
(18)
(22) −1
2
𝜓𝜎𝑘 = (𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕 𝑟𝑖 𝜕 ) (𝜓) . 𝑖=1
We briefly call (11a), (11b), and (11c) and (21a) and (21b) as (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH. It is easy to see that (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH ((11a), (11b), and (11c) and (21a) and (21b)) for 𝛼𝑛 = 𝛽𝑘 = 0 reduces to (2+1)-HDH ((6) and (7)) and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH for 𝛼𝑛 = 0, 𝛽𝑘 = 1 reduces to (𝑡𝑛 , 𝜏𝑘 )-HDH [18]. So (2+1)(𝛾𝑛 , 𝜎𝑘 )-HDH ((11a), (11b), and (11c) and (21a) and (21b)) presents a more generalized (2+1)-HDH which contains the (2+1)-HDH and (2+1)-(𝑡𝑛 , 𝜏𝑘 )-HDH as the special cases.
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Advances in Mathematical Physics
Example 3. Let us take 𝑛 = 2 and 𝑘 = 3, and set 𝛾2 = 𝑦, 𝜎3 = 𝑡. Then (21a) and (21b) become 𝑁
𝐵2,𝑡 − 𝐵3,𝑦 + [𝐵2 , 𝐵3 ] + 𝛽3 [𝐵2 , ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ] 𝑖=1
𝑁
−1
+
(23a)
2
+ 𝛼2 [∑𝑞𝑖 𝜕 𝑟𝑖 𝜕 , 𝐵3 ] = 0, 𝑖=1
+
2.2. Reduction. Consider the constraint given by
𝛼2 (𝑞𝑖,𝑡 − 𝐵3 (𝑞𝑖 )) − 𝛽3 (𝑞𝑖,𝑦 − 𝐵2 (𝑞𝑖 )) = 0, ∗
𝛼2 (𝑟𝑖,𝑡 + 𝜕−2 𝐵3 𝜕2 (𝑟𝑖 ))
Particularly, when taking 𝛼2 = 𝛽3 = 0; 𝛼2 = 0, 𝛽3 = 1; 𝛼2 = 1, 𝛽3 = 0; and 𝛼2 = 1, 𝛽3 = 1, respectively, (24a), (24b), and (24c) and (26) reduce to the (2+1)-HD equation, the first type of (2+1)-HD equation with self-consistent sources ((2+1)-HDESCS), the second type of (2+1)-HDESCS, and the mixed type of (2+1)-HDESCS and their Lax representations, respectively.
∗
− 𝛽3 (𝑟𝑖,𝑦 + 𝜕−2 𝐵2 𝜕2 (𝑟𝑖 )) = 0,
𝑖 = 1, 2, . . . , 𝑁, Then (11b) yields
2𝑤𝑤𝑡 − 3 [𝑤2 (𝑤𝑥 + 𝑤0 )]𝑦
𝑁
𝑘
(𝐿 )
𝜎𝑘
3
+ 𝑤 (𝑤𝑤𝑥𝑥𝑥 + 6𝑤𝑥 𝑤𝑥𝑥 + 3𝑤𝑤0𝑥𝑥 + 12𝑤𝑥 𝑤0𝑥 ) 𝑁
𝑘
= [𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 , 𝐿 ] = 0,
(28)
𝑖=1
𝑘
𝐵𝑘,𝜎𝑘 = (𝐿𝜎𝑘 ) = 0,
𝑁
+
+ 2𝛽3 𝑤 ∑ (𝑞𝑖 𝑟𝑖 )𝑥 − 2𝛽3 𝑤𝑤𝑥 ∑𝑞𝑖 𝑟𝑖 𝑖=1
(27)
𝑖=1
which gives the following nonlinear equation:
2
𝑁
𝑘
𝐿 = 𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 .
(23b)
𝑁
𝑖=1
𝑁
(24a)
− 3𝛼2 𝑤3 ∑ (𝑞𝑖,𝑥𝑥 𝑟𝑖 + 𝑞𝑖,𝑥 𝑟𝑖𝑥 )
𝑖=1
(29)
𝑘
(∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 )
𝜎𝑘
= (𝐿𝜎𝑘 ) = 0, −
𝑖=1
which imply that 𝐿, 𝐵𝑘 , 𝑞𝑖 , and 𝑟𝑖 under (27) are independent of 𝜎𝑘 . Subsequently, 𝑞𝑖,𝜎𝑘 and 𝑟𝑖,𝜎𝑘 in (11c) should be replaced by 𝜆 𝑖 𝑞𝑖 and −𝜆 𝑖 𝑟𝑖 as in the case of constrained flow of KP [19, 20]; namely, (11c) under the constraint (27) should be replaced by
𝑁
2
− 6𝛼2 𝑤 (𝑤𝑥 + 𝑤0 ) ∑ (𝑞𝑖 𝑟𝑖 )𝑥 𝑖=1
𝑁
+ 3𝛼2 [𝑤2 (𝑤𝑥 + 𝑤0 )]𝑥 ∑𝑞𝑖 𝑟𝑖 = 0, 𝑖=1
𝛼𝑛 (𝜆 𝑖 𝑞𝑖 − 𝐵𝑘 (𝑞𝑖 )) − 𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐵𝑛 (𝑞𝑖 )) = 0,
𝑁
𝑤𝑦 − 𝑤2 (𝑤𝑥𝑥 + 2𝑤0𝑥 ) + 𝛼2 𝑤∑ (𝑞𝑖 𝑟𝑖 )𝑥 𝑖=1
∗
(24b)
𝑁
∗
𝛼𝑛 (−𝜆 𝑖 𝑟𝑖 + 𝜕−2 𝐵𝑘 𝜕2 (𝑟𝑖 )) − 𝛽𝑘 (𝑟𝑖,𝛾𝑛 + 𝜕−2 𝐵𝑛 𝜕2 (𝑟𝑖 )) = 0.
− 𝛼2 𝑤𝑥 ∑𝑞𝑖 𝑟𝑖 = 0,
(30) (31)
𝑖=1
We will show that constraint (27) is invariant under 𝛾𝑛 -flow (11a) and (31). In fact, making use of (11a), (17), and (31), we have
𝛼2 [𝑞𝑖,𝑡 − 𝑤3 𝑞𝑖,𝑥𝑥𝑥 − 3𝑤2 (𝑤𝑥 + 𝑤0 ) 𝑞𝑖,𝑥𝑥 ] − 𝛽3 (𝑞𝑖,𝑦 − 𝑤2 𝑞𝑖,𝑥𝑥 ) = 0,
(24c)
𝛼2 (𝑟𝑖,𝑡 − 𝑤3 𝑟𝑖,𝑥𝑥𝑥 − 3𝑤2 𝑤0 𝑟𝑖,𝑥𝑥 )
𝑘
𝛾𝑛
2
− 𝛽3 (𝑟𝑖,𝑦 + 𝑤 𝑟𝑖,𝑥𝑥 ) = 0, 𝑖 = 1, 2, . . . , 𝑁,
𝑁
𝑖=1
N
𝜓𝑦 = (𝑤2 𝜕2 + 𝛼2 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ) (𝜓) ,
𝑘
−
−
𝑁
(𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ) = 𝛽𝑘 ∑ (𝑞𝑖,𝛾𝑛 𝜕−1 𝑟𝑖 𝜕2
with the Lax representation as follows: (25)
𝑖=1
𝑖=1
𝛾𝑛
𝑁
+ 𝑞𝑖 𝜕−1 𝑟𝑖,𝛾𝑛 𝜕2 ) = ∑ [𝛽𝑘 𝐵𝑛 (𝑞𝑖 ) 𝜕−1 𝑟𝑖 𝜕2 𝑖=1
𝑁
𝜓𝑡 = (𝑤3 𝜕3 + 3𝑤2 (𝑤𝑥 + 𝑤0 ) 𝜕2 + 𝛽3 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ) 𝑖=1
⋅ (𝜓) .
