A new extended q-deformed KP hierarchy

A new extended q -deformed KP hierarch y Runliang Lin † , Xiao jun Liu ‡ , Y un b o Z eng † ∗ † Departmen t of M athematical Sciences, Tsingh ua Univ er sit y , Be ijing 10008 4, P . R. China ‡ Departmen t of Applied Mathematics, China Agricultural Univ ersit y , Beijing 100083, P .R. C hina Abstract A metho d is prop osed in this pap er to construct a new extended q -deformed KP ( q -KP) hiearc h y and its Lax r epresen tation. This new extended q -KP h ie rarc hy conta ins t w o t yp es of q -deformed KP equa- tion with self-consisten t sources, and its tw o kinds of red uctio ns give the q -deformed Gelfand-Dic k ey hierarch y with self-consisten t sources and the constrained q -deformed KP hierarc h y , w hic h include t w o t yp es of q -deformed KdV equation w it h sources and tw o t yp es of q -deformed Boussinesq equation with sources. All of th e se results reduce to the classical ones wh en q goes to 1. This pro vides a general wa y to construct (2+1)- and (1+1)-dimensional q -deformed soliton equations with sour ce s an d their Lax representa tions. P A CS num b ers: 02.30.Ik 1 In tro duction In recen t y ears, the q -deformed integrable systems attra cte d man y interests b oth in mathematics and in ph ysics [1–20]. The deformation is p erformed b y using the q -deriv a t iv e ∂ q to t a k e the place of ordinary de riv ative ∂ x in the classical systems, where q is a para meter, and t he q -deformed inte grable systems reco v er the class ical ones as q → 1. The q -deformed N -th KdV ∗ E-mail addr esses: rlin@math.tsing h ua .edu.cn (R.L. Lin), lxj98@mails .tsingh ua.edu.cn (X.J. Liu), yzeng@ math.tsingh ua.edu.cn (Y.B. Ze ng). 1 ( q -NKdV or q -Gelfa nd- Dic k ey) hierarc h y , the q -deformed KP ( q - KP) hierar- c h y , and the q -AKNS-D hierarc h y were constructed, and some of their in- tegrable structures were also studied, suc h as the infinite conserv ation la ws, bi-Hamiltonian struc ture, tau function, symmetries, B¨ ac klund transforma- tion (see [5 , 11, 14, 15, 19, 2 0] and the references therein). Multi-comp onen t g eneralizatio n of an in tegrable mo del is a v ery impo r- tan t sub jec t [21–29]. F or example, the multi-component KP (mcKP) hier- arc h y giv en in [21] con tains many ph ysically rele v ant nonlinear in tegrable systems , suc h as Dav ey-Stew artson equation, t w o-dimensional T o da la t t ice and three-w a v e resonan t inte raction ones. Another ty p e of coupled in tegrable systems is the soliton equation with self-consisten t sources, which has many ph ysical applications and can b e obta ined b y coupling some suitable differ- en tial equations to the original soliton equation [30–37]. V ery recen tly , w e prop osed a systematical pro cedure to construct a new extended K P hierarc h y and its Lax represen tatio n [38]. This new extended KP hierarc h y con tains t w o types of KP equation with self-consisten t sources (KPSCS-I and KPSCS- I I), and its t w o kinds of reductions giv e t he Gelfand-D ic k ey hierarc h y with self-consisten t sources [39] and the k -constrained