Research Article. Yehui Huang, 1 Yuqin Yao, 2 and Yunbo Zeng Introduction

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1 Advances in Mathematical Physics Volume 015 Article ID pages Research Article Links between (γ n σ k )-KP Hierarchy (γ n σ k )-mkp Hierarchy and (1)-(γ n σ k )-Harry Dym Hierarchy Yehui Huang 1 Yuqin Yao and Yunbo Zeng 3 1 School of Mathematics and Physics orth China Electric Power University Beijing 1006 China Department of Applied Mathematics China Agricultural University Beijing China 3 Department of Mathematical Science Tsinghua University Beijing China Correspondence should be addressed to Yuqin Yao; yyqinw@16com Received 1 July 015; Accepted 16 ovember 015 Academic Editor: Boris G Konopelchenko Copyright 015 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 (1)-(γ n σ k )-Harry Dym hierarchy and (γ n σ k )-mkp hierarchy with two new time series γ n and σ k which consist of γ n -flow σ k -flow and mied γ n and σ k evolution equations of eigenfunctions are proposed Gauge transformations and reciprocal transformations between (γ n σ k )-KP hierarchy (γ n σ k )-mkp hierarchy and (1)-(γ n σ k )-Harry Dym hierarchy are studied Their soliton solutions are presented 1 Introduction Generalizations of soliton hierarchies are important topics since last century In 008 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 [ The Kadomtsev-Petviashvili (KP) hierarchy is an important 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 etended KP hierarchy (ekph) which is constructed by introducing another time series {τ k } [1 14 The ekph consists of t n -flow of KP hierarchy τ k -flow and t n -evolution equations of eigenfunctions To make difference we may call the ekph as (t n τ k )-KPH Recently we generalize the (t n τ k )-KPH to (γ n σ k )-KPH by introducing two new time series γ n and σ k with two parameters α n and β k [15: L γn L σk =[B n α n 1 r i L B n =L n 0 (1a) =[B k β k 1 r i L α n (σk B k ( )) β k (γn B n ( )) = 0 α n (r iσk B k (r i)) β k (r iγn B n (r i)) = 0 (1b) (1c) where the pseudodifferential operator L with potential functions u i is defined as L= u 1 1 u ()

2 Advances in Mathematical Physics and denotes the adjoint operator The compatibility of γ n - flow (1a) and σ k -flow (1b) under (1c) gives rise to the zerocurvature representation for (1a) (1b) and (1c) B nσk B kγn [B n B k β k [B n α n [ 1 r i B k 1 r i α n (σk B k ( )) β k (γn B n ( )) = 0 α n (r iσk B k (r i)) β k (r iγn B n (r i)) = 0 with the La representation ψ γn ψ σk =(B n α n 1 r i )(ψ) =(B k β k 1 r i )(ψ) (3a) (3b) The (γ n σ k )-KPH consists of γ n -flow σ k -flow and one mied γ n and σ k evolution equation of eigenfunctions The (γ n σ k )- KPH can be reduced to the KPH and (t n τ k )-KPH and contains first type and second type as well as mied type of KPESCS as special cases We also develop the dressing method to solve the (γ n σ k )-KPH [15 The (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 (1)-(γ n σ k )-Harry Dym hierarchy ((1)-(γ n σ k )-HDH) which consists of γ n -flow σ k -flow and one mied γ n and σ k evolution equation of eigenfunctions The (1)-(γ n σ k )-HDH