𝑘
(𝐿 − 𝐵𝑘 ) = (𝐿𝛾𝑛 ) = [𝐵̂𝑛 , 𝐿 ] ,
+ 𝛼𝑛 (𝜆 𝑖 𝑞𝑖 − 𝐵𝑘 (𝑞𝑖 )) 𝜕−1 𝑟𝑖 𝜕2 − 𝛽𝑘 𝑞𝑖 𝜕−1 𝐵𝑛 (𝑟𝑖 ) 𝜕2 (26)
∗
+ 𝛼𝑛 𝑞𝑖 𝜕−1 (−𝜆 𝑖 𝑟𝑖 + 𝜕−2 𝐵𝑘 𝜕2 (𝑟𝑖 )) 𝜕2 ] = [𝐵𝑛 ,
Advances in Mathematical Physics 𝑁
5 𝑁
𝑁
𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ] − [𝐵𝑘 , 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ] = [𝐵̂𝑛 , 𝑖=1
𝑖=1
−
𝑁
𝑤𝑦 − 𝑤2 (𝑤𝑥𝑥 + 2𝑤0𝑥 ) + 𝛼2 𝑤∑ (𝑞𝑖 𝑟𝑖 )𝑥 𝑖=1
−
𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ] − [𝐵𝑘 , 𝛼𝑛 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ] = [𝐵̂𝑛 , 𝑖=1
𝑖=1
−
(35b)
𝑁
𝑁
− 𝛼2 𝑤𝑥 ∑𝑞𝑖 𝑟𝑖 = 0, 𝑖=1
−
𝛼2 [𝜆 𝑖 𝑞𝑖 − 𝑤3 𝑞𝑖,𝑥𝑥𝑥 − 3𝑤2 (𝑤𝑥 + 𝑤0 ) 𝑞𝑖,𝑥𝑥 ]
𝑘
𝐿 ] − [𝐵̂𝑛 , 𝐵𝑘 ]− − [𝐵𝑘 , 𝐵̂𝑛 ]− + [𝐵𝑘 , 𝐵𝑛 ]− = [𝐵̂𝑛 , −
− 𝛽3 (𝑞𝑖,𝑦 − 𝑤2 𝑞𝑖,𝑥𝑥 ) = 0,
𝑘
𝐿 ] . −
(32)
𝛼2 (−𝜆 𝑖 𝑟𝑖 − 𝑤3 𝑟𝑖,𝑥𝑥𝑥 − 3𝑤2 𝑤0 𝑟𝑖,𝑥𝑥 ) − 𝛽3 (𝑟𝑖,𝑦 + 𝑤2 𝑟𝑖,𝑥𝑥 ) = 0,
Then 𝑁
𝑘
(𝐿 − 𝐵𝑘 − 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ) = 0. 𝑖=1
(33)
𝛾𝑛
This means that the submanifold determined by 𝑘-constraint (27) is invariant under the 𝛾𝑛 -flow (11a) and (31). Therefore, the constrained flow of (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH ((11a), (11b), and (11c) and (21a) and (21b)) under (27) reads 𝑁
−1
𝑖=1
+
𝑁
3. The (𝛾𝑛 , 𝜎𝑘 )-mKPH In the same way, the (𝛾𝑛 , 𝜎𝑘 )-mKPH can be formulated. The ̃ operator of mKP hierarchy is defined by 𝐿 ̃ = 𝜕 + V0 + V1 𝜕−1 + V2 𝜕−2 + ⋅ ⋅ ⋅ . 𝐿 ̃ 𝑡 = [𝐵̃𝑛 , 𝐿] ̃ , 𝐿 𝑛 ̃𝑛 , 𝐵̃𝑛 = 𝐿 ≥1
+
(37)
𝛼𝑛 (𝜆 𝑖 𝑞𝑖 − 𝐵𝑘 (𝑞𝑖 )) − 𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐵𝑛 (𝑞𝑖 )) = 0, ∗
𝛼𝑛 (−𝜆 𝑖 𝑟𝑖 + 𝜕−2 𝐵𝑘 𝜕2 (𝑟𝑖 ))
𝑛 ≥ 1. (34b)
∗
− 𝛽𝑘 (𝑟𝑖,𝛾𝑛 + 𝜕−2 𝐵𝑛 𝜕2 (𝑟𝑖 )) = 0,
(36)
The Lax equation of mKP hierarchy is given by (34a)
+ 𝛼𝑛 [𝐵𝑘 , ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 ] = 0, 𝑖=1
𝑖 = 1, 2, . . . , 𝑁,
which is the 𝑘-constraint (2+1)-(𝛾𝑛 , 𝜎𝑘 )-Harry Dym equation.
2
𝐵𝑘,𝛾𝑛 + [𝐵𝑘 , 𝐵𝑛 ] + 𝛽𝑘 [∑𝑞𝑖 𝜕 𝑟𝑖 𝜕 , 𝐵𝑛 ]
(35c)
𝑖 = 1, 2, . . . , 𝑁.
The commutativity of 𝜕𝑡𝑛 and 𝜕𝑡𝑘 flows gives the zerocurvature equation (7). Since the squared eigenfunction symmetry constraint, given by
with
𝑁
𝑁
𝑛/𝑘
𝑖=1
+
𝐵𝑛 = (𝐵𝑘 + 𝛽𝑘 ∑𝑞𝑖 𝜕−1 𝑟𝑖 𝜕2 )
̃ 𝑘 = 𝐵̃𝑘 + ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐿
(34c)
.
𝑖=1
𝑞̃𝑖,𝑡𝑛 = 𝐵̃𝑛 (̃ 𝑞𝑖 ) ,
Example 4. When 𝑛 = 2, 𝑘 = 3, 𝛾2 = 𝑦, (34a), (34b), and (34c) give 3 [𝑤2 (𝑤𝑥 + 𝑤0 )]𝑦
(38)
𝑟̃𝑖,𝑡𝑛 = −𝜕−1 𝐵̃𝑛∗ 𝜕 (̃𝑟𝑖 ) , is compatible with mKP hierarchy [21, 22], we have the following.
