Two new multi-component BKP hierarchies
aa r X i v : . [ n li n . S I] N ov Two new multi-component BKP hierarchies ∗† Hongxia Wu
1. Department of Mathematics, Jimei University, Xiamen, 361021, China2. Department of Mathematics, Beijing Institute of Technology, Beijing 100081,China
Xiaojun Liu
Department of Mathematics, Chinese Agriculture University, Beijing, PR China
Yunbo Zeng
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China
Abstract
We firstly propose two kinds of new multi-component BKP (mcBKP) hierarchy based on theeigenfunction symmetry reduction and nonstandard reduction, respectively. The first one containstwo types of BKP equation with self-consistent sources which Lax representations are presented.The two mcBKP hierarchies both admit reductions to the k − constrained BKP hierarchy and tointegrable (1+1)-dimensional hierarchy with self-consistent sources, which include two types of SKequation with self-consistent sources and of bi-directional SK equations with self-consistent sources.PACS: 02.30. IkKeywords: multi-component BKP hierarchy; BKP equation with self-consistent sources; k − constrained BKP hierarchy; n − reduction of BKP; Lax representation The multi-component KP (mcKP) hierarchy given in [1] contains many physically relevant nonlinear integrablesystems, such as Davey-Stewartson equation, two-dimensional Toda lattice and three-wave resonant interactionones, and attracts a lot of interests from both physical and mathematical points of view [1-8]. Another kindof multi-component KP equation is the so-called KP equation with self-consistent sources, which was initiatedby V.K. Mel’nikov [9-11].The first type of KP equation with self-consistent sources (KPSCS) arises in somephysical modes describing the interaction of long and short wave [8-10,12], and the second type of KPSCS ispresented in [8,11,13]. However, little attention has been paid to the multi-component BKP hierarchy. Thoughthe first type of the BKP equation with self-consistent sources (BKPSCS) is constructed by source generatingmethod [14], the Lax representation for the first type of BKPSCS and the second type of the BKPSCS havenot been investigated yet.It is known that the Lax equation of KP hierarchy is given by [15] L t n = [ B n , L ] (1 . ∗ Corresponding author:+8610 68916895 † E-mail address:[email protected] where L = ∂ + u ∂ − + u ∂ − + · · · (1 . ∂ denotes ∂ / ∂ x , u i , i = 1 , , · · · , are functions in infinitely many variables t = ( t , t , t , · · · ) with t = x , and B n = L n + stands for the differential part of L n .Owing to the commutativity of ∂ t n flows, we obtain zero-curvature equations of KP hierarchy B n,t k − B k,t n + [ B n , B k ] = 0 (1 . ∗ )satisfy the linear evolution equationsΦ t n = B n (Φ) (Φ ∗ t n = − B ∗ n (Φ ∗ )) (1 . L ∗ ∂ + ∂L = 0 (1 . t = t = · · · = 0)and of the constantterms B n , n = 3 , · · · , as well as that u = − u ′ , u = − u ′ + u (3)1 , · · · , and Φ ∗ = Φ ′ for n odd. Taking k = 3 , n = 5, (1.3) and (1.5) gives rise to the BKP equation u t + 19 u (5) − u (2) t + 53 uu (3) + 53 u ′ u (2) − uu t + 5 u u ′ − u ′ ∂ − x u t − ∂ − x u t t = 0 (1 . u ( i ) = ∂ i ∂x i , u ′ = ∂∂x in this paper.In this paper, following the idea in [8] and using the eigenfunction symmetry constraint, we firstly introducea new type of Lax equations which consist of the new time τ k − flow and the evolutions of wave functions.Under the evolutions of wave functions, the commutativity of the evolutions of τ k − flow and t n − flow gives riseto the first kind of new mcBKP hierarchy. This hierarchy enables us to obtain the first and the second typesof BKPSCS and their related Lax representations directly. This implies that the new mcBKP hierarchy canbe regarded as BKP hierarchy with self-consistent sources (BKPHSCS). Moreover, this new mcBKP hierarchycan be reduced to two integrable equation hierarchies: a (1+1)-dimensional soliton equation hierarchy withself-consistent sources and the k − constrained BKP hierarchy ( k − BKPH), which contain the first type andthe second type of SK equation with self-consistent sources and of bi-direction SK equation with self-consistentsources, respectively. Similar to the construction of the first kind of mcBKP hierarchy, we can also constructthe second kind of mcBKP hierarchy based on nonstandard reduction to obtain some new (2+1)-dimensionalsoliton equation with self-consistent sources. It is noted that the second kind of mcBKP hierarchy just as thefirst kind also admits the n − reduction and the k − constraint, which lead to some new (1+1)-dimensionalsoliton equations with self-consistent sources. Thus, these two mcBKP hierarchies provide an effective way tofind (1+1)-dimensional and (2+1)-dimensional soliton equations with self-consistent sources as well as their Laxrepresentations. Our paper is organized as follows. In section 2, we construct the first kind of new mcBKPhierarchy based on eigenfuction symmetry constraint and show that it contains the first and the second typesof BKPSCS. In section 3, the mcBKP hierarchy is reduced to a (1+1)-dimensional soliton hierarchy with self-consistent source and the k − constrained BKP hierarchy, respectively. In section 4, the second kind of newmcBKP hierarchy is also proposed based on nonstandard reduction. In addition, the n − reduction and the k − constraint of it are also considered. In section 5, some conclusions are given.
