A new extended discrete KP hierarchy and generalized dressing method

A new extended discrete KP hie rarc h y and generalized dressing metho d Y uqin Y ao 1) 1 , X iaojun Liu 2) 2 and Y un b o Zeng 1) 3 1) Dep artment of Mathematic al Scienc e, Tsinghua University, Beijing, 100084 , PR China 2) Dep artment of Applie d Mathematics, China A gricultur al Un iversity, Beijing, 100083, PR China Abstract Inspired by the squar ed eigenfunction symmetry constra in t, we intro- duce a new τ k -flow by “extending” a sp ecific t n -flow o f discrete KP hierarch y (DKPH). W e construct extended discrete KPH (exDKP H), which consists of t n - flow, τ k -flow and t n evolution of eig enfunction and adjoin t eigenfunctions, and its Lax representation. The exDKPH con tains t wo t yp es of discrete KP equation with self-c o nsisten t sources (DKP ESCS). Two reductions of ex DKPH are ob- tained. The gener alized dressing approach for solving the exDKPH is prop osed and the N-soliton solutions of t wo types of the DKPESCS are pres en ted. 1 In tro d uction Generalizations o f soliton hierarch y attract a lot of interests from b oth phys- ical and mathematical p oints and there w ere so me metho ds to gener alize the soliton hierarch y[1 ]-[4]. Recently , a s y stematic appro a c h inspired by squared eigenfunction symmetry cons train t was prop osed to cons truct the extended KP hierarch y[5 ]. By this method, the extended t wo-dimensional T o da lattice hier- arch y , the extended CKP hierar chy and the extended q-defo r med KP hierarch y hav e b e en o btained[6]-[8]. The discre te KP hiera rc h y(DKPH) [9]-[1 2] is an interesting ob ject in the resear ch of the discr e te integrable systems and the dis cretization of the int e- grable systems[13]. The Sa to’s appr oach for the discrete KPH was presented in [1 1]. Naturally , there a re some similar prop erties b etw een discrete KP H a nd KPH[14], such a s tau function[12, 14], Hamiltonian structure[1 2] and gauge transformatio n[10 , 1 5, 16], etc. In [10], O ev el has given explicitly tw o types of g auge trans fo rmation op erators o f the discr ete KPH. In [1 6], the combined gauge op erator and the deter minan t repr esen tation of the op erator hav e b een obtained. In this pap er, we will construct the extension of the discrete KPH(exDKPH). Inspired by the squa r ed eigenfunction symmetry constr ain t of discrete KP hi- erarch y [10], we introduce d the new τ k -flow by “extending” a sp ecific t n -flow of discrete KP hiera rc h y . Then we find the exDKPH c onsisting o f t n -flow of 1 yqy ao@math.tsingh ua.edu.cn 2 tigertooth4@gmail.com 3 