− 𝑤3 (𝑤𝑤𝑥𝑥𝑥 + 6𝑤𝑥 𝑤𝑥𝑥 + 3𝑤𝑤0𝑥𝑥 + 12𝑤𝑥 𝑤0𝑥 ) 𝑁
Definition 5. The (𝛾𝑛 , 𝜎𝑘 )-mKPH is defined by
𝑁
𝑁
− 2𝛽3 𝑤2 ∑ (𝑞𝑖 𝑟𝑖 )𝑥 + 2𝛽3 𝑤𝑤𝑥 ∑𝑞𝑖 𝑟𝑖 𝑖=1
̃ 𝛾 = [𝐵̃𝑛 + 𝛼𝑛 ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐿] ̃ , 𝐿 𝑛
𝑖=1
𝑁
+ 3𝛼2 𝑤3 ∑ (𝑞𝑖,𝑥𝑥 𝑟𝑖 + 𝑞𝑖,𝑥 𝑟𝑖𝑥 ) 𝑖=1
(39a)
𝑖=1
(35a)
𝑁
̃ 𝜎 = [𝐵̃𝑘 + 𝛽𝑘 ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐿] ̃ , 𝐿 𝑘
(39b)
𝑖=1
𝑁
+ 6𝛼2 𝑤2 (𝑤𝑥 + 𝑤0 ) ∑ (𝑞𝑖 𝑟𝑖 )𝑥 𝑖=1
𝑁
− 3𝛼2 [𝑤2 (𝑤𝑥 + 𝑤0 )]𝑥 ∑𝑞𝑖 𝑟𝑖 = 0, 𝑖=1
𝛼𝑛 (̃ 𝑞𝑖,𝜎𝑘 − 𝐵̃𝑘 (̃ 𝑞𝑖 )) − 𝛽𝑘 (̃ 𝑞𝑖,𝛾𝑛 − 𝐵̃𝑛 (̃ 𝑞𝑖 )) = 0, 𝛼𝑛 (̃𝑟𝑖,𝜎𝑘 + 𝜕−1 𝐵̃𝑘∗ 𝜕 (̃𝑟𝑖 )) − 𝛽𝑘 (̃𝑟𝑖,𝛾𝑛 + 𝜕−1 𝐵̃𝑛∗ 𝜕 (̃𝑟𝑖 )) = 0,
𝑖 = 1, 2, . . . , 𝑁.
(39c)
6
Advances in Mathematical Physics (39a), (39b), and (39c) have the Lax representation
which gives the following nonlinear equation: 4V𝑡 − V𝑥𝑥𝑥 + 6V2 V𝑥 − 3𝜕−1 V𝑦𝑦 − 6V𝑥 𝜕−1 V𝑦
𝑁
̃ , 𝜓̃𝛾𝑛 = (𝐵̃𝑛 + 𝛼𝑛 ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕) (𝜓) 𝑖=1
(40)
𝑁
𝜓̃𝜎𝑘 = (𝐵̃𝑘 + 𝛽𝑘 ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕) (𝜓) . In the same way as in Section 2, we can verify the compatibility of (39a) and (39b) under (39c). By using the formula
𝑖=1
−
−
(41)
𝑁
∑𝑞̃𝑖 𝜕−2 𝐵̃𝑛∗ 𝜕 (̃𝑟𝑖 ) 𝜕, 𝑖=1
the zero-curvature equation for (𝛾𝑛 , 𝜎𝑘 )-mKPH can be written as 𝑁
𝐵̃𝑛,𝜎𝑘 − 𝐵̃𝑘,𝛾𝑛 + [𝐵̃𝑛 , 𝐵̃𝑘 ] + 𝛽𝑘 [𝐵̃𝑛 , ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕] 𝑖=1
+
𝑁
(42a)
+ 𝛼𝑛 [∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐵̃𝑘 ] = 0, 𝑖=1
+
𝛼𝑛 (̃ 𝑞𝑖,𝜎𝑘 − 𝐵̃𝑘 (̃ 𝑞𝑖 )) − 𝛽𝑘 (̃ 𝑞𝑖,𝛾𝑛 − 𝐵̃𝑛 (̃ 𝑞𝑖 )) = 0, 𝛼𝑛 (̃𝑟𝑖,𝜎𝑘 + 𝜕−1 𝐵̃𝑘∗ 𝜕 (̃𝑟𝑖 ))
(42b)
− 𝛽𝑘 (̃𝑟𝑖,𝛾𝑛 + 𝜕−1 𝐵̃𝑛∗ 𝜕 (̃𝑟𝑖 )) = 0, 𝑖 = 1, 2, . . . , 𝑁. It is easy to see that (𝛾𝑛 , 𝜎𝑘 )-mKPH ((39a), (39b), and (39c) and (42a) and (42b)) for 𝛼𝑛 = 𝛽𝑘 = 0 reduces to mKPH (37) and (𝛾𝑛 , 𝜎𝑘 )-mKPH for 𝛼𝑛 = 0, 𝛽𝑘 = 1 reduces to exmKPH [23]. Example 6. Let us take 𝑛 = 2 and 𝑘 = 3, and set 𝛾2 = 𝑦, 𝜎3 = 𝑡, and V0 = V. Then (42a) and (42b) become 𝑁
𝐵̃2,𝑡 − 𝐵̃3,𝑦 + [𝐵̃2 , 𝐵̃3 ] + 𝛽3 [𝐵̃2 , ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕] 𝑖=1
+
𝑁
(43a)
+ 𝛼2 [∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐵̃3 ] = 0, 𝑖=1
𝑖=1
𝑖=1
(44a)
3 3 𝛼2 [𝑞̃𝑖,𝑡 − 𝑞̃𝑖,𝑥𝑥𝑥 − 3Ṽ 𝑞𝑖,𝑥𝑥 − (𝜕−1 V𝑦 ) 𝑞̃𝑖,𝑥 − V𝑥 𝑞̃𝑖,𝑥 2 2 [ 3 3𝑁 − V2 𝑞̃𝑖,𝑥 − ∑𝑞̃𝑗 𝑟̃𝑗 𝑞̃𝑖𝑥 ] − 𝛽3 (̃ 𝑞𝑖,𝑦 − 𝑞̃𝑖,𝑥𝑥 2 2 𝑗=1 ]
𝑁
𝑞𝑖 ) 𝜕−1 𝑟̃𝑖 𝜕 [𝐵̃𝑛 , ∑𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕] = ∑𝐵̃𝑛 (̃ 𝑖=1
𝑁
𝑞𝑖 𝑟̃𝑖 )𝑥 ] = 0, − 3 (̃ 𝑞𝑖 𝑟̃𝑖 )𝑦 − 6 (Ṽ
𝑖=1
𝑁
𝑁
+ 4𝛽3 ∑ (̃ 𝑞𝑖 𝑟̃𝑖 )𝑥 + 𝛼2 ∑ [3 (̃ 𝑞𝑖 𝑟̃𝑖,𝑥𝑥 − 𝑞̃𝑖,𝑥 𝑟̃𝑖 )
𝛼2 (̃ 𝑞𝑖,𝑡 − 𝐵̃3 (̃ 𝑞𝑖 )) − 𝛽3 (̃ 𝑞𝑖,𝑦 − 𝐵̃2 (̃ 𝑞𝑖 )) = 0,
𝑖 = 1, 2, . . . , 𝑁,
(44b)
3 3 𝛼2 [𝑟̃𝑖,𝑡 − 𝑟̃𝑖,𝑥𝑥𝑥 − 3Ṽ𝑟𝑖,𝑥𝑥 − (𝜕−1 V𝑦 ) 𝑟̃𝑖,𝑥 − V𝑥 𝑟̃𝑖,𝑥 2 2 [ 3 3𝑁 − V2 𝑟̃𝑖,𝑥 − ∑𝑞̃𝑗 𝑟̃𝑗 𝑟̃𝑖𝑥 ] − 𝛽3 (̃𝑟𝑖,𝑦 + 𝑟̃𝑖,𝑥𝑥 2 2 𝑗=1 ] − 2Ṽ𝑟𝑖𝑥 ) = 0,
𝑖 = 1, 2, . . . , 𝑁.