2. The first kind of new mcBKP hierarchy
Following the idea in [8] and using the eigenfunction symmetry constraint for BKP hierarchy [16], we define e B k by e B k = B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i ) (2 . q i , r i satisfy (1.4). Then we may introduce a new Lax equation given by L τ k = [ B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i ) , L ] (2 . a ) q i,t n = B n ( q i ) , r i,t n = B n ( r i ) , i = 1 , · · · , N (2 . b )where n, k are odd. Lemma 1 [ B n , r∂ − q ′ − q∂ − r ′ ] − = ( r∂ − q ′ − q∂ − r ′ ) t n Proof:
Set B n = n P i =1 a i ∂ i ( i ≥ B n , r∂ − q ′ − q∂ − r ′ ] − = n X i =1 ( a i r ( i ) ∂ − q ′ − a i q ( i ) ∂ − r ′ ) − n X i =1 ( r∂ − q ′ a i ∂ i − q∂ − r ′ a i ∂ i ) − = B n ( r ) ∂ − q ′ − B n ( q ) ∂ − r ′ − n X i =1 ( r∂ − q ′ a i ∂ i − q∂ − r ′ a i ∂ i ) − Applying integration by parts to the second term n X i =1 ( r∂ − q ′ a i ∂ i − q∂ − r ′ a i ∂ i ) − = · · · = n X i =1 ( − i [ r∂ − ( a i q ′ ) ( i ) − q∂ − ( a i r ′ ) ( i ) ] = r∂ − B ∗ n ( q ′ ) − q∂ − B ∗ n ( r ′ )Noticing the facts that q ∗ = q ′ , r ∗ = r ′ , q ∗ t n = − B ∗ n ( q ∗ ) and r ∗ t n = − B ∗ n ( r ∗ ), we can complete the proofimmediately. Theorem 1.
The commutativity of (1.1) and (2.2a) under (2.2b) leads to the following first kind of newintegrable multi-component BKP (mcBKP) hierarchy B n,τ k − ( B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )) t n + [ B n , B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )] = 0 (2 . a )or equivalently B n,τ k − B k,t n + [ B n , B k ] + N X i =1 { [ B n , r i ∂ − q ′ i − q i ∂ − r ′ i ] + B n ( q i ) ∂ − r ′ i + q i ∂ − B ′ n ( r i ) − B n ( r i ) ∂ − q ′ i − r i ∂ − B ′ n ( q i ) } = 0 (2 . a ′ ) q i,t n = B n ( q i ) , r i,t n = B n ( r i ) , i = 1 , · · · , N (2 . b )where n and k are odd.Under (2.3b), the Lax pair for (2.3a) is given by ψ t n = B n ( ψ ) , ψ τ k = [ B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )]( ψ ) (2 . Proof:
We will show that under (2.3b), (1.1) and (2.2a) lead to (2.3a). For convenience, we assume N = 1 anddenote q , r by q, r . By (1.1), (2.2) and lemma 1, we have B n,τ k = ( L nτ k ) + = [ B k + r∂ − q ′ − q∂ − r ′ , L n ] + = [ B k + r∂ − q ′ − q∂ − r ′ , L n + ] + + [ B k + r∂ − q ′ − q∂ − r ′ , L n − ] + = [ B k + r∂ − q ′ − q∂ − r ′ , L n + ] − [ B k + r∂ − q ′ − q∂ − r ′ , L n + ] − + [ B k , L n − ] + = [ B k + r∂ − q ′ − q∂ − r ′ , B n ] − [ r∂ − q ′ − q∂ − r ′ , B n ] − + [ B n , L k ] + = [ B k + r∂ − q ′ − q∂ − r ′ , B n ] + ( r∂ − q ′ − q∂ − r ′ ) t n + ( B k ) t n = [ B k + r∂ − q ′ − q∂ − r ′ , B n ] + ( B k + r∂ − q ′ − q∂ − r ′ ) t n Remark 1. (2.3a’) and (2.4) indicate that the new mcBKP hierarchy can be regarded as the BKP hierarchywith self-consistent sources and that it is Lax integrable.Next we will list some examples in the first kind of mcBKP hierarchy.