KP hierarc hy [40 , 41]. In fact, the approac h whic h w e prop osed in [3 8] in the framew or k o f Sato t heory can b e applied to construct many other extended (2+1)-dimensional soliton hierarc hies, suc h as BKP hierarch y , CKP hierarc h y and DKP hierarc hy , and pro vides a g e neral w ay to obta in (2+1)-dimensional and (1 +1)-dimensional in tegrable soliton hierarc hies with self-consisten t sources. The KdV equation with self-consisten t sources and the KP equation with self-consiste n t sources can describ e the in teraction of long and short w a v es (see [30 –37] and the referenc es therein). In con trast with the w ell- studied KdV and KP equation with self-consisten t sources, the q -Gelfand- Dic k ey hierarch y with self-consisten t sources a nd the q -KP hierarch y with self-consisten t sources ha v e not b een in v estigated ye t. It is ineresting to con- sider the case of the algebra of q -pseudo-differen tial op erator, and to see if our approach could b e generalized to construct new extended q -deformed in- tegrable systems, which w ould enable us to find tw o types of new q -deformed soliton equation with sources in a systematic w ay . In this pap er, w e will give a systematical pro cedure to construct a new extended q -defo r med KP ( q -KP) hierarc h y and its L a x represen tat io n. First, we define a new v ector filed ∂ τ k b y a linear com bination of all v ector fields ∂ t n in ordinary q -deformed KP hierarc h y , then w e introduce a new Lax t yp e equation whic h consists of the τ k -flo w and the ev olutio ns of w a v e functions. Under the ev olutions of w av e 2 functions, the comm utativit y of ∂ τ k -flo w and ∂ t n -flo ws giv es rise to a new extended q -KP hierarch y . This new extended q - KP hierar ch y con tains tw o t yp es of q -deformed KP equation with self-consisten t sources ( q -KPSCS-I and q -KPSCS-I I), and its t w o kinds of reductions giv e t he q -deformed Gelfand- Dic k ey hierar ch y with self-consisten t sources and the constrained q -deformed KP hierarc h y , whic h are some (1 + 1)-dimensional q - defo r med soliton equa- tion with self-consisten t sources, e.g., tw o types o f q -deformed KdV equation with self-consisten t sources ( q -KdVSCS-I and q -KdVSCS-I I) and t w o types of q -deformed Boussines q equation with self-consisten t sources ( q -BESCS-I and q -BESCS-I I). The q -KdVSCS-II is just the q -deformed Y a jima- Oik aw a equation. All of these results reduce to the classical ones when q → 1. T h us, the metho d pro posed in this pap er is a general wa y to find the (1 + 1) - and (2 + 1)-dimensional q -deformed soliton equation with self-consisten t sources and their Lax represen tations. It should b e noticed that a general setting of “pseudo-differen tial” op erators on regular time scales has been prop osed to construct some in tegrable systems [42 , 43], where the q -differen tial op erator is just a particular case. Our pap er will b e organized as follo ws. In section 2, w e will recall some notatio ns in the q -calculus and construct the new ex- tended q -KP hierarch y , and then t w o t yp es of q -deformed KP equation with sources will b e presen ted. In section 3, the tw o kinds of reductions for the new extended q -KP hierarch y will b e considered, a nd some (1 + 1)- dimensional q -deformed soliton equation with self-consisten t sources will b e deduced. In section 4, some conclusions will b e giv en. 2 New extended q -defor m ed KP hierarc h y In this section, w e will give a pro cedure to construct a new extended q - KP hierarc h y and its Lax represen tation. Then, as the examples, tw o types of q -deformed KP equation with self-consisten t sources ( q -KPSCS-I and q - KPSCS-I I) will b e presen t e d explicitly . The q -deformed differential op erator ∂ q is defined a s ∂ q ( f ( x )) = f ( q x ) − f ( x ) x ( q − 1) , whic h reco vers the ordinary differen tiation ∂ x ( f ( x )) as q → 1. Let us define the q -shift op erator θ as θ ( f ( x )) = f ( q x ) . 3 Then w e hav e the q - defor me d Leibnitz rule ∂ n q f = X k ≥ 0  n k  q θ n − k ( ∂ k q f ) ∂ n − k q , n ∈ Z , where the q -n um b er and the q - binomial are defined by ( n ) q = q n − 1 q − 1 ,  n k  q = ( n ) q ( n − 1) q · · · ( n − k + 1) q (1) q (2) q · · · ( k ) q ,  n 0  q = 1 . F or a q -pseudo-differen tia l op erator ( q -PDO) of the form P = n X i = −∞ p i ∂ i q , w e decomp ose P in to the differen tial part and the integral part as follo ws P + = X i ≥ 0 p i ∂ i q , P − = X i ≤− 1 p i ∂ i q . The conjugate o peration “ ∗ ” fo r P is defined by P ∗ = X i ( ∂ ∗ q ) i p i , ∂ ∗ q = − ∂ q θ − 1 = − 1 q ∂ 1 q . The q -KP hierarch y is defined b y t he Lax equation (see, e.g., [1 7 ]) ∂ t n L = [ B n , L ] , B n = L n + , (1) with Lax op erator of the for m L = ∂ q + ∞ X i =0 u i ∂ − i q . (2) According t o the Sato theory , w e can express the Lax op erator as a dressed op erator L = S ∂ q S − 1 , (3) where S = 1 + ∞ P i =1 S i ∂ − i q is called the Sato op erator and S − 1 is its formal in v erse. The Lax equation (1) is equiv alen t to the Sato equation S t n = − ( L n ) − S. (4) 4 The q -w av e function w q ( x, t ; z ) and q -adjoin t wa v e function w ∗ ( x, t ; z ) (here t = ( t 1 , t 2 , t 3 , . . . )) are defined as f o llo ws w q = S e q ( xz ) exp ∞ X i =1 t i z i ! , (5a) w ∗ = ( S ∗ ) − 1 | x/q e 1 /q ( − xz ) exp − ∞ X i =1 t i z i ! , (5b) where the notation P | x/t = P i p i ( x/t ) t i ∂ i q (for P = P i p i ( x ) ∂ i q ) is used, and e q ( x ) = exp ∞ X k =1 (1 − q ) k k (1 − q k ) x k ! . It is easy to sho w that w q and w ∗ q satisfy the following linear systems Lw q = z w q , ∂ w q ∂ t n = B n w q , L ∗ | x/q w ∗ q = z w ∗ q , ∂ w ∗ q ∂ t n = − ( B n | x/q ) ∗ w ∗ q . It can b e prov ed that [14] T ( z ) − ≡ X i ∈ Z L i − z − i − 1 = w q ∂ − 1 q θ ( w ∗ q ) . (6) F or any fixed k ∈ N , w e define a new v aria ble τ k whose v ector field is ∂ τ k = ∂ t k − N X i =1 X s ≥ 0 ζ − s − 1 i ∂ t s , where ζ i ’s are arbitrary distinct non-zero parameters. The τ k -flo w is giv en b y L τ k = ∂ t k L − N X i =1 X s ≥ 0 ζ − s − 1 i ∂ t s L = [ B k , L ] − N X i =1 X s ≥ 0 ζ − s − 1 i [ B s , L ] = [ B k , L ] + N X i =1 X s ∈ N ζ − s − 1 i [ L s − , L ] = [ B k , L ] + N X i =1 X s ∈ Z ζ − s − 1 i [ L s − , L ] . 