Particularly, when taking 𝛼2 = 𝛽3 = 0; 𝛼2 = 0, 𝛽3 = 1; 𝛼2 = 1, 𝛽3 = 0; and 𝛼2 = 1, 𝛽3 = 1, respectively, (44a) and (44b) reduce to the mKP equation, the first type of mKP equation with self-consistent sources, the second type of mKP equation with self-consistent sources, and the mixed type of mKP equation with self-consistent sources, respectively.
4. Links between (𝛾𝑛 , 𝜎𝑘 )-KPH and (𝛾𝑛 , 𝜎𝑘 )-mKPH and (𝛾𝑛 , 𝜎𝑘 )-mKPH and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH In [17] the gauge transformations between KP and mKP hierarchies and the reciprocal transformation between mKP and HD hierarchies are proposed. The KP, mKP, and HD hierarchy are intimately related under these gauge transformations and reciprocal links. In [22], the constrained KP hierarchy and the constrained modified KP hierarchy are studied. And in [24], the gauge transformation between KP and mKP hierarchies with self-consistent sources is given. It is natural to think whether these transformations can be generalized to the (𝛾𝑛 , 𝜎𝑘 )-KPH, (𝛾𝑛 , 𝜎𝑘 )-mKPH, and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH. The answer is positive and the main results are as follows. Theorem 7. (a) Suppose 𝐿, 𝑞𝑖 , 𝑟𝑖 satisfy (𝛾𝑛 , 𝜎𝑘 )-KPH (1a), (1b), and (1c) and 𝑞 and 𝑓 are independent eigenfunctions for Lax pair (4); then
+
𝛼2 (̃𝑟𝑖,𝑡 + 𝜕−1 𝐵̃3∗ 𝜕 (̃𝑟𝑖 )) − 𝛽3 (̃𝑟𝑖,𝑦 + 𝜕−1 𝐵̃2∗ 𝜕 (̃𝑟𝑖 )) = 0,
− 2Ṽ 𝑞𝑖𝑥 ) = 0,
(43b)
̃ = 𝑞−1 𝐿𝑞, 𝐿 𝑞̃𝑖 = 𝑞−1 𝑞𝑖 ,
Advances in Mathematical Physics
7
𝑟̃𝑖 = −𝜕−1 (𝑞𝑟𝑖 ) ,
̃ 𝑘 𝑞−1 (𝑞) 𝑞𝑖 − 𝛼𝑛 𝛽𝑘 𝑞−2 𝑞𝑖 𝜕−1 (𝑟𝑖 𝑞) 𝑞𝑖 = −𝛼𝑛 𝑞−1 𝐿 ≥0
𝑞̃ = 𝑞−1 𝑓
̃ 𝑘 (𝑞𝑖 𝑞−1 ) + 𝛽𝑘 𝑞−1 (𝑞𝑖,𝛾𝑛 − 𝐿𝑛≥0 (𝑞𝑖 )) + 𝛼𝑛 𝐿 ≥0 (45)
satisfy the (𝛾𝑛 , 𝜎𝑘 )-mKPH ((39a), (39b), and (39c)) and its Lax pair (40). ̃ 𝑞̃𝑖 , 𝑟̃𝑖 satisfy the (𝛾𝑛 , 𝜎𝑘 )-mKPH ((39a), (b) Suppose 𝐿, (39b), and (39c)); 𝑞̃ is eigenfunction for Lax pair (40). After the transformation 𝑥 = 𝑞̃(𝑥, 𝛾𝑛 , 𝜎𝑘 ), 𝛾𝑛 = 𝛾𝑛 , 𝜎𝑘 = 𝜎𝑘 ; then
̃ 𝑘 (̃ − 𝛼𝑛 𝐿 ≥1 𝑞𝑖 ) ̃ 𝑘 (1) 𝑞𝑖 + 𝛼𝑛 𝐿 ̃ 𝑘 (𝑞𝑖 𝑞−1 ) = −𝛼𝑛 𝑞−1 𝐿 ≥0 ≥0 − 𝛼𝑛 𝛽𝑘 𝑞−2 𝑞𝑖 𝜕−1 (𝑟𝑖 𝑞) 𝑞𝑖 ̃ 𝑘 (̃ + 𝛽𝑘 𝑞−1 (𝑞𝑖,𝛾𝑛 − 𝐿𝑛≥0 (𝑞𝑖 )) − 𝛼𝑛 𝐿 ≥1 𝑞𝑖 ) ̃ 𝑘 (̃ ̃ 𝑘 𝑞𝑖 ) − 𝛼𝑛 𝛽𝑘 𝑞−2 𝑞𝑖 𝜕−1 (𝑟𝑖 𝑞) 𝑞𝑖 = 𝛼𝑛 𝐿 ≥1 𝑞𝑖 ) − 𝛼𝑛 𝐿 ≥1 (̃
̃ 𝐿 = 𝐿, 𝑞𝑖 = 𝑞̃𝑖 ,
+ 𝛽𝑘 𝑞−1 (𝑞𝑖,𝛾𝑛 − 𝐿𝑛≥0 (𝑞𝑖 ))
(46)
= −𝛼𝑛 𝛽𝑘 𝑞−2 𝑞𝑖 𝜕−1 (𝑟𝑖 𝑞) 𝑞𝑖
𝑟𝑖 = 𝜕−1 (̃ 𝑞𝑟̃𝑖𝑥 ) − 𝑟̃𝑖 𝑞̃
+ 𝛽𝑘 𝑞−1 (𝑞𝑖,𝛾𝑛 − 𝐿𝑛≥0 (𝑞𝑖 )) .
satisfy the (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH ((11a), (11b), and (11c)).
(48)
Proof. For convenience, we omit ∑.
