A new extended matrix KP hierarchy and its solutions

A new extended matrix KP hierarc h y and its solutio ns Y ehui Huang ∗ 1 , Xiaojun Liu ∗∗ 2 , Y uqin Y ao ∗∗ 3 and Y unb o Zeng ∗ 4 ∗ Dep artment of Mathematic al Scienc e, Tsinghua University, Beijing, 100084 , PR China ∗∗ Dep artment of Applie d Mathematics, China A gricultur al University, Beijing, 10008 3, PR China Abstract With the square eigenfunctions symmetry constraint, w e intro duce a new extended matrix KP hierarch y and its Lax representation from the matrix KP hierarch y b y adding a new τ B flow. The extended KP hiera rch y con tains t wo time series t A and τ B and eig e nfunctions and a djo int eigenfunctions a s comp onents. The extended matrix KP hier arch y and its t A -reduction and τ B reduction include tw o types of matr ix KP hierarch y with self-consistent sources and t wo t yp es of (1+1)-dimensiona l reduced matrix KP hierarch y with self-consis tent sources. In pa rticular, the first type and second type of the 2+1 AKNS equation and the Davey-Stew ar ts o n equatio n with self-consistent sources are deduced from the extended matrix KP hier arch y . The generalize d dressing approach for solving the extended matrix K P hierarch y is pro p o sed and s o me solutio ns are presented. The soliton solutions of tw o types of 2 +1-dimensiona l AK NS eq uation with self- consistent sour ces a nd tw o types o f Dav ey- Stew a rtson equation with self-consis ten t sourc e s ar e studied. P A CS num b ers: 02.30.IK KEYW ORDS: Extended matrix KP hierarch y; Lax represen tation; Generalized d ressing metho d; 2+1 AKNS equation with self-consisten t sources; DS equ ation with self-consisten t sour ces 1 In tro du ction Generalizati ons of Kadomtse v-Petvia shvili (KP) hierarch y attracts lots of interests from p h ysical and mathematical p oints of view 1 − 9 . On e kind of generalizatio n is the so called m ulti-comp onent KP (mcKP) hierarch y or matrix KP h ierarc hy 2 − 4 , which conta ins man y p h ysical r elev an t non- linear integrable systems, such as (2+1)-dimensional AKNS hierarc hy and Da v ey-Stew artson (DS) equation. An extended DS equation can b e d er ived from matrix KP hierarch y 4 . T he explicit solu- tions of the matrix KP equation w ere studied in 10 − 11 . Th e relation b et wee n the 1+1 dimensional 1 Corresponding auth or: Y ehui Huang, T el: +86-13810446 869, e-mail: huangyh@mails.tsingh ua.edu.cn 2 tigertooth4@gmail.com 3 yyqinw@126.co m 4 yzeng@math.tsinghua.edu.cn 1 C-in tegrable B ¨ urgers hierarc hy and the matrix KP hierarc hy is discussed in 12 . The constrained matrix KP fl o ws w as obtained from th e matrix KP hierac hy 3 . Another kin d of generalizat ion of KP equation is the so called KP equation with self-consisten t sources initiate d by Mel’niko v 5 − 7 . The fi rst t yp e and second t yp e of K P equation with self-consisten t sour ces were studied in 5 . The KP equation with self-consisten t sources describ es the inte r action of a long wa ve w ith a short-w a ve pac k et p ropagating on the x , y plane at an angle to eac h other. Recen tly , a s ystematic approac h inspired b y squared eigenfun ction symmetry constrain t w as prop osed to constru ct a n ew extended KP h ierarc hy 13 . The extended K P h ierarc hy extended the KP hierarch y by con taining t wo times series t n and τ k and m ore comp onen ts giv en b y eigenfunctions and adjoin t eigenfunctions. This extended KP hierarc hy and its t n - and τ k -reduction p ro vide an un ifi ed w ay to fi nd the t wo t yp es of KP equation w ith self-consisten t sources and some (1+1)-dimensional soliton equ ations with self-consisten t sources. T h e solutions of the extended K P hierarch y and extend ed mKP hierarc hy can b e deriv ed under a generalized dressing appr oac h 14 . In this pap er, w e w ill construct the extension of the matrix KP hierarch y . Inspired