can be reduced to the (1)-Harry Dym hierarchy ((1)-HDH) and (t n τ k )-HDH [18 and contains first type and second type as well as mied type of Harry Dym equation with self-consistent sources (HDESCS) as special cases Then the (γ n σ k )-mkph is proposed The generalized gauge transformations and generalized reciprocal links between these three kinds of (γ n σ k ) hierarchies are studied Further based on these transformations and the solutions of the (γ n σ k )-KPH [15 the solutions of (γ n σ k )- mkph and (1)-(γ n σ k )-HDH are obtained respectively The paper is organized as follows In Section we propose a new (γ n σ k )-HDH and present its reduction A new (γ n σ k )-mkph is proposed in Section 3 Section 4 is devoted to studying the links between the three kinds of (γ n σ k ) hierarchies Section 5 presents the soliton solutions and a conclusion is given in the last section (4) A ew (1)-(γ n σ k )-Harry Dym Hierarchy 1 A ew (1)-(γ n σ k )-Harry Dym Hierarchy It is well known that the pseudodifferential operator L for (1)-HDH with potential functions w i is defined as L=w w 0 w 1 1 w (5) The (1)-HDH is given by [17 L tn =[B n L n 1 (6) where B n = L n =(Ln ) n 1 The compatibility of the t n -flow and t k -flow of (6) leads to the zero-curvature representation of (1)-HDH: B ntk B ktn [B n B k =0 (7) In particular when taking B = w and B 3 = w 3 3 3w (w w 0 ) and setting t = y t 3 = tandw 0 = (1/) 1 (w y /w )(1/)w (7) yields the (1)-HD equation 4w t =w 3 w 3 1 w [w 1 ( w y w ) (8) y which is reduced to the Harry Dym equation Consider 4w t =w 3 w (9) by dropping the y-dependence Basedonthesquaredeigenfunctionsymmetryconstraint L k = B k tn = B n ( ) 1 r i r itn = B n (r i ) (10) which is compatible with (1)-HDH [17 we propose the following generalized (1)-HDH with two generalized time series γ n and σ k : L γn L σk =[B n α n 1 r i L =[B k β k 1 r i L α n (σk B k ( )) β k (γn B n ( )) = 0 α n (r iσk B k (r i )) β k (r iγn B n (r i )) (11a) (11b) (11c) We will prove the compatibility of (11a) and (11b) under (11c) in the following theorem Theorem 1 The γ n -flow (11a) and σ k -flow (11b) under (11c) are compatible

3 Advances in Mathematical Physics 3 Proof Denote Then (15) (16) and (18) under (11c) yield B n = B n α n 1 r i B k = B k β k 1 r i (1) B nσk B kγn [ B n B k =[α n (q σk B k (q)) β k (q γn B n (q)) 1 r q 1 [α n (r σk B k (r)) (19) In order to prove L γn σ k = L σk γ n thatis β k (r γn B n (r)) =0 [ B nσk B kγn [ B n B k L = 0 (13) we only need to prove B nσk B kγn [ B n B k =0 (14) For convenience we omit Wecanfindthat The compatibility of γ n -flow (11a) and σ k -flow (11b) under (11c) gives rise to the zero-curvature representation for (11a) (11b) and (11c): (B n α n 1 r i (B k β k )σ k 1 r i ) γ n B nσk and similarly = B nσk α n (q 1 r ) σk =[B k β k (q 1 r )L n α n (q 1 r ) σk =[B k L n β k [q 1 r L n α n q σk 1 r α n q 1 r σk B kγn =[B n L k α n [q 1 r L k β k q γn 1 r β k q 1 r γn Moreover on the other hand under the formula [18 [B n 1 r i = B n ( ) 1 r i we have [ B n B k =[B n B k [B n β k q 1 r [α n q 1 r B k 3 B n (r i) 4 (15) (16) (17) =0 [B n α n 1 r i B k β k 1 r i (0) which