5 Define ˜ B k b y ˜ B k = B k + N X i =1 X s ∈ Z ζ − s − 1 i L s − , whic h, according to (6), can b e written as ˜ B k = B k + N X i =1 w q ( x, t ; ζ i ) ∂ − 1 q θ ( w ∗ q ( x, t ; ζ i )) . By setting φ i = w q ( x, t ; ζ i ), ψ i = θ ( w ∗ q ( x, t ; ζ i )), w e hav e ˜ B k = B k + N X i =1 φ i ∂ − 1 q ψ i , (7a) where φ i and ψ i satisfy the following equations φ i,t n = B n ( φ i ) , ψ i,t n = − B ∗ n ( ψ i ) , i = 1 , · · · , N . (7b) No w w e in tro duce a new La x t yp e equation giv en by L τ k = [ B k + N X i =1 φ i ∂ − 1 q ψ i , L ] . (8a) with φ i,t n = B n ( φ i ) , ψ i,t n = − B ∗ n ( ψ i ) , i = 1 , · · · , N . (8b) W e hav e the following lemma Lemma 1. [ B n , φ ∂ − 1 q ψ ] − = B n ( φ ) ∂ − 1 q ψ − φ∂ − 1 q B ∗ n ( ψ ) . Pr o of. Without lo ss o f g enerality , we consider a monomial: P = a∂ n q ( n ≥ 1). Then [ P , φ∂ − 1 q ψ ] − = a ( ∂ n q ( φ )) ∂ − 1 q ψ − ( φ ∂ − 1 q ψ a∂ n q ) − . Notice that the second term can b e rewritten in the following w a y ( φ∂ − 1 q ψ a∂ n q ) − = φ ( θ − 1 ( ψ a )) ∂ n − 1 q − φ∂ − 1 q ( ∂ q θ − 1 ( aψ )) ∂ n − 1 q ) − = ( φ∂ − 1 q ( − ∂ q θ − 1 ( aψ )) ∂ n − 1 q ) − = · · · = φ∂ − 1 q  ( − ∂ q θ − 1 ) n ( aψ )  = φ ∂ − 1 q P ∗ ( ψ ) , then the lemma is prov ed. 6 Prop osition 1. (1) and (8 ) give rise to the fo l lowing new extende d q -deforme d KP hier ar chy B n,τ k − ( B k + P N i =1 φ i ∂ − 1 q ψ i ) t n + [ B n , B k + P N i =1 φ i ∂ − 1 q ψ i ] = 0 (9a) φ i,t n = B n ( φ i ) , (9b) ψ i,t n = − B ∗ n ( ψ i ) , i = 1 , · · · , N . (9c) Pr o of. W e will show that under (8b), (1) and (8a) give rise to (9a). F or con- v enience, w e a ssume N = 1, a nd denote φ 1 and ψ 1 b y φ and ψ , resp ectiv ely . By (1), (8 ) and Lemma 1, we hav e B n,τ k = ( L n τ k ) + = [ B k + φ∂ − 1 q ψ , L n ] + = [ B k + φ∂ − 1 q ψ , L n + ] + + [ B k + φ∂ − 1 q ψ , L n − ] + = [ B k + φ∂ − 1 q ψ , L n + ] − [ B k + φ∂ − 1 q ψ , L n + ] − + [ B k , L n − ] + = [ B k + φ∂ − 1 q ψ , B n ] − [ φ∂ − 1 q ψ , B n ] − + [ B n , L k ] + = [ B k + φ∂ − 1 q ψ , B n ] + B n ( φ ) ∂ − 1 q ψ − φ∂ − 1 q B ∗ n ( ψ ) + B k ,t n = [ B k + φ∂ − 1 q ψ , B n ] + ( B k + φ∂ − 1 q ψ ) t n . Under (9b) and (9c), the Lax represen tation for (9 a) is giv en by Ψ τ k = ( B k + N X i =1 φ i ∂ − 1 q ψ i )(Ψ) , (10a) Ψ t n = B n (Ψ) . (10b) Remark 1 The main step in our approa ch is to define a new Lax equation (8). F or the extended KP hierarch y in [38], a similar form ula lik e (8) can b e motiv ated by the w ell- k now n k -constrain t of KP hierarch y , whic h is obtained b y imp osing L k = B k + N P i =1 φ i ∂ − 1 ψ i . Here, the form ula (8) can also b e moti- v ated by the k -constrain t of q -KP hierarc hy as giv en in [14]. This enables us to obtain the k -constrained q -KP hierarc hy and the q -Gelfand-Dick ey hier- arc h y with sources by dropping the τ k -dep endenc e and t n -dep endenc e in the new extended q -KP hierarc h y (9) resp ectiv ely (see Section 3). Remark 2 When taking φ i = ψ i = 0, i = 1 , . . . , N , then the extended q -KP hierarc h y (9) reduces to the q -KP hierarch y . 