Similarly, we have −2 −1 ̃ 𝑛 (̃ 𝛽𝑘 (̃ 𝑞𝑖,𝛾𝑛 − 𝐿 ≥1 𝑞𝑖 )) = −𝛼𝑛 𝛽𝑘 𝑞 𝑞𝑖 𝜕 (𝑟𝑖 𝑞) 𝑞𝑖
(a) Consider ̃ 𝛾 = −𝑞 𝑞𝛾 𝐿𝑞 + 𝑞 𝐿 𝛾 𝑞 + 𝑞 𝐿𝑞𝛾 = −𝑞 𝐿 𝑛 𝑛 𝑛 𝑛 −2
−1
−1
−2
+ 𝛼𝑛 𝑞−1 (𝑞𝑖,𝜎𝑘 − 𝐿𝑘≥0 (𝑞𝑖 )) .
(𝐿𝑛≥0
+ 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝑞) 𝐿𝑞 + 𝑞−1 [𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 , 𝐿] 𝑞
Thus, we have ̃ 𝑘 (̃ ̃ 𝑛 (̃ 𝑞𝑖,𝜎𝑘 − 𝐿 𝑞𝑖,𝛾𝑛 − 𝐿 𝛼𝑛 (̃ ≥1 𝑞𝑖 )) − 𝛽𝑘 (̃ ≥1 𝑞𝑖 ))
+ 𝑞−1 𝐿 (𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝑞)
= 𝑞−1 [𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐿𝑛≥0 (𝑞𝑖 )) − 𝛼𝑛 (𝑞𝑖,𝜎𝑘 − 𝐿𝑘≥0 (𝑞𝑖 ))] (50)
= [𝑞−1 𝐿𝑞, 𝑞−1 (𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝑞)] + 𝑞−1 [𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 , 𝐿] 𝑞 = [𝑞−1 (𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 ) 𝑞
(47)
̃ = [𝑞−1 𝐿𝑛 𝑞 − 𝑞−1 (𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝑞) , 𝐿] ≥0 ̃ + 𝛼𝑛 [𝑞−1 𝑞𝑖 𝜕−1 𝑟𝑖 𝑞 − 𝑞−1 𝐿𝑛≥0 (𝑞) , 𝐿]
= 0. In the following, we will prove that 𝑞̃ = 𝑞−1 𝑓 satisfies Lax pair (40). Consider 𝑞̃𝛾𝑛 = −𝑞−2 𝑞𝛾𝑛 𝑓 + 𝑞−1 𝑓𝛾𝑛
̃ = [𝐿 ̃ 𝑛 , 𝐿] ̃ + 𝛼𝑛 [̃ ̃ 𝑞𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐿] − 𝑞−1 𝑞𝑖 𝜕−1 (𝑟𝑖 𝑞) , 𝐿] ≥1 ̃ . ̃ 𝑛 + 𝛼𝑛 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐿] = [𝐿 ≥1
= −𝑞−1 (𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝑞) 𝑞−1 𝑓 + 𝑞−1 (𝐿𝑛≥0 + 𝛼𝑛 𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝑓) ̃ 𝑛 𝑞−1 (𝑞) 𝑞−1 𝑓 + 𝑞−1 𝑞𝐿 ̃ 𝑛 𝑞−1 (𝑓) = −𝑞−1 𝑞𝐿 ≥0 ≥0
In the same way, (39b) can be proved. The two equations in (39c) can be proved similarly. So we only prove the first one. Consider
̃ 𝑘 (̃ = −𝛼𝑛 𝑞−2 𝑞𝜎𝑘 𝑞𝑖 + 𝛼𝑛 𝑞−1 𝑞𝑖,𝜎𝑘 − 𝛼𝑛 𝐿 ≥1 𝑞𝑖 ) = −𝛼𝑛 𝑞−2 (𝐿𝑘≥0 + 𝛽𝑘 𝑞𝑖 𝜕−1 𝑟𝑖 ) (𝑞) 𝑞𝑖 + 𝑞 [𝛽𝑘 (𝑞𝑖,𝛾𝑛 − ̃ 𝑘 (̃ − 𝛼𝑛 𝐿 ≥1 𝑞𝑖 )
(51)
− 𝛼𝑛 𝑞−1 𝑞𝑖 𝜕−1 𝑟𝑖 (𝑞) 𝑞−1 𝑓 + 𝛼𝑛 𝑞−1 𝑞𝑖 𝜕−1 𝑟𝑖 (𝑓) ̃ 𝑛 (1) 𝑞̃ + 𝐿 ̃ 𝑛 (̃ ̃𝑞̃𝑖 𝑟̃𝑖 + 𝛼𝑛 𝑞̃𝑖 𝜕−1 (𝑟𝑖 𝑓) = −𝐿 ≥0 ≥0 𝑞) + 𝛼𝑛 𝑞 ̃ 𝑛 + 𝛼𝑛 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕) (̃ = (𝐿 𝑞) . ≥1
̃ 𝑘 (̃ 𝑞𝑖,𝜎𝑘 − 𝐿 𝛼𝑛 (̃ ≥1 𝑞𝑖 ))
−1
(49)
𝐿𝑛≥0
(𝑞𝑖 )) +
𝛼𝑛 𝐿𝑘≥0
(𝑞𝑖 )]
Similarly, we can prove that 𝑞̃ = 𝑞−1 𝑓 satisfies the second equation in (40). 𝑛 ̃ 𝑛 is (b) Since the first order coefficient of 𝐿 = 𝐿 𝑛
given by [𝐿≥1 𝑥], we have 𝑛 ̃𝑛 − 𝐿 ̃ 𝑛 (̃ 𝐿≥2 = 𝐿 ≥1 ≥1 𝑞) 𝜕𝑥 .
≥1
≥1
(52)
8
Advances in Mathematical Physics ̃ 𝛾 = [𝜕𝛾 , 𝐿] ̃ = [𝜕𝛾 + 𝑞̃𝛾 𝜕𝑥 , 𝐿] = 𝐿𝛾 + [̃ ̃ 𝐿 𝑞𝛾𝑛 𝑞̃𝑥−1 𝜕𝑥 , 𝐿] 𝑛 𝑛 𝑛 𝑛 𝑛 yields ̃ 𝛾 − [̃ ̃ 𝐿𝛾𝑛 = 𝐿 𝑞𝛾𝑛 𝑞̃𝑥−1 𝜕𝑥 , 𝐿] 𝑛 ̃ ̃ 𝑛 + 𝛼𝑛 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕, 𝐿] = [𝐿 ≥1
5. Solutions for (𝛾𝑛 , 𝜎𝑘 )-mKPH and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH
̃ ̃ 𝑛 + 𝛼𝑛 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕) (̃ 𝑞) 𝑞̃𝑥−1 𝜕𝑥 , 𝐿] − [(𝐿 ≥1 ̃ 𝑛 (̃ ̃𝑛 − 𝐿 = [𝐿 ≥1 ≥1 𝑞) 𝜕𝑥 , 𝐿] ̃ + [𝛼𝑛 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕 − 𝛼𝑛 𝑞̃𝑖 𝜕−1 (̃𝑟𝑖 𝑞̃𝑥 ) 𝜕𝑥 , 𝐿] =
summarized in a diagram. In our generalized system, we can find that similar results remain correct after we add some constraints on the hierarchies. These provide us with a convenient way to obtain the solutions of (𝛾𝑛 , 𝜎𝑘 )-mKPH and (𝛾𝑛 , 𝜎𝑘 )-HDH from the solutions of (𝛾𝑛 , 𝜎𝑘 )-KPH.