under (11c) can be simplified as follows Then we have the following Theorem The commutativity of (11a) and (11b) under (11c) gives rise to the zero-curvature equation for the generalized (1)-HDH with two generalized time series: B nσk B kγn [B n B k β k [B n α n [ 1 r i B k 1 r i α n (σk B k ( )) β k (γn B n ( )) = 0 α n (r iσk B k (r i )) β k (r iγn B n (r i )) with the La representation (1a) (1b) =[L n (L n ) L k (L k ) β k [B n q 1 r α n [q 1 r B k β k [B n q 1 r α n [q 1 r B k (18) ψ γn ψ σk =(B n α n 1 r i )(ψ) =(B k β k 1 r i )(ψ) () =[B n L k [L n B k [(L n ) (L k ) β k [B n q 1 r α n [q 1 r B k β k B n (q) 1 r β k q 3 B n (r) 4 α n B k (q) 1 r α n q 3 B k (r) 4 We briefly call (11a) (11b) and (11c) and (1a) and (1b) as (1)-(γ n σ k )-HDH It is easy to see that (1)-(γ n σ k )-HDH ((11a) (11b) and (11c) and (1a) and (1b)) for α n =β k =0 reduces to (1)-HDH ((6) and (7)) and (1)-(γ n σ k )-HDH for α n β k =1reduces to (t n τ k )-HDH [18 So (1)- (γ n σ k )-HDH ((11a) (11b) and (11c) and (1a) and (1b)) presents a more generalized (1)-HDH which contains the (1)-HDH and (1)-(t n τ k )-HDH as the special cases

4 4 Advances in Mathematical Physics Eample 3 Let us take n=and k=3andsetγ =y σ 3 =t Then(1a)and(1b)become B t B 3y [B B 3 β 3 [B α [ 1 r i B 3 1 r i α (t B 3 ( )) β 3 (y B ( )) = 0 α (r it B 3 (r i )) β 3 (r iy B (r i )) which gives the following nonlinear equation: (3a) (3b) Particularly when taking α =β 3 =0; α β 3 =1; α =1 β 3 =0;andα =1 β 3 =1 respectively (4a) (4b) and(4c)and(6)reducetothe(1)-hdequationthe first type of (1)-HD equation with self-consistent sources ((1)-HDESCS) the second type of (1)-HDESCS and the mied type of (1)-HDESCS and their La representations respectively Reduction Consider the constraint given by Then (11b) yields L k = B k β k 1 r i (7) ww t 3[w (w w 0 ) y w 3 (ww 6w w 3ww 0 1w w 0 ) β 3 w ( r i ) β 3 ww r i 3α w 3 ( r i r i ) 6α w (w w 0 ) ( r i ) 3α [w (w w 0 ) r i w y w (w w 0 )α w α w r i ( r i ) α [t w 3 3w (w w 0 ) β 3 (y w ) α (r it w 3 r i 3w w 0 r i ) β 3 (r iy w r i ) with the La representation as follows: (4a) (4b) (4c) ( (L k ) =[B k β k q σk i 1 r i L k (8) B kσk =(L k σ k ) 1 r i )σ k =(L k σ k ) (9) which imply that L B k andr i under (7) are independent of σ k Subsequentlyσk and r iσk in (11c) should be replaced by λ i and λ i r i as in the case of constrained flow of KP [19 0; namely (11c) under the constraint (7) should be replaced by α n (λ i B k ( )) β k (γn B n ( )) = 0 (30) α n (λ i r i B k (r i )) β k (r iγn B n (r i )) =0 (31) We will show that constraint (7) is invariant under γ n -flow (11a) and (31) In fact making use of (11a) (17) and (31) we have (L k B k ) γn =(L k γ n ) =[ B n L k (β k 1 r i )γ n =β k (γn 1 r i ψ y = (w α 1 r i )(ψ) (5) ψ t =(w 3 3 3w (w w 0 ) β 3 1 r i ) (ψ) (6) 1 r iγn )= [β k B n ( ) 1 r i α n (λ i B k ( )) 1 r i β k 1 B n (r i ) α n 1 (λ i r i B k (r i )) =[B n