Corresp onding author: yzeng@math.tsingh ua.edu.cn 1 discrete K P hier arch y , τ k -flow a nd the t n -evolutions o f eigenfunctions and ad- joint eigenfunctions. The co mm uta tivity of t n -flow and τ k -flow gives rise to zer o curv ature r epresentation for exDKPH. Also the Lax representation of exDKPH is derived. Due to the introduction of τ k -flow the ex DK PH contains tw o time series { t n } and { τ k } and mo re comp onents by adding eigenfunctions and ad- joint eigenfunctions. The e x DKPH contains the fir st type and second type of discrete KP equation with self-cons is ten t sour ces(DKPESCS). The KP equation with self-co ns isten t sour ces a r ose in some physical mo de ls describing the inter- action of lo ng and short wa ves[4 ]. The similarity of KP eq uation and discr ete KP equation enables us to sp eculate on the po ten tial a pplication of discrete KP equatio n with self-consistent source s . By t n -reduction and τ k -reduction, the exDK P H r educes to a discrete 1 + 1-dimensional integrable hiera rc h y with self-consistent sour ces and constrained discre te KP hierar chy , resp ectively . The dressing metho d is an impor tan t to ol for solving soliton hierarch y [12]. How ever this metho d c an not b e applied directly for so lving the “extended” hierarch y . A ge neralized dress ing appr oach for e x KPH is prop osed in [17]. In this pap er, with the combination of dressing metho d and v ariation of constants metho d, a generaliza tion to the dr e ssing metho d for exDKPH is pre s en ted, which is bas e d o n the dressing metho d for discre te KPH [11] and the similar appro ac h for finding W r onskian solutions to co nstrained KP hiera rc hy [18]. In this wa y , we can solve the entire hierar c h y o f ex DKPH in an unified and simple manner . As the sp ecial cases, the N-soliton solutions of the b oth t yp es of DKP E SCS are obtained simult aneously . This pap er will b e orga nize d as follows. In Sec.2, we prese n t the exDKP H and its Lax pair, w hich includes tw o types of DKP ESCS. In Sec.3 , t n -reduction and τ k - reduction fo r the exDKP H are given. In Sec.4, we discuss the generalized dressing metho d for the exDKPH. In Sec.5, we prese nt the N-soliton solutions of the DKPESCS. 