Example 1 ( The first type of BKPSCS ) For n = 3 , k = 5, (2.3)with u = u leads to the first type of theBKP equation with self-consistent sources ((2+1)-dimensional CDGKS equation with self-consistent sources) u τ + 19 u (5) − u (2) t + 53 uu (3) + 53 u ′ u (2) − uu t + 5 u u ′ − u ′ ∂ − x u t − ∂ − x u t t + N X i =1 ( q (2) i r i − r (2) i q i ) = 0 ,q i,t = q (3) i + 3 u q ′ i , r i,t = r (3) i + 3 u r ′ i , i = 1 , · · · , N (2 . ψ t = ( ∂ + 3 u∂ )( ψ ) ,ψ τ = ( ∂ + 5 u∂ + 152 u ′ ∂ + ( 53 ∂ − x u t + 103 u (2) + 5 u ) ∂ + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )))( ψ ) (2 . Example 2 (The second type of BKPSCS)
For n = 5 , k = 3, (2.3) with u = u yields the second type ofBKP equation with self-consistent sources u t + 19 u (5) − u (2) τ + 53 uu (3) + 53 u ′ u (2) − uu τ + 5 u u ′ − u ′ ∂ − x u τ − ∂ − x u τ τ = 19 N X i =1 [5 q (3) i r ′ i − r (3) i q ′ i + 10 q (4) i r i − r (4) i q i +5( q ′ i r i − r ′ i q i ) τ + 30 uq (2) i r i − ur (2) i q i + 30 u ′ q ′ i r i − u ′ r ′ i q i ] ,q i,t = q (5) i + 5 uq (3) i + 5 u ′ q (2) i + [ 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )] q ′ i ,r i,t = r (5) i + 5 ur (3) i + 5 u ′ r (2) i + [ 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )] r ′ i (2 . ψ τ = [ ∂ + 3 u∂ + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )]( ψ ) ,ψ t = [ ∂ + 5 u∂ + 5 u ′ ∂ + ( 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )) ∂ ]( ψ ) (2 . Remark 2
The first type of BKPSCS (2.5) coincide with what obtained in [14] by source generating method.However the Lax representation for (2.5) and the second type of BKPSCS have not been found before.