(53)
We first briefly recall the generalized dressing method for the (𝛾𝑛 , 𝜎𝑘 )-KPH proposed in [15]. Let 𝑓𝑖 , 𝑔𝑖 satisfy 𝑓𝑖,𝛾𝑛 = 𝜕𝑛 (𝑓𝑖 ) ,
𝑛 [𝐿≥2 , 𝐿]
+ [𝛼𝑛 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕 − 𝛼𝑛 𝑞̃𝑖 (̃𝑟𝑖 𝑞̃ − 𝜕−1 (̃𝑟𝑖𝑥 𝑞̃)) 𝜕𝑥 , 𝐿] =
𝑛 [𝐿≥2 , 𝐿]
=
𝑛 [𝐿≥2 , 𝐿]
𝑔𝑖,𝛾𝑛 = 𝜕𝑛 (𝑔𝑖 ) ,
+ [−𝛼𝑛 𝑞𝑖 𝜕𝑥−1 𝑟𝑖𝑥 𝜕𝑥 + 𝛼𝑛 𝑞𝑖 𝑟𝑖 𝜕𝑥 , 𝐿] +
𝑔𝑖,𝜎𝑘 = 𝜕𝑘 (𝑔𝑖 ) ,
[𝛼𝑛 𝑞𝑖 𝜕𝑥−1 𝑟𝑖 𝜕𝑥2 , 𝐿] ,
and let ℎ𝑖 be the linear combination of 𝑓𝑖 and 𝑔𝑖 : ℎ𝑖 = 𝑓𝑖 + 𝐹𝑖 (𝛼𝑛 𝛾𝑛 + 𝛽𝑘 𝜎𝑘 ) 𝑔𝑖
𝑘
𝛼𝑛 (𝑞𝑖,𝜎𝑘 − 𝐿≥2 (𝑞𝑖 )) = 𝛼𝑛 𝑞̃𝑖,𝜎𝑘 − 𝛼𝑛 𝑞̃𝜎𝑘 𝑞𝑖,𝑥 −
𝑘 𝛼𝑛 𝐿≥2
(𝑞𝑖 )
𝑘 ̃ 𝑘 + 𝛽𝑘 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕𝑥 ) (̃ − 𝛼𝑛 (𝐿 𝑞) 𝑞𝑖𝑥 − 𝛼𝑛 𝐿≥2 (𝑞𝑖 ) ≥1
=
(̃ 𝑞𝑖 ) −
𝑖 = 1, . . . , 𝑁,
(58)
with 𝐹𝑖 (𝑋) being a differentiable function of 𝑋, 𝑋 = 𝛼𝑛 𝛾𝑛 + 𝛽𝑘 𝜎𝑘 . The dressing operator 𝑊 is defined by
̃ 𝑘 𝑞𝑖 ) ̃ 𝑛 (̃ = 𝛽𝑘 (̃ 𝑞𝑖,𝛾𝑛 − 𝐿 ≥1 𝑞𝑖 )) + 𝛼𝑛 𝐿 ≥1 (̃
̃𝑘 𝛼𝑛 𝐿 ≥1
(57b) 𝑖 = 1, . . . , 𝑁,
which implies that (11a) holds. In the same way, we can prove (11b). Consider
̃𝑘 𝛼𝑛 𝐿 ≥1
(57a)
𝑓𝑖,𝜎𝑘 = 𝜕𝑘 (𝑓𝑖 ) ,
𝑊= (54)
(̃ 𝑞) 𝑞𝑖𝑥
̃ 𝑛 (̃ − 𝛼𝑛 𝛽𝑘 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕𝑥 (̃ 𝑞) 𝑞𝑖𝑥 + 𝛽𝑘 (̃ 𝑞𝑖,𝛾𝑛 − 𝐿 ≥1 𝑞𝑖 )) 𝑘
− 𝛼𝑛 𝐿≥2 (𝑞𝑖 ) ̃ 𝑛 (̃ = −𝛼𝑛 𝛽𝑘 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕𝑥 (̃ 𝑞) 𝑞𝑖𝑥 + 𝛽𝑘 (̃ 𝑞𝑖,𝛾𝑛 − 𝐿 ≥1 𝑞𝑖 )) . Similarly 𝑛
𝛽𝑘 (𝑞𝑖,𝛾𝑛 − 𝐿≥2 (𝑞𝑖 )) = −𝛼𝑛 𝛽𝑘 𝑞̃𝑖 𝜕−1 𝑟̃𝑖 𝜕𝑥 (̃ 𝑞) 𝑞𝑖𝑥 ̃ 𝑘 (̃ + 𝛼𝑛 (̃ 𝑞𝑖,𝜎𝑘 − 𝐿 ≥1 𝑞𝑖 )) .
(55)
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝜕) , 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁)
(59)
where ℎ ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 1 ℎ ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 1 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁) = . .. .. .. , . . . . . (𝑁−1) (𝑁−1) (𝑁−1) ℎ1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 ℎ 1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 1 ℎ 1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 𝜕 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝜕) = . .. .. .. .. . . . . . . (𝑁) (𝑁) (𝑁) ℎ1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 𝜕𝑁
(60)
is the Wronskian determinant. Define So 𝛼𝑛 (𝑞𝑖,𝜎𝑘 −
𝑘 𝐿≥2
(𝑞𝑖 )) − 𝛽𝑘 (𝑞𝑖,𝛾𝑛 −
𝑛 𝐿≥2
(𝑞𝑖 ))
̃ 𝑛 (̃ ̃ 𝑘 (̃ = 𝛽𝑘 (̃ 𝑞𝑖,𝛾𝑛 − 𝐿 𝑞𝑖,𝜎𝑘 − 𝐿 ≥1 𝑞𝑖 )) − 𝛼𝑛 (̃ ≥1 𝑞𝑖 )) = 0.
𝑞𝑖 = −𝐹𝑖̇ 𝑊 (𝑔𝑖 ) , (56)
In this way, we present the connection between the solutions of (𝛾𝑛 , 𝜎𝑘 )-KPH, (𝛾𝑛 , 𝜎𝑘 )-mKPH, and (𝛾𝑛 , 𝜎𝑘 )-HDH under the gauge transformations and reciprocal transformations. In [17], the original three hierarchies are intimately
𝑟𝑖 = (−1)𝑁−𝑖
𝑊𝑟 (ℎ1 , . . . , ̂ℎ𝑖 , . . . , ℎ𝑁) 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁)
,
(61) 𝑖 = 1, . . . , 𝑁,
where the hat ̂ means ruling out this term from the Wronskian determinant, 𝐹𝑖̇ = 𝑑𝐹𝑖 /𝑑𝑋. Then we have the following.