2 New extended discrete KP hierarc h y W e denote the shift and the difference o perato rs acting on the asso ciative r ing F o f functions b y Γ and ∆, resp ectiv ely , as follows F = { f ( l ) = f ( l, t 1 , t 2 , · · · , t i , · · · ); l ∈ Z , t i ∈ R } Γ( f ( l )) = f ( l + 1) = f (1) ( l ) , ∆( f ( l )) = f ( l + 1) − f ( l ) . In this pap er, we use P ( f ) to denote an action of difference o pera tor P o n the function f , w hile P f means the multiplication of difference o pera tor P and zero order difference op erator f . Define the following op eration ∆ j f = ∞ X i =0 „ j i « (∆ i ( f ( l + j − i )))∆ j − i , „ j i « = j ( j − 1) · · · ( j − i + 1) i ! . (1) Also, w e define the adjoint op erator to the ∆ op erator by ∆ ∗ ∆ ∗ ( f ( l )) = (Γ − 1 − I )( f ( l )) = f ( l − 1) − f ( l ) , (2) 2 ∆ ∗ j f = ∞ X i =0  j i  (∆ ∗ i ( f ( l + i − j )))∆ ∗ j − i . (3) Let P = P k j = −∞ f j ( l )∆ j , the adjoint op erato r P ∗ is defined by P ∗ = P k j = −∞ ∆ ∗ j f j ( l ) . The Lax equation of the DKP hiera rc h y is given by[9, 11] L t n = [ B n , L ] , (4) where L = ∆ + f 0 + f 1 ∆ − 1 + f 2 ∆ − 2 + · · · is a pseudo- difference op erator with po ten tial functions f i ∈ F , B n = L n + stands fo r the difference part of L n . The commutativit y of t n - and t m -flow gives r ise to the zero-c urv ature equa tions for DKP hierarch y: B n,t m − B m,t n + [ B n , B m ] = 0 . (5) with the Lax pair given by ψ t n = B n ( ψ ) , ψ t m = B m ( ψ ) . (6) The t n evolutions of eigenfunction ψ and adjoint eigenfunction φ read ψ t n = B n ( ψ ) , φ t n = − B ∗ n ( φ ) . (7) F or n = 2 , m = 1 , (5) gives rise to the DKP equa tio n[9] ∆( f 0 t 2 + 2 f 0 t 1 − 2 f 0 f 0 t 1 ) = (∆ + 2) f 0 t 1 t 1 . (8) It is known that the squared eigenfunction symmetry co nstraint given by[10] ˜ B k = B k + N X i =1 ψ i ∆ − 1 φ i ψ i,t n = B n ( ψ i ) , φ i,t n = − B ∗ n ( φ i ) , i = 1 , · · · , N , is compatible with DKP hierar c h y . Here N is an arbitrar y natural num ber , ψ i and φ i are N different eigenfunctions and adjoint eigenfunctions of the equations (9c). This c o mpatibilit y enables us to construct a new extended discrete K P hierarch y (exDKPH) as L t n = [ B n , L ] , (9a) L τ k = [ B k + N X i =1 ψ i ∆ − 1 φ i , L ] , (9b) ψ i,t n = B n ( ψ i ) , φ i,t n = − B ∗ n ( φ i ) , i = 1 , · · · , N . (9c) W e have the following lemma. Lemma 1. L et Q = a ∆ k , k ≥ 1 , then (∆ − 1 φQ ) − = ∆ − 1 Q ∗ ( φ ) (10a) [ B n , ψ ∆ − 1 φ ] − = B n ( ψ )∆ − 1 φ − ψ ∆ − 1 B ∗ n ( φ ) . (10b) 3 Pr o of. Using f ∆ = ∆Γ − 1 ( f ) − ∆(Γ − 1 ( f )) , ∆ ∗ = − ∆Γ − 1 , we hav e (∆ − 1 φa ∆ k ) − = (∆ − 1 ∆Γ − 1 ( φa )∆ k − 1 − ∆ − 1 ∆(Γ − 1 ( φa ))∆ k − 1 ) − = − (∆ − 1 ∆(Γ − 1 ( φa ))∆ k − 1 ) − = · · · = ( − 1) k ∆ − 1 ∆ k (Γ − k ( φa )) = ∆ − 1 ∆ ∗ k ( φa ) = ∆ − 1 Q ∗ ( φ ) which yields to (10a) and (10b). Prop osition 1. The c ommut ativi ty of (9a) and (9b) under (9c) gives rise to the fol lowing zer o-curvatu r e r epr esentation for exDKPH (9) B n,τ k − ( B k + N X i =1 ψ i ∆ − 1 φ i ) t n + [ B n , B k + N X i =1 ψ i ∆ − 1 φ i ] = 0 , (11a) ψ i,t n = B n ( ψ i ) , φ i,t n = − B ∗ n ( φ i ) , i = 1 , 2 , · · · , N , (11b) with t he L ax r epr esentation given by Ψ t n = B n (Ψ) , Ψ τ k = ( B k + N X i =1 ψ i ∆ − 1 φ i )(Ψ) . (12) Pr o of. F or co n v enience, we omit P . By (9) and Lemma 1, we hav e B n,τ k = ( L n τ k ) + = [ B k + ψ ∆ − 1 φ, L n ] + = [ B k + ψ ∆ − 1 φ, L n + ] + +[ B k + ψ ∆ − 1 φ, L n − ] + = [ B k + ψ ∆ − 1 φ, L n + ] − [ B k + ψ ∆ − 1 φ, L n + ] − + [ B k , L n − ] + = [ B k + ψ ∆ − 1 φ, L n + ] − [ ψ ∆ − 1 φ, B n ] − + [ B n , L k ] + = [ B k + ψ ∆ − 1 φ, B n ] + ( B k + ψ ∆ − 1 φ ) t n . Remark. The exDKP H (11) ex tends the DKPH (5 ) by containing tw o time series { t n } and { τ k } and more comp onent s ψ i and φ i , i = 1 , · · · , N . Example 1. The first typ e of DKPSCS is given by (11) with n = 1 , k = 2 ∆( f 0 τ 2 + 2 f 0 t 1 − 2 f 0 f 0 t 1 ) = (∆ + 2) f 0 t 1 t 1 − ∆ 2 N X i =1 ( ψ i φ ( − 1) i ) , (13a) ψ i,t 1 = ∆( ψ i ) + f 0 ψ i , φ i,t 1 = − ∆ ∗ ( φ i ) − f 0 φ i , i = 1 , 2 , · · · , N . (13b) Its L ax r epr esentation is Ψ t 1 = (∆ + f 0 )(Ψ) (14a) Ψ τ 2 = (∆ 2 + ( f 0 + f (1) 0 )∆ + ∆( f 0 ) + f (1) 1 + f 1 + f 2 0 + N X i =1 ψ i ∆ − 1 φ i )(Ψ) . (14b) 4 Example 2. The se c ond typ e of DKPSCS is given by (11) with n = 2 , k = 1 ∆( f 0 t 2 + 2 f 0 τ 1 − 2 f 0 f 0 τ 1 ) = (∆ + 2) f 0 τ 1 τ 1 + N X i =1 [∆ 2 (( f 0 + f − 1 0 − 2) ψ i φ − 1 i ) + ∆( ψ (2) i φ i − ψ i φ ( − 2) i ) + ∆((Γ + 1)( ψ i φ ( − 1) i ) τ 1 )] , (15a) ψ i,t 2 = ∆ 2 ( ψ i ) + ( f 0 + f (1) 0 )∆( ψ i ) + (∆( f 0 ) + f (1) 1 + f 1 + f 2 0 ) ψ i , (15b) φ i,t 2 = − ∆ ∗ 2 ( ψ i ) − ∆ ∗ (( f 0 + f (1) 0 ) ψ i ) − (∆( f 0 ) + f (1) 1 + f 1 + f 2 0 ) ψ i . (15c) Its L ax r epr esentation is Ψ t 2 = (∆ 2 + ( f 0 + f (1) 0 )∆ + ∆( f 0 ) + f (1) 1 + f 1 + f 2 0 )(Ψ) (16a) Ψ τ 1 = (∆ + f 0 + N X i =1 ψ i ∆ − 1 φ i )(Ψ) . (16b) 3 Reductions of the exDKPH 3.1 The t n - reduction The t n -reduction is given by L n = B n or L n − = 0 . (17) Then we hav e ( L n ) t n = [ B n , L n ] = 0 , B n,t n = 0 . So L is independent of t n and w e have B n ( ψ i ) = L n ( ψ i ) = λ n i ψ i , B ∗ n ( φ i ) = λ n i φ i . ( 18) Then we can drop t n depe ndency from (11) and obtain B n,τ k = [( B n ) k n + + N X i =1 ψ i ∆ − 1 φ i , B n ] , (19a) B n ( ψ i ) = λ n i ψ i , B ∗ n ( φ i ) = λ n i φ i , i = 1 , 2 , · · · , N , (19b) with the Lax pair given by Ψ τ k = (( B n ) k n + + N X i =1 ψ i ∆ − 1 φ i )(Ψ) , B n (Ψ) = λ n Ψ . 