7 Remark 3 Inte grable systems can b e constructed from the algebra of “pseudo- differen tial” op erators on regular t ime scales in [4 2 , 43], where the algebra of q -“pseudo-differen tial” op erator is a pa r t ic ular case. In fact, our a ppro ac h for constructing new extended in tegrable systems can also b e generalized to the general setting as in [42, 43]. F or conv enience, w e write out some o perators here B 1 = ∂ q + u 0 , B 2 = ∂ 2 q + v 1 ∂ q + v 0 , B 3 = ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 , φ i ∂ − 1 q ψ i = r i 1 ∂ − 1 q + r i 2 ∂ − 2 q + r i 3 ∂ − 3 q + . . . , i = 1 , . . . , N , where v 1 = θ ( u 0 ) + u 0 , v 0 = ( ∂ q u 0 ) + θ ( u 1 ) + u 2 0 + u 1 , v − 1 = ( ∂ q u 1 ) + θ ( u 2 ) + u 0 u 1 + u 1 θ − 1 ( u 0 ) + u 2 , s 2 = θ ( v 1 ) + u 0 , s 1 = ( ∂ q v 1 ) + θ ( v 0 ) + u 0 v 1 + u 1 , s 0 = ( ∂ q v 0 ) + θ ( v − 1 ) + u 0 v 0 + u 1 θ − 1 ( v 1 ) + u 2 . r i 1 = φ i θ − 1 ( ψ i ) , r i 2 = − 1 q φ i θ − 2 ( ∂ q ψ i ) , r i 3 = 1 q 3 φ i θ − 3 ( ∂ 2 q ψ i ) . and v − 1 comes from L 2 = B 2 + v − 1 ∂ − 1 q + v − 2 ∂ − 2 q + · · · . Then, one can compute the follow ing commutators [ B 2 , B 3 ] = f 2 ∂ 2 q + f 1 ∂ q + f 0 , [ B 2 , φ i ∂ − 1 q ψ i ] = g i 1 ∂ q + g i 0 + . . . , [ B 3 , φ i ∂ − 1 q ψ i ] = h i 2 ∂ 2 q + h i 1 ∂ q + h i 0 + . . . , i = 1 , . . . , N , where f 2 = ∂ 2 q s 2 + ( q + 1) θ ( ∂ q s 1 ) + θ 2 ( s 0 ) + v 1 ∂ q s 2 + v 1 θ ( s 1 ) + v 0 s 2 − ( q 2 + q + 1) θ ( ∂ 2 q v 1 ) − ( q 2 + q + 1) θ 2 ( ∂ q v 0 ) − ( q + 1) s 2 θ ( ∂ q v 1 ) − s 2 θ 2 ( v 0 ) − s 1 θ ( v 1 ) − s 0 , f 1 = ∂ 2 q s 1 + ( q + 1) θ ( ∂ q s 0 ) + v 1 ∂ q s 1 + v 1 θ ( s 0 ) + v 0 s 1 − ∂ 3 q v 1 − ( q 2 + q + 1) θ ( ∂ 2 q v 0 ) − s 2 ∂ 2 q v 1 − ( q + 1) s 2 θ ( ∂ q v 0 ) − s 1 ∂ q v 1 − s 1 θ ( v 0 ) − s 0 v 1 , 8 f 0 = ∂ 2 q s 0 + v 1 ∂ q s 0 − ∂ 3 q v 0 − s 2 ∂ 2 q v 0 − s 1 ∂ q v 0 , g i 1 = θ 2 ( r i 1 ) − r i 1 , g i 0 = ( q + 1) θ ( ∂ q r i 1 ) + θ 2 ( r i 2 ) + v 1 θ ( r i 1 ) − r i 1 θ − 1 ( v 1 ) − r i 2 , h i 2 = θ 3 ( r i 1 ) − r i 1 , h i 1 = ( q 2 + q + 1) θ 2 ( ∂ q r i 1 ) + θ 3 ( r i 2 ) + s 2 θ 2 ( r i 1 ) − r i 1 θ − 1 ( s 2 ) . h i 0 = ( q 2 + q + 1) θ ( ∂ 2 q r i 1 ) + ( q 2 + q + 1) θ 2 ( ∂ q r i 2 ) + θ 3 ( r i 3 ) + ( q + 1) s 2 θ ( ∂ q r i 1 ) + s 2 θ 2 ( r i 2 ) + s 1 θ ( r i 1 ) − r i 1 θ − 1 ( s 1 ) + 1 q r i 1 θ − 2 ( ∂ q s 2 ) − r i 2 θ − 2 ( s 2 ) − r i 3 . No w, w e list some examples in t he new extended q -KP hierarc h y ( 9). Example 1 (The first type of q -KPSCS ( q - KPSC S-I)) . F or n = 2 and k = 3 , (9) yields the first typ e of q -deforme d KP e quation with self-c ons i s t ent so ur c es ( q -KPSCS-I) as fol l o ws − ∂ s 2 ∂ t 2 + f 2 = 0 , (11a) ∂ v 1 ∂ τ 3 − ∂ s 1 ∂ t 2 + f 1 + P N i =1 g i 1 = 0 , (11b) ∂ v 0 ∂ τ 3 − ∂ s 0 ∂ t 2 + f 0 + P N i =1 g i 0 = 0 , (11c) φ i,t 2 = B 2 ( φ i ) , ψ i,t 2 = − B ∗ 2 ( ψ i ) , i = 1 , . . . , N . (11d) The L ax r epr esentation for (11) is Ψ τ 3 = ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 + N X i =1 φ i ∂ − 1 q ψ i )(Ψ) , (12a) Ψ t 2 = ( ∂ 2 q + v 1 ∂ q + v 0 )(Ψ) . (12b) L et q → 1 and u 0 ≡ 0 , then