3. The n − reduction and k − constraint of (2.3)3.1 The n − reduction of (2.3) The n − reduction of (2.3)is given by [15] L n = B n , or L n − = 0 (2 . L t n = [ B n , L ] = [ L n , L ] = 0 , B k,t n = ( L k + ) t n = 0 , and q i,t n = r i,t n = 0 (2 . q i and r i are wave function, they have to satisfy [15] B n ( q i ) = L n ( q i ) = λ ni q i , B n ( r i ) = L n ( r i ) = λ ni r i (2 . n − reduction case. By using the Lemma 1 and (2.10), wecan conclude that the constraint (2.9) is invariant under the τ k − flow. Due to (2.10) and (2.11), one can drop t n − dependency from (2.3) and get the following (1+1)-dimensional integrable hierarchy with self-consistentsources B n,τ k + [ B n , B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )] = 0 ,B n ( q i ) = λ ni q i , B n ( r i ) = λ ni r i , i = 1 , · · · , N (2 . B n ( ψ ) = λ n ψ,ψ τ k = [ B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )]( ψ ) (2 . Example 3 (The first type of SKSCS)
For n = 3 , k = 5, (2.12) present the first type of SK equationwith self-consistent sources (SKSCS) u τ + 19 u (5) + 53 u ′ u (2) + 53 uu (3) + 5 u u ′ + N X i =1 ( q (2) i r i − q i r (2) i ) = 0 ,q (3) i + 3 uq ′ i = λ i q i r (3) i + 3 ur ′ i = λ i q i , i = 1 , · · · , N (2 . n = 3 , k = 5 leads to the Lax pair of (2.14)( ∂ + 3 u∂ )( ψ ) = λ ψ,ψ τ = [ ∂ + 5 u∂ + 5 u ′ ∂ + ( 103 u (2) + 5 u ) ∂ + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )]( ψ ) (2 . Example 4 (The first type of bi-directional SKSCS)
For n = 5 , k = 3, (2.12) presents the firsttype of bi-directional SK equation with self-consistent sources (bi-directional SKSCS)19 u (5) − u (2) τ + 53 uu (3) + 53 u ′ u (2) − uu τ + 5 u u ′ − u ′ ∂ − x u τ − ∂ − x u τ τ = 19 N X i =1 [5 q (3) i r ′ i − r (3) i q ′ i + 10 q (4) i r i − r (4) i q i +5( q ′ i r i − r ′ i q i ) τ + 30 uq (2) i r i − ur (2) i q i + 30 u ′ q ′ i r i − u ′ r ′ i q i ] ,q (5) i + 5 uq (3) i + 5 u ′ q (2) i + [ 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )] q ′ i = λ i q i ,r (5) i + 5 ur (3) i + 5 u ′ r (2) i + [ 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )] r ′ i = λ i r i , i = 1 , · · · , N (2 . ψ τ = [ ∂ + 3 u∂ + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )]( ψ ) , [ ∂ + 5 u∂ + 5 u ′ ∂ + ( 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )) ∂ ]( ψ ) = λ ψ (2 . q i = r i = 0, then (2.14)and (2.16) reduces to the SK equation and bi-directional SK equation [18]. k − constraint of (2.3) The k − constraint of (2.3)is given by [16] L k = B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i ) (2 . L τ k = 0 and B n,τ k = 0. Then (2.3)becomes k − constrainedBKP hierarchy( B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )) t n = [( B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )) nk + , B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )] ,q i,t n = ( B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )) nk + ( q i ) , r i,t n = ( B k + N X i =1 ( r i ∂ − q ′ i − q i ∂ − r ′ i )) nk + ( r i ) , i = 1 , · · · , N (2 . Example 5 (The second type of SKSCS)
For n = 5 , k = 3, (2.19) presents the second type of SK equationwith self-consistent sources u t + 19 u (5) + 53 uu (3) + 53 u ′ u (2) + 5 u u ′ = 19 N X i =1 [5 q (3) i r ′ i − r (3) i q ′ i + 10 q (4) i r i − r (4) i q i + 30 uq (2) i r i − ur (2) i q i + 30 u ′ q ′ i r i − u ′ r ′ i q i ] ,q i,t = q (5) i + 5 uq (3) i + 5 u ′ q (2) i + [ 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )] q ′ i ,r i,t = r (5) i + 5 ur (3) i + 5 u ′ r (2) i + [ 103 u (2) + 5 u + 53 N X i =1 ( q ′ i r i − q i r ′ i )] r ′ i , i = 1 , · · · , N (2 . Example 6 (The second type of bi-directional SKESCS)