5 Advances in Mathematical Physics 5 β k β k 1 r i 1 r i [B k α n 1 r i [B k α n 1 r i =[ B n =[ B n L k [ B n B k [B k B n [B k B n =[ B n w y w (w w 0 )α w α w r i ( r i ) α [λ i w 3 3w (w w 0 ) (35b) Then L k (3) (L k B k β k 1 r i =0 (33) )γ n This means that the submanifold determined by k-constraint (7) is invariant under the γ n -flow (11a) and (31) Therefore the constrained flow of (1)-(γ n σ k )-HDH ((11a) (11b) and (11c) and (1a) and (1b)) under (7) reads B kγn [B k B n β k [ α n [B k 1 r i 1 r i B n α n (λ i B k ( )) β k (γn B n ( )) = 0 α n (λ i r i B k (r i )) with β k (r iγn B n (r i )) B n =(B k β k 1 r i ) n/k (34a) (34b) (34c) Eample 4 When n= k=3 γ =y (34a) (34b) and (34c) give 3[w (w w 0 ) y w 3 (ww 6w w 3ww 0 1w w 0 ) β 3 w ( r i ) β 3 ww r i 3α w 3 ( r i r i ) 6α w (w w 0 ) ( r i ) 3α [w (w w 0 ) r i (35a) β 3 (y w ) α (λ i r i w 3 r i 3w w 0 r i ) β 3 (r iy w r i ) (35c) which is the k-constraint (1)-(γ n σ k )-Harry Dym equation 3 The (γ n σ k )-mkph In the same way the (γ n σ k )-mkph can be formulated The L operator of mkp hierarchy is defined by L = V 0 V 1 1 V (36) The La equation of mkp hierarchy is given by L tn =[ B n L B n = L n 1 n 1 (37) The commutativity of tn and tk flows gives the zerocurvature equation (7) Since the squared eigenfunction symmetry constraint given by L k = B k tn = B n ( ) r itn 1 r i = 1 B n ( r i) (38) is compatible with mkp hierarchy [1 we have the following Definition 5 The (γ n σ k )-mkph is defined by L γn L σk = [ B n α n 1 r i L = [ B k β k 1 r i L α n ( σk B k ( )) β k ( γn B n ( )) = 0 α n ( r iσk 1 B k ( r i)) β k ( r iγn 1 B n ( r i)) (39a) (39b) (39c)

6 6 Advances in Mathematical Physics (39a) (39b) and (39c) have the La representation which gives the following nonlinear equation: ψ γn ψ σk =( B n α n 1 r i ) ( ψ) =( B k β k 1 r i ) (ψ) (40) 4V t V 6V V 3 1 V yy 6V 1 V y 4β 3 ( r i ) α [3 ( r i r i ) 3( r i ) y 6(V r i ) (44a) In the same way as in Section we can verify the compatibility of (39a) and (39b) under (39c) By using the formula [ B n 1 r i = B n ( ) 1 r i B n ( r i) (41) the zero-curvature equation for (γ n σ k )-mkph can be written as B nσk B kγn [ B n B k β k [ B n α n [ 1 r i B k 1 r i α n ( σk B k ( )) β k ( γn B n ( )) = 0 α n ( r iσk 1 B k ( r i)) β k ( r iγn 1 B n ( r i)) (4a) (4b) It is easy to see that (γ n σ k )-mkph ((39a) (39b) and (39c) and (4a) and (4b)) for α n =β k =0reduces to mkph (37) and (γ n σ k )-mkph for α n β k =1reduces to emkph [3 Eample 6 Let us take n=and k=3andsetγ =y σ 3 =t and V 0 = V Then (4a) and (4b) become B t B 3y [ B B 3 β 3 [ B α [ 1 r i B 3 1 r i α ( t B 3 ( )) β 3 ( y B ( )) = 0 α ( r it 1 B 3 ( r i)) β 3 ( r iy 1 B ( r i)) = 0 (43a) (43b) α [ t 3V 3 ( 1 V y ) 3 V [ 3 V 3 V ) j=1 q j r j β 3 ( y α [ r it r i 3V r i 3 ( 1 V y ) r i 3 V r i [ 3 V r i 3 V r i ) j=1 q j r j r i β 3 ( r iy r i (44b) Particularly when taking α = β 3 = 0; α = 0 β 3 = 1; α =1 β 3 =0;andα =1 β 3 =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 mied type of mkp equation with self-consistent sources respectively 4 Links between (γ n σ k )-KPH and (γ n σ k )-mkph and (γ n σ k )-mkph and (1)-(γ n σ k )-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 [ the constrained KP hierarchy and the constrained modified KP hierarchy are studied And in [4 the gauge transformation between KP and mkp hierarchieswithself-consistentsourcesisgivenitisnatural to think whether these transformations can be generalized to the (γ n σ k )-KPH (γ n σ k )-mkph and (1)-(γ n σ k )-HDH The answer is positive and the main results are as follows Theorem 7 (a) Suppose L r i satisfy (γ n σ k )-KPH (1a) (1b) and (1c) and q and f are independent eigenfunctions for La pair (4); then L =q 1 Lq =q 1