the q -KPSCS-I (11) r e duc es to the first typ e of KP e quation w i th self-c onsistent sour c es (KPSCS-I) which r e ads as [30 , 31] u 1 ,t 2 − u 1 ,xx − 2 u 2 ,x = 0 , (13a) 2 u 1 ,τ 3 − 3 u 2 ,t 2 − 3 u 1 ,x,t 2 + u 1 ,xxx + 3 u 2 ,xx − 6 u 1 u 1 ,x + 2 ∂ x N X i =1 φ i ψ i = 0 , (13b) φ i,t 2 − φ i,xx − 2 u 1 φ i = 0 , (13c) ψ i,t 2 + ψ i,xx + 2 u 1 ψ i = 0 , i = 1 , . . . , N . (1 3 d) 9 Example 2 (The second ty p e of q -deformed KPSC S ( q -KPSCS-I I)) . F or n = 3 and k = 2 , (9) yields the se c ond typ e of q -deforme d KP e quation with self-c ons i s tent sour c es ( q -KPSCS-II ) a s fol lows ∂ s 2 ∂ τ 2 − f 2 + P N i =1 h i 2 = 0 , (14a) ∂ s 1 ∂ τ 2 − ∂ v 1 ∂ t 3 − f 1 + P N i =1 h i 1 = 0 , (14b) ∂ s 0 ∂ τ 2 − ∂ v 0 ∂ t 3 − f 0 + P N i =1 h i 0 = 0 , ( 1 4c) φ i,t 3 = B 3 ( φ i ) , ψ i,t 3 = − B ∗ 3 ( ψ i ) , i = 1 , . . . , N . (14d) The L ax r epr esentation for (14) is Ψ τ 2 = ( ∂ 2 q + v 1 ∂ q + v 0 + N X i =1 φ i ∂ − 1 q ψ i )(Ψ) , (15a) Ψ t 3 = ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 )(Ψ) . (15b) L et q → 1 and u 0 ≡ 0 , then the q -KPSCS-II (14) r e duc es to the se c ond typ e of KP e q uation with se lf-c onsistent sour c es (KPSCS-II) wh i c h r e ads as [30] u 1 ,τ 2 − u 1 ,xx − 2 u 2 ,x + ∂ x N X i =1 φ i ψ i = 0 , (16a) 3 u 2 ,τ 2 + 3 u 1 ,x,τ 2 − 2 u 1 ,t 3 − u 1 ,xxx + 6 u 1 u 1 ,x − 3 u 2 ,xx + 3 ∂ x N X i =1 φ i,x ψ i = 0 , (16b) φ i,t 3 − φ i,xxx − 3 u 1 φ i,x − 3( u 1 ,x + u 2 ) φ i = 0 , (16c) ψ i,t 3 − ψ i,xxx − 3 u 1 ψ i,x + 3 u 2 ψ i = 0 , i = 1 , . . . , N . (16d) 3 Reduction s The new extended q -deformed KP hierarch y (9) admits reductions to sev eral w ell-kno wn q -deformed (1 + 1)- dime nsional systems. 3.1 The n -reduction of (9 ) The n -reduction is giv en by L n = B n or L n − = 0 , (1 7 ) 10 then (5) implies that B n ( φ i ) = L n φ i = ζ n i φ i , (18a) − B ∗ n ( ψ i ) = − L n ∗ ψ i = − ζ n i ψ i . (18b) By using Lemma 1 and (18), w e can see that the constrain t (17) is inv a rian t under the τ k flo w ( L n − ) τ k = [ B k , L n ] − + N X i =1 [ φ i ∂ − 1 q ψ i , L n ] − = [ B k , L n − ] − + N X i =1 [ φ i ∂ − 1 q ψ i , L n + ] − + N X i =1 [ φ i ∂ − 1 q ψ i , L n − ] − = N X i =1 [ φ i ∂ − 1 q ψ i , B n ] − = − N X i =1 ( φ i,t n ∂ − 1 q ψ i + φ i ∂ − 1 q ψ i,t n ) = − N X i =1 ( ζ n i φ i ∂ − 1 q ψ i − ζ n i φ i ∂ − 1 q ψ i ) = 0 . (19) The equations (17) and (4) imply that S t n = 0 , so ( L k ) t n = 0 , whic h together with (19) means that one can drop t n dep ende ncy from (9) and obtain B n,τ k = [( B n ) k n + + N X i =1 φ i ∂ − 1 q ψ i , B n ] , (20a) B n ( φ i ) = ζ n i φ i , (20b) B ∗ n ( ψ i ) = ζ n i ψ i , i = 1 , · · · , N . (20c) The system (20) is the q -deformed Gelfa nd-Dic ke y hierarch y with self-consisten t sources. Example 3 (The firs ty ep of q -deformed KdVSCS ( q -KdVSCS-I)) . F o r n = 2 and k = 3 , (20) pr esents the first typ e of q -deforme d Kd V e q u ation with self- c onsistent sour c es ( q -KdVSCS- I ) v 1 ,τ 3 + f 1 + P N i =1 g i 1 = 0 , (21a) v 0 ,τ 3 + f 0 + P N i =1 g i 0 = 0 , (21b) u 2 + θ ( u 2 ) + ∂ q ( u 1 ) + u 0 u 1 + u 1 θ − 1 ( u 0 ) = 0 , (21c) ( ∂ 2 q + v 1 ∂ q + v 0 )( φ i ) − ζ 2 φ i = 0 , (21d) ( ∂ 2 q + v 1 ∂ q + v 0 ) ∗ ( ψ i ) − ζ 2 ψ i = 0 , i = 1 , · · · , N , (21e) 11 with the L ax r epr esentation Ψ τ 3 = ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 + N X i =1 φ i ∂ − 1 q ψ i )(Ψ) , ( ∂ 2 q + v 1 ∂ q + v 0 )(Ψ) = λ Ψ , u 2 + θ ( u 2 ) + ∂ q ( u 1 ) + u 0 u 1 + u 1 θ − 1 ( u 0 ) = 0 . L et q → 1 an d u 0 ≡ 0 , then the q -KdVSCS-I (21) r e duc es to the first typ e o f KdV e quation wi th self-c onsistent sour c es (KdVSCS-I) whi c h r e ads as u 2 = − 1 2 u 1 ,x , u 1 ,τ 3 − 3 u 1 u 1 ,x − 1 4 u 1 ,xxx + ∂ x P N i =1 φ i ψ i = 0 , φ i,xx + 2 u 1 φ i − ζ 2 φ i = 0 , ψ i,xx + 2 u 1 ψ i − ζ 2 ψ i = 0 , i = 1 , · · · , N . The first typ e of KdV e quation w ith self-c ons i s t ent sour c es (KdVSCS-I) c an b e solve d by the invers e sc attering metho d [32, 33] or by the Darb oux tr ans- formation (se e [3 5] and the r efer enc es ther ei n). Example 4 (The first ty p e of q -BESCS ( q -BESCS-I)) . F or n = 3 and k = 2 , (20) p r ese nts the first typ e of q -deforme d Boussinesq e quation with self- c onsistent sour c es ( q -BESC S -I) s 2 ,τ 2 − f 2 + P N i =1 h i 2 = 0 , (22a) s 1 ,τ 2 − f 1 + P N i =1 h i 1 = 0 , (22b) s 0 ,τ 2 − f 0 + P N i =1 h i 0 = 0 , (22c) ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 )( φ i ) − ζ 3 φ i = 0 , (22d) ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 ) ∗ ( ψ i ) − ζ 3 ψ i = 0 , i = 1 , · · · , N , (22e) with the L ax r epr esentation Ψ τ 2 = ( ∂ 2 q + v 1 ∂ q + v 0 + N X i =1 φ i ∂ − 1 q ψ i )(Ψ) , ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 )(Ψ) = λ Ψ . L et q → 1 and u 0 ≡ 0 , then the q -BES C S-I (22) r e duc es to the first typ e of Boussinesq e quation with self-c onsis tent sour c es (BESCS-I ) which r e ads as − 2 u 2 ,x − u 1 ,xx + u 1 ,τ 2 + ∂ x P N i =1 φ i ψ i = 0 , 3 u 2 ,τ 2 − 3 u 2 ,xx + 3 u 1 ,x,τ 2 + 6 u 1 u 1 ,x − u 1 ,xxx + 3 ∂ x P N i =1 φ i,x ψ i = 0 , φ i,xxx + 3 u 1 φ i,x + 3( u 1 ,x + u 2 ) φ i − ζ 3 φ i = 0 , ψ i,xxx + 3 u 1 ψ i,x − 3 u 2 ψ i + ζ 3 ψ i = 0 , i = 1 , · · · , N . 12 3.2 The k -constrained h i erarc h y of (9) The k -constrain t is giv en by [40, 41] L k = B k + N X i =1 φ i ∂ − 1 q ψ i . (23) By using the ab ov e k -constrain t, it can b e pro v ed that L a nd B n are inde- p enden t of τ k . By dropping τ k dep ende ncy from (9), w e get  B k + P N i =1 φ i ∂ − 1 q ψ i  t n = h ( B k + P N i =1 φ i ∂ − 1 q ψ i ) n k + , B k + P N i =1 φ i ∂ − 1 q ψ i i , (24a) φ i,t n = ( B k + P N j =1 φ j ∂ − 1 q ψ j ) n k + ( φ i ) , (24b) ψ i,t n = − ( B k + P N j =1 φ j ∂ − 1 q ψ j ) n k ∗ + ( ψ i ) , i = 1 , · · · , N , (24c) whic h is the constrained q -deformed KP hierarc h y . Some solutions of the con- strained q -deformed KP hierarch y can b e represen ted b y q -deformed W ron- skian determinan t (see [20] and the r eferences therein). Remark 4 In [42, 43], the k -constrained q -KP hierarc h y can b e constructed from the q -KP hierarc h y b y imp osing the k -constrain t. Here, the k -constrained q -KP hierarc h y is obta ine d directly from the extende d q -KP hierarch y (9) b y dropping the τ k dep ende nce due to the k -constraint. Example 5 (The second t yp e of q -K dVS CS ( q -KdVSCS-I I)) . F or n = 3 and k = 2 , (24) gives rise to the se c ond typ e of q -def o rm e d KdV e quation with self-c ons i s tent sour c es ( q -KdVSCS-II ). v 1 ,t 3 + f 1 − P N i =1 h i 1 = 0 , (25a) v 0 ,t 3 + f 0 − P N i =1 h i 0 = 0 , (25b) u 2 + θ ( u 2 ) + ∂ q ( u 1 ) + u 0 u 1 + u 1 θ − 1 ( u 0 ) − P N i =1 r i 1 = 0 , (25c) φ i,t 3 = ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 )( φ i ) , (25d) ψ i,t 3 = − ( ∂ 3 q + s 2 ∂ 2 q + s 1 ∂ q + s 0 ) ∗ ( ψ i ) , i = 1 , · · · , N . (25e) L et q → 1 and u 0 ≡ 0 , then the q -KdVSCS-II (25) r e duc es to the se c ond typ e of KdV e quation with se l f - c onsistent sour c es (KdVSCS-II or Y ajima-Oikaw a 13 e quation) which r e ads as u 2 = − 1 2 u 1 ,x + 1 2 P N i =1 φ i ψ i , u 1 ,t 3 = 1 4 u 1 ,xxx + 3 u 1 u 1 ,x + 3 4 P N i =1 ( φ i,xx ψ i − φ i ψ i,xx ) , φ i,t 3 = φ i,xxx + 3 u 1 φ i,x + 3 2 u 1 ,x φ i + 3 2 φ i P N j =1 φ j ψ j , ψ i,t 3 = ψ i,xxx + 3 u 1 ψ i,x + 3 2 u 1 ,x ψ i − 3 2 ψ i P N i =1 φ j ψ j , i = 1 , · · · , N . Example 6 (The second type of q -BESCS ( q - BE SCS-I I)) . F or n = 2 and k = 3 , (24) gives rise to the se c ond typ e of q -deforme d Boussinesq e quation with self-c ons i s tent sour c es ( q -BES CS-II)) s 2 ,t 2 − f 2 = 0 , (26a) s 1 ,t 2 − f 1 − P N i =1 g i 1 = 0 , ( 2 6b) s 0 ,t 2 − f 0 − P N i =1 g i 0 = 0 , (26c) φ i,t 2 = ( ∂ 2 q + v 1 ∂ q + v 0 )( φ i ) , (26d) ψ i,t 2 = − ( ∂ 2 q + v 1 ∂ q + v 0 ) ∗ ( ψ i ) , i = 1 , · · · , N . (26e) L et q → 1 and u 0 ≡ 0 , then the q -BESCS-II (2 6) r e d uc es to the se c ond typ e of Boussinesq e quation with se l f - c onsistent sour c es (BESCS-I I) which r e ads as − 2 u 2 ,x − u 1 ,xx + u 1 ,t 2 = 0 , 3 u 2 ,t 2 − 3 u 2 ,xx + 3 u 1 ,x,t 2 + 6 u 1 u 1 ,x − u 1 ,xxx − 2 ∂ x P N i =1 φ i ψ i = 0 , φ i,t 2 = φ i,xx + 2 u 1 φ i , ψ i,t 2 = − ψ i,xx − 2 u 1 ψ i , i = 1 , · · · , N . 4 Conclus ions A metho d is prop osed in this pap er to construct a new extended q -deformed KP ( q -KP) hiearc hy and its Lax r epresen tation. This new extended q -KP hierarc h y con tains t w o t yp es of q -deformed K P equation with self-consisten t sources ( q -KPSCS-I and q -KPSCS-I I), and its t wo kinds of reductions giv e the q -deformed Gelfand- D ic k ey hierarch y with self-consisten t sources a nd the constrained q -deformed KP hierarc h y . Th us, the reductions of the new 14 extended q -KP hierarc h y may giv e some q -deformed (1 + 1)-dimens ional soli- ton equation with self-consisten t sources, e.g., the tw o types of q -deformed KdV equation with self-consisten t sources (including q -deformed Y a jima- Oik a w a equation) a nd t w o t yp es of q -deformed Boussinesq equation with self-consisten t sources. All of these results reduce to the classical ones when q → 1. The metho d pro posed in this pap er is a general w a y to find (1 + 1)- a nd (2 + 1)-dimensional q -deformed soliton equation with self-consisten t sources and their Lax represen ta t io ns . 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