For n = 3 , k = 5, (2.19) gives rise to thesecond type of bi-directional SK equation with self-consistent sources19 u (5) − u (2) t + 53 uu (3) + 53 u ′ u (2) − uu t + 5 u u ′ − u ′ ∂ − x u t , − ∂ − x u t t + N X i =1 ( q (2) i r i − r (2) i q i ) = 0 ,q i,t = q (3) i + 3 u q ′ i , r i,t = r (3) i + 3 u r ′ i , i = 1 , · · · , N (2 . q i = 1 and R i = − r ′ i yield the following mcBKP hierarchy B n,τ k − ( B k + N X i =1 ∂ − R i ) t n + [ B n , B k + N X i =1 ∂ − R i ] = 0 ,R t n = − B ∗ n ( R ) (2 . ψ t n = B n ( ψ ) , ψ τ k = ( B k + N X i =1 ∂ − R i )( ψ ) (2 . k and k are both odd. Example 7
For n = 3 , k = 5, (2.22)gives rise to u τ + 19 u (5) − u (2) t + 53 uu (3) + 53 u ′ u (2) − uu t + 5 u u ′ − u ′ ∂ − x u t − ∂ − x u t t + N X i =1 R ′ i = 0 ,R i,t = R (3) i + 3( uR i ) ′ , i = 1 , · · · , N (2 . ψ τ = [ ∂ + 5 u∂ + 5 u ′ ∂ + ( 53 ∂ − x u t + 103 u (2) + 5 u ) ∂ + N X i =1 ∂ − R i )]( ψ ) ,ψ t = ( ∂ + 3 u∂ )( ψ ) (2 . Example 8
For n = 5 , k = 3, (2.22)presents u t + 19 u (5) − u (2) τ + 53 uu (3) + 53 u ′ u (2) − uu τ + 5 u u ′ − u ′ ∂ − x u τ − ∂ − x u τ τ = 19 N X i =1 [ 109 R (3) i + 103 ( uR i ) ′ + 5 R i,τ ] ,R i,t = R (5) i + 5( uR (2) i ) ′ + 5( u ′ R ′ i ) ′ + ([ 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 R i ]) R i ) ′ (2 . ψ τ = [ ∂ + 3 u∂ + N X i =1 ∂ − R i )]( ψ ) ,ψ t = [ ∂ + 5 u∂ + 5 u ′ ∂ + ( 53 ∂ − x u τ + 103 u (2) + 5 u + 53 N X i =1 R i ) ∂ ]( ψ ) (2 . k − constraint of (2.22)is given by [19] L k = B k + ∂ − R (2 . B k + ∂ − R ) t n = [ { B k + ∂ − R } nk + , B k + ∂ − R ] ,R t n = − [( B k + ∂ − R ) nk + ] ∗ ( R ) , i = 1 , · · · , N (2 . Example 9
For n = 3 , k = 5, we obtain 5-constrained equation from (2.29)19 u (5) − u (2) t + 53 uu (3) + 53 u ′ u (2) − uu t + 5 u u ′ − u ′ ∂ − x u t − ∂ − x u t t + N X i =1 R ′ i = 0 ,R i,t = R (3) i + 3( uR i ) ′ (2 . Example 10
For n = 5 , k = 3, we obtain 3-constrained equation from (2.29) u t = − u (5) − uu (3) − u ′ u (2) − u u ′ + 109 R (3) + 103 ( uR ) ′ ,R t = R (5) + 5[( uR ) (3) − ( u ′ R ) (2) + ( u R ) ′ ] + 53 (2 u (2) R + R ) (2 .
4. The second kind of new mcBKP hierarchy
In the previous section, the first kind of new mcBKP hierarchy is constructed based on the eigenfunctionsymmetry reduction. In fact, we find that we can construct the second kind of new mcBKP hierarchy by usingits nonstandard reduction given by [20], which is completely different from the first kind.It is show in [20] that the non-symmetry constraint L k = B k + N X i =1 ( r i ∂ − q ′ i + q i ∂ − r ′ i ) , k is even (2 . L τ k = [ B k + N X i =1 ( r i ∂ − q ′ i + q i ∂ − r ′ i ) , L ] (2 . a ) q i,t n = B n ( q i ) , r i,t n = B n ( r i ) , i = 1 , · · · , N (2 . b )where n is odd and k is even.In the exactly same way as for Lemma 1, we can find Lemma 2. [ B n , r∂ − q ′ + q∂ − r ′ ] − = ( r∂ − q ′ + q∂ − r ′ ) t n Theorem 2 (1.1) and (2.33) lead to the second kind of new integrable mcBKP hierarchy B n,τ k − ( B k + N X i =1 ( r i ∂ − q ′ i + q i ∂ − r ′ i )) t n + [ B n , B k + N X i =1 ( r i ∂ − q ′ i + q i ∂ − r ′ i )] = 0 (2 . a ) q i,t n = B n ( q i ) , r i,t n = B n ( r i ) , i = 1 , · · · , N (2 . b )where n is odd and k is even. With the Lax pair for (2.34a) under (2.34b) given by ψ τ k = ( B k + N X i =1 ( r i ∂ − q ′ i + q i ∂ − r ′ i ))( ψ ) ,ψ t n = B n ( ψ ) (2 . Example 11