7 Advances in Mathematical Physics 7 r i = 1 (qr i ) q =q 1 f (45) =α n q 1 L k 0 q1 (q) α n β k q 1 (r i q) β k q 1 (γn L n 0 ()) α n L k 0 (q 1 ) α n L k 1 ( ) satisfy the (γ n σ k )-mkph ((39a) (39b) and (39c)) and its La pair (40) (b) Suppose L r i satisfy the (γ n σ k )-mkph ((39a) (39b) and (39c)); s eigenfunction for La pair (40) After the transformation = q( γ n σ k ) γ n =γ n σ k =σ k ;then =α n q 1 L k 0 (1) α n L k 0 (q 1 ) α n β k q 1 (r i q) β k q 1 (γn L n 0 ()) α n L k 1 ( ) L= L = (46) =α n L k 1 ( )α n L k 1 ( )α n β k q 1 (r i q) β k q 1 (γn L n 0 ()) r i = 1 ( q r i ) r i q satisfy the (1)-(γ n σ k )-HDH ((11a) (11b) and (11c)) Proof For convenience we omit (a) Consider L γn =q q γn Lq q 1 L γn qq 1 Lq γn =q (L n 0 =α n β k q 1 (r i q) β k q 1 (γn L n 0 ()) Similarly we have β k ( γn L n 1 ( )) = α n β k q 1 (r i q) α n q 1 (σk L k 0 ()) (48) (49) α n 1 r i )(q)lqq 1 [L n 0 α n 1 r i Lq q 1 L(L n 0 α n 1 r i )(q) =[q 1 Lq q 1 (L n 0 α n 1 r i )(q)q 1 [L n 0 α n 1 r i Lq=[q 1 (L n 0 α n 1 r i )q q 1 (L n 0 α n 1 r i )(q) L = [q 1 L n 0 q q 1 L n 0 (q) L α n [q 1 1 r i q q 1 1 (r i q) L = [ L n 1 L α n [ 1 r i L =[ L n 1 α n 1 r i L (47) Thus we have α n ( σk L k 1 ( )) β k ( γn L n 1 ( )) q γn =q 1 [β k (γn L n 0 ()) α n (σk L k 0 ()) =0 (50) Inthefollowingwewillprovethat q =q 1 f satisfies La pair (40) Consider =q q γn fq 1 f γn =q 1 (L n 0 α n 1 r i )(q)q 1 f q 1 (L n 0 α n 1 r i )(f) In the same way (39b) can be proved The two equationsin(39c)canbeprovedsimilarlysoweonly prove the first one Consider α n ( σk L k 1 ( )) =q 1 q L n 0 q1 (q) q 1 fq 1 q L n 0 q1 (f) α n q 1 1 r i (q) q 1 fα n q 1 1 r i (f) = L n 0 (1) q L n 0 ( q) αn q r i α n 1 (r i f) =( L n 1 αn 1 r i ) ( q) (51) =α n q q σk α n q 1 σk α n L k 1 ( ) =α n q (L k 0 β k 1 r i )(q) q 1 [β k (γn L n 0 ()) α n L k 0 () α n L k 1 ( ) Similarly we can prove that q = q 1 f satisfies the second equation in (40) (b) Since the first order coefficient of L n 1 = L n 1 is given by [L n 1 wehave L n = L n 1 L n 1 ( q) (5)

8 8 Advances in Mathematical Physics L γn L γn =[ γn L = [ γn q γn L = L γn [ q γn q 1 L yields = L γn [ q γn q 1 L =[ L n 1 α n 1 r i L [( L n 1 α n 1 r i ) ( q) q 1 L =[ L n 1 L n 1 ( q) L [α n 1 r i α n 1 ( r i q ) L =[L n L [α n 1 r i α n ( r i q 1 ( r i q)) L =[L n L [α n 1 r i α n r i L =[L n L [α n 1 r i L (53) which implies that (11a) holds In the same way we can prove (11b) Consider α n (σk L k ( )) =α n σk α n q σk α n L k ( ) =β k ( γn L n 1 ( )) α n L k 1 ( ) α n ( L k 1 β k 1 r i )( q) α n L k ( ) =α n L k 1 ( )α n L k 1 ( q) (54) 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 (γ n σ k )-mkph and (γ n σ k )-HDH from the solutions of (γ n σ k )-KPH 5 Solutions for (γ n σ k )-mkph and (1)-(γ n σ k )-HDH We first briefly recall the generalized