Advances in Mathematical Physics
9
Theorem 8 (see [15]). Let 𝑊 be defined by (59) and (58), let 𝐿 = 𝑊𝜕𝑊−1 , and let 𝑞𝑖 and 𝑟𝑖 be given by (61); then 𝑊, 𝐿, 𝑞𝑖 , 𝑟𝑖 satisfy (𝛾𝑛 , 𝜎𝑘 )-KPH ((1a), (1b), and (1c) and (3a) and (3b)). Choose 𝑞 = 𝑊 (1) = (−1)𝑁
𝑊𝑟 (ℎ1 , ℎ2 , . . . , ℎ𝑁 )
𝑊𝑟 (ℎ1 , ℎ2 , . . . , ℎ𝑁)
(62)
as the particular eigenfunction for (4); based on Theorem 7, we have the Wronskian solution for (𝛾𝑛 , 𝜎𝑘 )-mKPH: ̃= 𝐿
−1
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝜕) 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝜕) 𝜕[ ] , ) ) 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁
𝑞̃𝑖 = −𝐹𝑖̇ 𝑟̃𝑖 =
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝑔𝑖 ) , ) 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁
𝑊𝑟 (ℎ1 , . . . , ̂ℎ𝑖 , . . . , ℎ𝑁 )
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁)
(63a)
(63b)
.
(63c)
ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 1 ℎ ℎ ⋅⋅⋅ ℎ 𝜕𝑥 𝑞̃𝑥𝑁(𝑁+1)/2 ..1 ..2 .. ..𝑁 .. . . . . . ℎ(𝑁) ℎ(𝑁) ⋅⋅⋅ ℎ(𝑁) 𝜕𝑁 1 2 𝑁 𝑥 = ℎ ℎ ⋅⋅⋅ ℎ 𝑁 1 2 𝑁(𝑁+1)/2 ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 𝑞̃𝑥 . . . . .. .. .. .. ℎ(𝑁) ℎ(𝑁) ⋅⋅⋅ ℎ(𝑁) 1 2 𝑁 ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 1 −1 [ ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 𝜕𝑥 ] [ . . . . . ] [ .. .. .. .. .. ] ] [ (𝑁) (𝑁) 𝑁 ] [ ℎ1 ℎ2 ⋅⋅⋅ ℎ(𝑁) 𝑁 𝜕𝑥 ] [ ⋅ 𝑞̃𝑥 𝜕𝑥 [ ] [ ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 ] [ ℎ ℎ ⋅⋅⋅ ℎ ] ] [ 1 2 𝑁 [ .. .. .. .. ] . . . . (𝑁) (𝑁) (𝑁) [ ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 ] =
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝜕𝑥 )
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁)
Choose
−1
[ 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝜕𝑥 ) ] ⋅ 𝑞̃𝑥 𝜕𝑥 [ ] , 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁) [ ]
𝜂𝑁
𝑓 = 𝑊 (𝑒 ) ℎ 1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 1 ℎ ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 𝜇𝑁 𝜂𝑁 1 𝑒 = .. .. .. .. 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁) ... . . . . (𝑁) (𝑁) (𝑁) 𝑁 ℎ ℎ ⋅ ⋅ ⋅ ℎ 𝜇 1 𝑁 2 𝑁
where the elements ℎ𝑖 = ℎ𝑖 (𝑥) and ℎ𝑖 denotes 𝜕ℎ𝑖 /𝜕𝑥. Similarly, we have
as another eigenfunction for (4). Based on Theorem 7, we have ℎ 1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 1 ℎ ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 𝜇𝑁 𝑁 𝜂𝑁 1 (−1) 𝑒 𝑞̃ = .. .. .. .. . ) .. 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁 . . . . . (𝑁) (𝑁) (𝑁) 𝑁 ℎ 1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 𝜇𝑁
(67)
(64)
(65)
𝑞𝑖 = −𝐹𝑖̇
=
−1 ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 1 [ 𝑞̃𝑥 𝜕𝑥 ℎ1 𝑞̃𝑥 𝜕𝑥 ℎ2 ⋅⋅⋅ 𝑞̃𝑥 𝜕𝑥 ℎ𝑁 𝑞̃𝑥 𝜕𝑥 ] [ . .. ] .. .. .. [ .. ] [ 𝑞̃𝑁 𝜕𝑁 ℎ 𝑞̃𝑁 𝜕.𝑁 ℎ ⋅⋅⋅. 𝑞̃𝑁 𝜕.𝑁 ℎ 𝑞̃𝑁.𝜕𝑁 ] [ 𝑥 𝑥 1 𝑥 𝑥 2 𝑥 𝑥 𝑁 𝑥 𝑥 ] ] [ [ 𝑞̃𝑥 𝜕𝑥 ℎ1 𝑞̃𝑥 𝜕𝑥 ℎ2 ⋅⋅⋅ 𝑞̃𝑥 𝜕𝑥 ℎ𝑁 ] [ 𝑞̃2 𝜕2 ℎ 𝑞̃2 𝜕2 ℎ ⋅⋅⋅ 𝑞̃2 𝜕2 ℎ ] [ 𝑥 𝑥 1 𝑥 𝑥 2 𝑥 𝑥 𝑁 ] [ .. . . . ] 𝑁 .𝑁 𝑁 ..𝑁 .. 𝑁 ..𝑁 [ 𝑞̃𝑥 𝜕𝑥 ℎ1 𝑞̃𝑥 𝜕𝑥 ℎ2 ⋅⋅⋅ 𝑞̃𝑥 𝜕𝑥 ℎ𝑁 ]
(66)
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁)
,
(68a)
𝑟𝑖 = −𝜕𝑥−1 (̃ 𝑞𝑥 𝑟̃𝑖 ) = −𝜕𝑥−1 (̃𝑟𝑖 (𝑥))
Using 𝑥 = 𝑞̃, 𝜕𝑥 = 𝑞̃𝑥 𝜕𝑥 , based on Theorem 7, we have ℎ1 ℎ2 ⋅⋅⋅ ℎ𝑁 1 𝑞̃𝑥 𝜕𝑥 ℎ1 𝑞̃𝑥 𝜕𝑥 ℎ2 ⋅⋅⋅ 𝑞̃𝑥 𝜕𝑥 ℎ𝑁 𝑞̃𝑥 𝜕𝑥 . . . . .. .. . 𝑁 𝑁 𝑁 ..𝑁 .. 𝑁 ..𝑁 𝑁 𝑁 ̃ (𝑥) = 𝑞̃𝑥 𝜕𝑥 ℎ1 𝑞̃𝑥 𝜕𝑥 ℎ2 ⋅⋅⋅ 𝑞̃𝑥 𝜕𝑥 ℎ𝑁 𝑞̃𝑥 𝜕𝑥 𝑞̃𝑥 𝜕𝑥 𝐿=𝐿 𝑞̃𝑥 𝜕𝑥 ℎ1 𝑞̃𝑥 𝜕𝑥 ℎ2 ⋅⋅⋅ 𝑞̃𝑥 𝜕𝑥 ℎ𝑁 2 2 𝑞̃𝑥 𝜕𝑥 ℎ1 𝑞̃𝑥2 𝜕𝑥2 ℎ2 ⋅⋅⋅ 𝑞̃𝑥2 𝜕𝑥2 ℎ𝑁 . .. .. .. .. . . . 𝑞̃𝑁 𝜕𝑁 ℎ 𝑞̃𝑁 𝜕𝑁 ℎ ⋅⋅⋅ 𝑞̃𝑁 𝜕𝑁 ℎ 𝑥 𝑥 1 𝑥 𝑥 2 𝑥 𝑥 𝑁
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁, 𝑔𝑖 )
−𝜕𝑥−1
̂ 𝑊𝑟 (ℎ1 , . . . , ℎ𝑖 , . . . , ℎ𝑁)
𝑊𝑟 (ℎ1 , . . . , ℎ𝑁)
(68b) ,