5 (19) can b e rega rded as discr ete (1 +1)-dimensional integrable hierar c h y with self-consistent sour ces. When n = 2 , k = 1 , (19) g iv es rise to 2∆( f 0 τ 1 − f 0 f 0 τ 1 ) = (∆ + 2) f 0 τ 1 τ 1 + N X i =1 [∆ 2 ( f 0 + f ( − 1) 0 − 2) ψ i φ − 1 i + ∆( ψ (2) i φ i − ψ i φ ( − 2) i ) + ∆(Γ + 1)( ψ i φ ( − 1) i ) τ 1 ] (20a) ∆ 2 ( ψ i ) + ( f 0 + f (1) 0 )∆( ψ i ) + (∆( f 0 ) + f (1) 1 + f 1 + f 2 0 ) ψ i = λ 2 i ψ i , (20b) ∆ ∗ 2 ( ψ i ) + ∆ ∗ (( f 0 + f (1) 0 ) ψ i ) + (∆( f 0 ) + f (1) 1 + f 1 + f 2 0 ) ψ i = λ 2 i φ i , (20c) which can b e transfor med to the first type of V eselov-Shabat equation[19] with self-consistent sour ces (VSESCS). 3.2 The τ k - r eduction The τ k -reduction is given by[10 ] L k = B k + N X i =1 ψ i ∆ − 1 φ i . By dropping τ k depe ndency from (11), w e obtain ( B k + N X i =1 ψ i ∆ − 1 φ i ) t n = [( B k + N X i =1 ψ i ∆ − 1 φ i ) n k + , B k + N X i =1 ψ i ∆ − 1 φ i ] , (21a ) ψ i,t n = ( B k + N X i =1 ψ i ∆ − 1 φ i ) n k + ( ψ i ) , (21b) φ i,t n = − ( B k + N X i =1 ψ i ∆ − 1 φ i ) n k ∗ + ( φ i ) , i = 1 , 2 , · · · , N , (21c) which is the k-constr ained DKP hierarchy . When n = 1 , k = 2, (21) leads to 2∆( f 0 t 1 − f 0 f 0 t 1 ) = (∆ + 2) f 0 t 1 t 1 + ∆ 2 N X i =1 ( ψ i φ ( − 1) i ) , (22a) ψ i,t 1 = ∆( ψ i ) + f 0 ψ i , φ i,t 1 = − ∆ ∗ ( φ i ) − f 0 φ i , i = 1 , 2 , · · · , N , (22b) which can be transformed to the second type of VSESCS. 4 Dressing approac h for exDKPH 4.1 Dressing approac h for discrete KP hierarc hy W e first br iefly recall the dressing approach for DKPH [1 1]. Assume tha t op er- ator L o f DKP H (4) can b e written as a dressing form L = W ∆ W − 1 , (23) 6 W = ∆ N + w 1 ∆ N − 1 + w 2 ∆ N − 2 + · · · + w N . It is known [12] that if W satisfies W t n = − L n − W , (24) then L satisfies (4). It is easy to ch eck the following Le mma . Lemma 2. If h t n = ∆ n ( h ) , W satisfies (24), then ψ = W ( h ) satisfies (7) , i.e. ψ t n = B n ( ψ ) . (25) If there are N independent functions h 1 , . . . , h N solving W ( h ) = 0 i.e. W ( h i ) = 0, then w 1 , . . . , w N are completely deter mined from these h i , by solving the linear equation:      h 1 ∆( h 1 ) · · · ∆ N − 1 ( h 1 ) h 2 ∆( h 2 ) · · · ∆ N − 1 ( h 2 ) . . . . . . . . . . . . h N ∆( h N ) · · · ∆ N − 1 ( h N )           w N w N − 1 . . . w 1      = −      ∆ N ( h 1 ) ∆ N ( h 2 ) . . . ∆ N ( h N )      . Then the o pera tor W can b e written as W = 1 W r d ( h 1 , · · · , h N )          h 1 h 2 · · · h N 1 ∆( h 1 ) ∆( h 2 ) · · · ∆( h N ) ∆ . . . . . . . . . . . . . . . ∆ N ( h 1 ) ∆ N ( h 2 ) · · · ∆ N ( h N ) ∆ N          (26) where W r d ( h 1 , · · · , h N ) =          h 1 h 2 · · · h N ∆( h 1 ) ∆( h 2 ) · · · ∆( h N ) . . . . . . . . . . . . ∆ N − 1 ( h 1 ) ∆ N − 1 ( h 2 ) · · · ∆ N − 1 ( h N )          . Prop osition 2. Assu me that h i satisfies h i,t n = ∆ n ( h i ) , i = 1 , · · · , N (27) W and L ar e c onst r u cte d by (26) and (23), then W and L satisfy (24) and (4), r esp e ctively. Pr o of. T a k ing par tial deriv ative ∂ t n to the equation W ( h i ) = 0: W t n ( h i ) + W ∆ n ( h i ) = ( W t n + L n + W + L n − W )( h i ) = ( W t n + L n − W )( h i ) = 0 , i = 1 , · · · , N , since L n − W = L n W − L n + W = W ∆ n − L n + W , L n − W is a non-neg ativ e difference op erator of o rder < N , W t n + L n − W is a lso of or der < N . Then acco rding to the difference equation’s theory , W t n + L n − W is a zero op erator. 7 4.2 Dressing approac h for exDKPH W e now g eneralized the dres sing approa c h to exDKPH (9). W e hav e Lemma 3. Un der (23), if W satisfies (24) and W τ k = − L k − W + N X i =1 ψ i ∆ − 1 φ i W (28) then L satisfies (9a) and (9b). Pr o of. It is known that L satisfies (9a). W e hav e L τ k = W τ k ∆ W − 1 − W ∆ W − 1 W τ k W − 1 =( − L k − + X i ψ i ∆ − 1 φ i ) L + L ( L k − − X i ψ i ∆ − 1 φ i ) = [ B k + N X i =1 ψ i ∆ − 1 φ i , L ] . This dressing op erator W is constructed as follows: Let g i , ¯ g i satisfy g i,t n = ∆ n ( g i ) , g i,τ k = ∆ k ( g i ) (29a) ¯ g i,t n = ∆ n ( ¯ g i ) , ¯ g i,τ k = ∆ k ( ¯ g i ) , i = 1 , . . . , N . (29b) And let h i be the linear combination of g i and ¯ g i h i = g i + α i ( τ k ) ¯ g i i = 1 , . . . , N , (30) with the co efficient α i being a differentiable function of τ k . Suppo se h 1 , . . . , h N are still linearly indep enden t. Define ψ i = − ˙ α i W ( ¯ g i ) , φ i = ( − 1) N − i W rd(Γ h 1 , · · · , ˆ Γ h i , · · · , Γ h N ) W rd(Γ h 1 , · · · , Γ h N ) , i = 1 , . . . , N (31) where the ha t ˆ means rule out this term from the discrete W ronskian determi- nant, ˙ α i = dα i dτ k . W e have Prop osition 3. L et W b e define d by (26) and (30), L = W ∆ W − 1 , ψ i and φ i b e given by (31), then W , L , ψ i , φ i satisfy (24),(28) and ex DKPH (9). T o prov e it, we need several lemmas under the ab ov e assumptions. The firs t one is : Lemma 4. (The discr et e version of Oevel and Str ampp’s lemma [18]) W − 1 = N X i =1 h i ∆ − 1 φ i . 8 Pr o of. Note that φ 1 , . . . , φ N defined in (31) satisfy the linear equation N X i =1 ∆ j (Γ h i ) · φ i = δ j,N − 1 , j = 0 , 1 , · · · , N − 1 (32) where δ j,N − 1 is the Kr one cker’s delta symbol. Using pr o perties f ∆ − 1 = P j ≥ 0 ∆ − j − 1 ∆ j (Γ f ), we hav e N X i =1 h i ∆ − 1 φ i = N X i =1 ∞ X j =0 ∆ − j − 1 ∆ j (Γ( h i )) · φ i = ∞ X j =0 ∆ − j − 1 N X i =1 ∆ j (Γ( h i )) · φ i = N − 1 X j =0 ∆ − j − 1 δ j,N − 1 + ∞ X j = N ∆ − j − 1 N X i =1 ∆ j (Γ( h i )) · φ i = ∆ − N + O (∆ − N − 1 ) , So w e ha ve W X i h i ∆ − 1 φ i = 1 + ( W X i h i ∆ − 1 φ i ) − = 1 + X i W ( h i )∆ − 1 φ i = 1 . (33) This complete the pro of. Lemma 5. W ∗ ( φ i ) = 0 , for i = 1 , . . . , N . Pr o of. Lemma 1 implies that (∆ − 1 φ i W ) − = ∆ − 1 W ∗ ( φ i ) . (34) Using Lemma 4 and (10a), we hav e 0 = (∆ j W − 1 W ) − 1 = (∆ j N X i =1 h i ∆ − 1 φ i W ) − = ( N X i =1 ∆ j ( h i )∆ − 1 φ i W ) − = N X i =1 ∆ j ( h i )∆ − 1 W ∗ ( φ i ) , j = 0 , · · · , N − 1 . Solving the equatio ns with resp ect to ∆ − 1 W ∗ ( φ i ), we find ∆ − 1 W ∗ ( φ i ) = 0 . This implies W ∗ ( φ i ) = 0 . Lemma 6. The op er ator ∆ − 1 φ i W is a non-ne gative differ enc e op er ator and (∆ − 1 φ i W )( h j ) = δ ij , 1 ≤ i, j ≤ N . (35) Pr o of. Lemma 5 and (34) implies that ∆ − 1 φ i W is a non-nega tiv e difference op erator. W e define functions c ij = (∆ − 1 φ i W )( h j ), then ∆( c ij ) = φ i W ( h j ) = 0, which means c ij do es not dep end on the discr ete v a r iable n . F ro m Lemma 4, we find that N X i =1 ∆ k ( h i ) c ij = ∆ k ( X i ( h i ∆ − 1 φ i W )( h j )) = ∆ k ( W − 1 W )( h j ) = ∆ k ( h j ) , so c ij = δ ij . 9 Pro of of Pr opo sition 3. The pro of o f (2 4) is analo gous to the pro of of in the previous sectio n. F or (28), taking ∂ τ k to the identit y W ( h i ) = 0, using (29), (30), the definition (31) and Lemma 6, we find 0 =( W τ k )( h i ) + ( W ∆ k )( h i ) + ˙ α i W ( ¯ g i ) = ( W τ k )( h i ) + ( L k W )( h i ) − N X j =1 ψ j δ j i =( W τ k + L k − W − N X j =1 ψ j ∆ − 1 φ j W )( h i ) . Since the no n-negative difference o p erato r acting on h i in the la s t e xpression has degree < N , it ca n not annihilate N indep enden t functions unless the op e rator itself v anishes. Hence (28) is prov ed. Then Lemma 3 leads to (9b). The first equation in (9c) is easy to b e verified by a direct ca lculation, so it r emains to prov e the second equation in (9c). Firstly , w e see tha t ( W − 1 ) t n = − W − 1 W t n W − 1 = W − 1 ( L n − B n ) = ∆ n W − 1 − W − 1 B n . Then we substitute W − 1 = P h i ∆ − 1 φ i to this equality at b oth ends, w e hav e ( W − 1 ) t n = X ∆ n ( h i )∆ − 1 φ i + X h i ∆ − 1 φ i,t n = (∆ n W − 1 − W − 1 B n ) − = X ∆ n ( h i )∆ − 1 φ i − X h i ∆ − 1 B ∗ n ( φ i ) Then P h i ∆ − 1 φ i,t n = − P h i ∆ − 1 B ∗ n ( φ i ) implies that (9c) holds. 