dressing method for the (γ n σ k )-KPH proposed in [15 Let f i g i satisfy f iγn = n (f i ) f iσk = k (f i ) g iγn = n (g i ) g iσk = k (g i ) and let h i be the linear combination of f i and g i : (57a) (57b) h i =f i F i (α n γ n β k σ k ) g i (58) with F i (X) being a differentiable function of X X=α n γ n β k σ k The dressing operator W is defined by where W= Wr (h 1h ) Wr (h 1 h ) (59) h 1 h h α n β k 1 r i ( q) β k ( γn L n 1 ( )) α n L k ( ) =α n β k 1 r i ( q) β k ( γn L n 1 ( )) Wr (h 1 h )= h 1 h h h (1) 1 h (1) h (1) h 1 h h 1 (60) Similarly β k (γn L n ( )) = α nβ k 1 r i ( q) α n ( σk L k 1 ( )) So α n (σk L k ( )) β k (γn L n ( )) =β k ( γn L n 1 ( )) α n ( σk L k 1 ( )) = 0 (55) (56) Wr (h 1 h )= h 1 h h h () 1 h () is the Wronskian determinant Define =F i W(g i ) h () r i = (1) i Wr (h 1ĥih ) Wr (h 1 h ) (61) In this way we present the connection between the solutions of (γ n σ k )-KPH (γ n σ k )-mkph and (γ n σ k )-HDH under the gauge transformations and reciprocal transformations In [17 the original three hierarchies are intimately where the hat means ruling out this term from the Wronskian determinant F i = df i /dx Thenwehavethe following

9 Advances in Mathematical Physics 9 Theorem 8 (see [15) Let W be defined by (59) and (58) let L=W W 1 andlet and r i be given by (61); then W L r i satisfy (γ n σ k )-KPH ((1a) (1b) and (1c) and (3a) and (3b)) Choose q=w(1) = (1) Wr (h 1 h h ) (6) Wr (h 1 h h ) as the particular eigenfunction for (4); based on Theorem 7 we have the Wronskian solution for (γ n σ k )-mkph: L = Wr (h 1h ) Wr (h1 h ) [Wr (h 1 1h ) Wr (h1 h ) (63a) = F i Wr (h 1 h g i ) Wr (h 1 h ) (63b) r i = Wr (h 1 ĥ i h ) (63c) Wr (h 1 h ) Choose f=w(e η ) = e η Wr (h 1 h ) h 1 h h 1 h 1 h h μ h () 1 h () h () μ (64) as another eigenfunction for (4) Based on Theorem 7 we have q = (1) e η Wr (h 1 h ) h 1 h h 1 h 1 h h μ h () 1 h () h () μ (65) Using = q = q based on Theorem 7 we have L= L () = h 1 h h 1 q h 1 q h q h q q h 1 q h q h q q h 1 q h q h q (66) = q (1)/ q (1)/ q [ [ h 1 h h 1 h 1 h h h () 1 h () h () h 1 h h h 1 h h h () 1 h () h () h 1 h h 1 h 1 h h h () 1 h () h () h 1 h h h 1 h h h () 1 h () h () = Wr (h 1h ) Wr (h 1 h ) q [ [ [ Wr (h 1 h ) 1 Wr (h 1 h ) 1 (67) where the elements h i = h i () and h i denotes h i / Similarly we have = F i Wr (h 1 h g i ) Wr (h 1 h ) r i = 1 = 1 ( q r i)= 1 ( r i ()) Wr (h 1 ĥ i h ) Wr (h 1 h ) (68a) (68b) where g i =g i () Equations (67) and (68a) and (68b) give the Wronskian solution for (1)-(γ n σ k )-HDH Let us illustrate it by solving (4a) (4b) and (4c) and (44a) and (44b) We take the solution of (57a) and (57b) as follows: [ [ q h 1 q h q h q h 1 q h q h h 1 h h 1 q h 1 q h q h q q h 1 q h q h q q h 1 q h q h q h 1 q h q h q h 1 q h q h 1 f i fl ep (λ i λ i yλ3 i t) = eξ i g i fl ep (μ i μ i yμ3 i t) = eη i h i fl f i F i (α yβ 3 t) g i = F i ep ( ξ i η i ) cosh (Ω i ) Ω i = 1 (ξ i η i ln F i ) (69) (70)