where 𝑔𝑖 = 𝑔𝑖 (𝑥). Equations (67) and (68a) and (68b) give the Wronskian solution for (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH. Let us illustrate it by solving (24a), (24b), and (24c) and (44a) and (44b). We take the solution of (57a) and (57b) as follows: 𝑓𝑖 fl exp (𝜆 𝑖 𝑥 + 𝜆2𝑖 𝑦 + 𝜆3𝑖 𝑡) = 𝑒𝜉𝑖 , 𝑔𝑖 fl exp (𝜇𝑖 𝑥 + 𝜇𝑖2 𝑦 + 𝜇𝑖3 𝑡) = 𝑒𝜂𝑖 ,
(69)
ℎ𝑖 fl 𝑓𝑖 + 𝐹𝑖 (𝛼2 𝑦 + 𝛽3 𝑡) 𝑔𝑖 = 2√𝐹𝑖 exp ( Ω𝑖 =
𝜉𝑖 + 𝜂𝑖 ) cosh (Ω𝑖 ) , 2
1 (𝜉 − 𝜂𝑖 − ln 𝐹𝑖 ) . 2 𝑖
(70)
10
Advances in Mathematical Physics
Example 9 (solutions for the mixed type of mKPESCS ((44a) ̃ = 𝑞−1 𝐿𝑞, then it is easy to find and (44b))). Since 𝐿 𝑞𝑥 . 𝑞
(71)
𝜆 1 + 𝜇1 𝜇1 − 𝜆 1 + tanh (Ω1 ) . 2 2
(72)
V0 =
6. Conclusion
For 𝑁 = 1, we have 𝑞=−
The one-soliton solution for (44a) and (44b) with 𝑁 = 1 is as follows: V = V0 = 𝑞̃1 =
(𝜆 1 − 𝜇1 ) [tanh (Ω1 + 𝜃1 ) − tanh (Ω1 )] , 2
𝐹1̇ (𝜇1 − 𝜆 1 ) 𝑒𝜉1 +𝜂1 sech (Ω1 + 𝜃1 ) , 2√𝐹1 𝜆 1 𝜇1
𝑟̃1 = −
(73)
1 −(𝜉1 +𝜂1 ) 𝑒 sech Ω1 , 2√𝐹1
where 𝜃1 = ln √𝜆 1 /𝜇1 . Example 10 (solutions for the mixed type of HDESCS ((24a), (24b), and (24c))). Based on Theorem 7, we have that ℎ 1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 1 ℎ ℎ ⋅ ⋅ ⋅ ℎ 𝜇 𝜂𝑁 𝑁 𝑁 1 2 𝑁 (−1) 𝑒 (74) 𝑞̃ = . . . . . .. .. .. .. 𝑊𝑟 (ℎ1 , . . . , ℎ𝑁) .. (𝑁) (𝑁) (𝑁) 𝑁 ℎ1 ℎ2 ⋅ ⋅ ⋅ ℎ𝑁 𝜇𝑁 is the eigenfunction of the mixed type of mKPESCS and we ̃ that is, have 𝐿 = 𝐿; 𝑤𝜕𝑥 + 𝑤0 + 𝑤1 𝜕𝑥−1 + ⋅ ⋅ ⋅ = 𝜕𝑥 + V0 + V1 𝜕𝑥−1 + ⋅ ⋅ ⋅ = 𝑞̃𝑥 𝜕𝑥 + V0 + ⋅ ⋅ ⋅ .
(75)
From the above equation, we obtain 𝑤 = 𝑞̃𝑥 , 𝑤0 = V0 .
𝜆 1 − 𝜇1 𝜂1 +Ω1 𝑒 sech (Ω1 + 𝜃1 ) , 2√𝜆 1 𝜇1
(76)
(77a)
(𝜆 1 − 𝜇1 ) 𝜂1 +Ω1 sech2 (Ω1 + 𝜃1 ) cosh (Ω1 ) , 𝑒 2
(77b)
𝑤0 =
(𝜆 1 − 𝜇1 ) [tanh (Ω1 + 𝜃1 ) − tanh (Ω1 )] , 2
(77c)
𝑞1 =
𝐹1̇ (𝜇1 − 𝜆 1 ) 𝑒𝜉1 +𝜂1 sech (Ω1 + 𝜃1 ) , 2√𝐹1 𝜆 1 𝜇1
(77d)
𝑟1 =
1 −1 −(𝜉1 +𝜂1 ) 𝜕 𝑒 sech (Ω1 ) , 2√𝐹1 𝑥
(77e)
𝑤=
In this paper, a new (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH is proposed by introducing new time series 𝛾𝑛 and 𝜎𝑘 and adding eigenfunctions as components. The (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH includes (2+1)-HD hierarchy and extended HD hierarchy and contains first type and second type as well as mixed type of (2+1)-HD equation with self-consistent sources as special cases. The reduction of the (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH is studied and the 𝑘-constrained (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH is given. In the same way as (2+1)(𝛾𝑛 , 𝜎𝑘 )-HDH, the (𝛾𝑛 , 𝜎𝑘 )-mKPH is formulated and its zerocurvature equation and Lax representation are presented. The gauge transformations and reciprocal transformations between (𝛾𝑛 , 𝜎𝑘 )-KPH, (𝛾𝑛 , 𝜎𝑘 )-mKPH, and (2+1)(𝛾𝑛 , 𝜎𝑘 )-HDH are given. They help us to obtain solutions of (𝛾𝑛 , 𝜎𝑘 )-mKPH and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH. By making use of the transformations, we find the soliton solutions of the (𝛾𝑛 , 𝜎𝑘 )-mKPH and (2+1)-(𝛾𝑛 , 𝜎𝑘 )-HDH. Particularly, the soliton solutions for the mixed type of mKPESCS and HDESCS are obtained.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments This work is supported by National Natural Science Foundation of China (11171175, 11201477, and 11301179) and the Fundamental Research Funds for the Central Universities (2014ZZD10).
References
The implicit solutions for the mixed type of HDESCS ((24a), (24b), and (24c)) with 𝑁 = 1 are 𝑥 = 𝑞̃ =
where 𝜃1 = ln √𝜆 1 /𝜇1 , 𝜉1 = 𝜆 1 𝑥+𝜆21 𝑦+𝜆31 𝑡, 𝜂1 = 𝜇1 𝑥+𝜇12 𝑦+ 𝜇13 𝑡, Ω1 = (1/2)(𝜉1 − 𝜂1 − ln 𝐹1 ), and the relation between 𝑥 and 𝑥 is given by (77a).
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