5 N-soliton solutions for exDKPH Using Pr opos ition 3, we c an find solutions to every equations in the exDKP H (9). Let us illustrate it by solving (13) and (15). F or (13), le t δ i = e λ i − 1 , κ i = e µ i − 1, we take the so lutio n of (29) as follows g i := ex p( lλ i + δ i t 1 + δ 2 i τ 2 ) = e ξ i , ¯ g i := exp( l µ i + κ i t 1 + κ 2 i τ 2 ) = e η i h i := g i + α i ( τ 2 ) ¯ g i = 2 √ α i exp( ξ i + η i 2 ) cosh(Ω i ) , Ω i = 1 2 ( ξ i − η i − ln α i ) . (36 ) Since L = W ∆ W − 1 = ∆ + f 0 + f 1 ∆ − 1 + · · · , we hav e f 0 = Res ∆ ( W ∆ W − 1 ∆ − 1 ) (37) where W is g iv en b y (26) and (3 6), then f 0 , ψ i and φ i given by (31) gives r ise to the N-soliton solution for (13). F or exa mple, we obtain 1-soliton solution for (13) with N = 1 as follows f 0 = exp( λ 1 + µ 1 2 )  cosh(Ω 1 + 2 θ 1 ) cosh(Ω 1 + θ 1 ) − cosh(Ω 1 + θ 1 ) cosh Ω 1  , θ 1 = λ 1 − µ 1 2 10 ψ 1 = − d √ α 1 dτ 2 ( e µ 1 − λ 1 ) exp ξ 1 + η 1 2 sech Ω 1 , φ 1 = e − ( λ 1 + µ 1 ) / 2 exp( − ξ 1 + η 1 2 ) 2 √ α 1 sech (Ω 1 + θ 1 ) . The 2-solito n s olution of (1 3) with N = 2 is given by f 0 = − ∆( w 1 ) = ( e λ 1 + e λ 2 )∆( v 1 v ) , ψ 1 = − ˙ α 1 v  1 + α 2 ( e µ 2 − e λ 1 )( e µ 1 − e µ 2 ) ( e λ 2 − e λ 1 )( e µ 1 − e λ 2 ) e χ 2  ( e µ 1 − e λ 1 )( e µ 1 − e λ 2 ) e η 1 , ψ 2 = − ˙ α 2 v  1 + α 1 ( e µ 1 − e λ 2 )( e µ 1 − e µ 2 ) ( e λ 2 − e λ 1 )( e µ 2 − e λ 1 ) e χ 2  ( e µ 2 − e λ 2 )( e µ 2 − e λ 1 ) e η 2 , φ 1 = Γ  1 + α 2 e χ 2 ( e λ 1 − e λ 2 ) v e − ξ 1  , φ 2 = Γ  1 + α 1 e χ 1 ( e λ 2 − e λ 1 ) v e − ξ 2  , with v = 1 + α 1 e λ 2 − e µ 1 e λ 2 − e λ 1 e χ 1 + α 2 e µ 2 − e λ 1 e λ 2 − e λ 1 e χ 2 + α 1 α 2 e µ 2 − e µ 1 e λ 2 − e λ 1 e χ 1 + χ 2 , v 1 = 1 + α 1 e 2 λ 2 − e 2 µ 1 e λ 2 − e λ 1 e χ 1 + α 2 e 2 µ 2 − e 2 λ 1 e λ 2 − e λ 1 e χ 2 + α 1 α 2 e 2 µ 2 − e 2 µ 1 e λ 2 − e λ 1 e χ 1 + χ 2 . It can be shown that the interaction b et w een the tw o so lutions is elastic. F or (15), we take the solution of (29) as follows g i := ex p( lλ i + δ i τ 1 + δ 2 i t 2 ) = e ξ i , ¯ g i := exp( l µ i + κ i τ 1 + κ 2 i t 2 ) = e η i h i := g i + α i ( τ 1 ) ¯ g i = 2 √ α i exp( ξ i + η i 2 ) cosh(Ω i ) . Then f 0 = Res ∆ ( W ∆ W − 1 ∆ − 1 ) , f 1 = Res ∆ ( W ∆ W − 1 ) together with ψ i and φ i given by (31) pr e sen ts the N-soliton solution for (15). Ac kn o wledgemen t This work is supp orted by Natio nal B a sic Resear c h Pr o gram of China (9 7 3 Pr o- gram) (200 7CB81480 0), China Postdocto r al Science F ounda tion funded pro ject (20080 430420) and National Natural Science F oundation of China (108 01083,10 671121). References [1] Da te E , Jimbo M, Kas hiwara M, and Miwa T 19 81 J. Phys. So c. J apan 50 3806. [2] V an de Leur J W 1998 J. Math. Phys. 39 28 33. 11 [3] K ac V G a nd v an de Leur J W 2003 J. Math. Phys. 4 4 3245. [4] Me l’nikov V K 198 7 Commun. Math. Phys. 112 639. [5] L iu X J 20 08 Phys. Le tt. A 372 38 19. 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