For n = 3 , k = 2, (2.34) leads to the following new integrable (2+1)-dimensional equations u τ + u (2) + N X i =1 ( q i r i ) (2) = 0 , − u t + u (3) + 3 uu ′ + 3 N X i =1 ( q ′ i r (2) i + r ′ i q (2) i ) = 0 ,q i,t = q (3) i + 3 uq ′ i , r i,t = r (3) i + 3 ur ′ i , i = 1 , · · · .N (2 . Example 12
For n = 3 , k = 4, (2.34) yields to another new integrable (2+1) dimensional equation3 u τ + 2 u ′ t + u (4) + 6( u ′ ) + 6 uu (2) + 3 N X i =1 ( q i r i ) (2) = 0 , u (2) t − ∂ − x u t t + 23 u (5) + 12 u ′ u (2) + 12 u ′ u + 6 uu (3) + 6 N X i =1 ( q ′ i r i (2) + r ′ i q (2) i ) = 0 ,q i,t = q (3) i + 3 uq ′ i , r i,t = r (3) i + 3 ur ′ i , i = 1 , · · · .N (2 . n − reduction and the k − constraint given by B n = L n and L k = B k + N P i =1 ( r i ∂ − q ′ i + q i ∂ − r ′ i ), respectively. Example 13
For n = 3 , k = 2, the 2-constrained (2.34) is given by q i,t = q (3) i − q i r i q ′ i , r i,t = r (3) i − q i r i r ′ i (2 . Example 14
For n = 3 , k = 4, the 4-constrained (2.34) is given by u t = − u (3) − uu ′ − N X i =1 ( q i r i ) ′ ,q i,t = q (3) i + 3 uq ′ i , r i,t = r (3) i + 3 ur ′ i , i = 1 , · · · .N (2 . Example 15
The 3-reduction of (2.34) for n = 3 , k = 2 reads u τ − u + N X i =1 ( q (2) i r i + r (2) i q i − q ′ i r ′ i ) = 0 ,q (3) i + 3 uq ′ i = λ i q i , r (3) i + 3 ur ′ i = λ i r i , i = 1 , · · · .N (2 . Example 16
The another 3-reduction of (2.34) for n = 3 , k = 4 reads3 u τ + 13 u (4) + 3( u ′ ) − u + 3 N X i =1 ( q i r (2) i + q (2) i r i ) = 0 ,q (3) i + 3 uq ′ i = λ i q i , r (3) i + 3 ur ′ i = λ i r i , i = 1 , · · · , N (2 .
5. Conclusion
We firstly propose two new kinds of multi-component BKP hierarchy based on the eigenfunction symmetryreduction and nonstandard reduction, respectively. The first kind of mcBKP hierarchy includes two types ofBKP equation with self-consistent sources. It admits reductions to the k − constrained BKP hierarchy containingthe second type of (1+1)-dimensional integrable soliton equation with self-consistent sources, and n − reductionof BKP hierarchy including the first type of (1+1)-dimensional integrable soliton equation with self-consistentsources. Then we construct the second kind of mcBKP hierarchy based on nonstandard reduction to obtainsome new integrable (2+1)-dimensional soliton equation with self-consistent sources. It is noted that the secondkind of mcBKP hierarchy also admits the n − reduction and k − constraint, which lead to some new integrable(1+1)-dimensional soliton equation with self-consistent sources. Thus, these two mcBKP hierarchies provide aneffective way to find (1+1)-dimensional and (2+1)-dimensional soliton equations with self-consistent sources aswell as their Lax representations. Though the solution for the first type of BKPSCS was constructed by sourcegenerating method [14], the solution structure for the second type of BKPSCS has not been investigated yet.We will solve the integrable equation in the forthcoming paper. Acknowledgment
This work is supported by National Basic Research Program of China (973 Program) (2007CB814800) andNational Natural Science Foundation of China (grant No. 10601028).