10 10 Advances in Mathematical Physics Eample 9 (solutions for the mied type of mkpescs ((44a) and (44b))) Since L =q 1 Lqthenitiseasytofind For =1wehave q= λ 1 μ 1 V 0 = q q (71) μ 1 λ 1 tanh (Ω 1 ) (7) The one-soliton solution for (44a) and (44b) with =1is as follows: V = V 0 = (λ 1 μ 1 ) [tanh (Ω 1 θ 1 )tanh (Ω 1 ) q 1 = F 1 F 1 λ 1 μ 1 (μ 1 λ 1 )e ξ 1η 1 sech (Ω 1 θ 1 ) r 1 = 1 F 1 e (ξ 1η 1 ) sech Ω 1 where θ 1 = ln λ 1 /μ 1 (73) Eample 10 (solutions for the mied type of HDESCS ((4a) (4b) and (4c))) Based on Theorem 7 we have that q = (1) e η Wr (h 1 h ) h 1 h h 1 h 1 h h μ h () 1 h () h () μ (74) is the eigenfunction of the mied type of mkpescs and we have L= L;thatis w w 0 w 1 1 = V 0 V 1 1 = q V 0 From the above equation we obtain (75) w= q (76) w 0 = V 0 The implicit solutions for the mied type of HDESCS ((4a) (4b) and (4c)) with =1are = q = λ 1 μ 1 λ 1 μ 1 e η 1Ω 1 sech (Ω 1 θ 1 ) w= (λ 1 μ 1 ) e η 1Ω 1 sech (Ω 1 θ 1 ) cosh (Ω 1 ) w 0 = (λ 1 μ 1 ) q 1 = r 1 = [tanh (Ω 1 θ 1 )tanh (Ω 1 ) F 1 F 1 λ 1 μ 1 (μ 1 λ 1 )e ξ 1η 1 sech (Ω 1 θ 1 ) 1 F 1 1 e(ξ 1η 1 ) sech (Ω 1 ) (77a) (77b) (77c) (77d) (77e) where θ 1 = ln λ 1 /μ 1 ξ 1 =λ 1 λ 1 yλ3 1 t η 1 =μ 1 μ 1 y μ 3 1 t Ω 1 = (1/)(ξ 1 η 1 ln F 1 ) and the relation between and is given by (77a) 6 Conclusion In this paper a new (1)-(γ n σ k )-HDH is proposed by introducing new time series γ n and σ k and adding eigenfunctions as components The (1)-(γ n σ k )-HDH includes (1)-HD hierarchy and etended HD hierarchy and contains first type and second type as well as mied type of (1)-HD equation with self-consistent sources as special cases The reduction of the (1)-(γ n σ k )-HDH is studied and the k-constrained (1)-(γ n σ k )-HDH is given In the same way as (1)- (γ n σ k )-HDH the (γ n σ k )-mkphisformulatedanditszerocurvature equation and La representation are presented The gauge transformations and reciprocal transformations between (γ n σ k )-KPH (γ n σ k )-mkph and (1)- (γ n σ k )-HDH are given They help us to obtain solutions of (γ n σ k )-mkph and (1)-(γ n σ k )-HDH By making use of the transformations we find the soliton solutions of the (γ n σ k )-mkph and (1)-(γ n σ k )-HDH Particularly the soliton solutions for the mied 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 ational atural Science Foundation of China ( and ) and the Fundamental Research Funds for the Central Universities (014ZZD10) References [1 B A Kupershmidt KdV6: an integrable system Physics Letters Avol37no15pp [ Y Q Yao and Y B Zeng The bi-hamiltonian structure and new solutions of KdV6 equation Letters in Mathematical Physicsvol86no-3pp [3 E Date M Jimbo M Kashiwara and T Miwa Operator approach to the Kadomtsev-Petviashvili equation transformation groups for soliton equations III the Physical Society of Japan vol 50 no 11 pp [4 M Jimbo and T Miwa Solitons and infinite dimensional Lie algebras Publications of the Research Institute for Mathematical Sciencesvol19no3pp [5 M Sato and Y Sato Soliton equations as dynamical systems on infinite dimensional Grassmann manifold in onlinear Partial Differential Equations in Applied Science Tokyo 198 pp59 71 orth-holland Publishing Amsterdam The etherlands 1983 [6 VGKacandJWvandeLeur Then-component KP hierarchy and representation theory Mathematical Physicsvol 44 no 8 article

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