by the square eigenfun ction symmetry constrain t of matrix KP hierarc hy , w e in tro d uce a new τ B flo w b y ”extending” a sp ecific t A -flo w of matrix K P hierarc hy . The extend ed matrix KP hierarc hy consists of t A -flo w, τ B -flo w and the t A -ev olutions of eigenfun ctions and adjoint eigenfunctions. W e get the zero curv ature r epresen tations for the extended matrix KP hierarch y from the comm utativit y of t A - flo w and τ B -flo w and its Lax represent ation. Th e extended matrix KP hierarc hy conta ins t wo time series t A and τ B and more comp onents by addin g eigenfunctions and adjoin t eigenfunctions, and admits t A -reduction and τ B -reduction. The extended matrix K P hierarc hy and its t w o reduction pro vide an un ified w a y to fi nd tw o types of matrix KP hierarch y with self-consisten t sources, and t wo t yp es of (1+1)-dimensional reduced matrix KP hierarc hy with self-consistent sources. By r estricting the elemen ts of the m atrix K P hierarc hy in 2 × 2 matrices, we deduce tw o t yp es of 2+1-dimens ional AKNS equation with self-consisten t sources and tw o types of DS equation with self-consisten t sources fr om the extended matrix KP hierarc hy as examples. W e can d educe tw o t yp es of 1+1 AKNS equation with self-consisten t sources under tw o t yp es of reductions. Th e DS equation with self-consisten t sources w ere also studied in 15 and 16 . W e would lik e to emph asize that ou r DS equation with self-consisten t sources is different from those in 15 and 16 as they hav e differen t t yp es of sour ces and the sources satisfy d ifferen t conditions. The dressin g metho d is an imp ortan t tool for solving Gelfand Dic ke y and KP hierarc hy 17 . Ho w ever the dr essing metho d for the matrix K P hierarc hy can not b e d irectly applied to the extended matrix KP hierarch y . With the com bination of a dr essin g app r oac h and the metho d of v ariation of constan ts, we p r op ose a generalized dr essing metho d for extended matrix KP h ierarc hy . By u sing this metho d, we can solv e the enti r e hierarc hy of extended matrix KP hierarch y . W e solv e t wo t yp es of 2+1 AKNS equation with self-consisten t sources and tw o t yp es of DS equation with self-consisten t sources as examples. This pap er is organized as follo ws. In section 2, we construct the extended matrix KP h ierar- c hy and derive its Lax pair, including tw o t yp es of the 2+1 AKNS equation with self-consisten t sources and DS equation with self-consisten t sources as examples. In section 3, t A -reduction and τ B -reduction of the extended matrix KP hierarc hy are giv en. In section 4, a generalized dr essing metho d for the extended matrix KP hierarc hy is discussed. In section 5, we giv e the N-soliton solutions for the extend ed matrix KP hierarc hy . Th e soliton s olution of 2+1 AKNS equation w ith self-consisten t sources and DS equation w ith self-consisten t sour ces are studied. In section 6, we 2 present the conclusion. 2 The extended matrix KP hierarc h y First w e review the we ll known matrix KP hierarch y 3 − 4 . Let g 0 b e a fin ite dimensional matrix algebra, which means that th e elemen ts in g 0 are N × N matrices. The matrix KP hierarch y can b e formulated b y a pseud o differen tial op er ator whic h is called th e dressing op erator 17 W = 1 + w 1 ∂ − 1 + w 2 ∂ − 2 + · · · (2.1) where ∂ = ∂ ∂ x , and the matrix v alued coefficients w i ∈ g o are fun ctions of x . w = ( w 1 , w 2 , · · · ) will b e the d ynamical fields of the matrix KP hierarch y . Define A = { c n ∂ n , n ∈ N , c n ∈ g 0 , [ c n , c m ] = 0 } . F or eac h elemen t A ∈ A , the ev olution of W is giv en by W t A = − P < 0 ( W AW − 1 ) W = − W A + M A W , (2.2) where M A = P ≥ 0 ( W AW − 1 ) , (2.3) P ≥ 0 ( L ) and P < 0 ( L ) denote the nonn egativ e parts and negativ e parts of psed o differen tial op erator L . If we fix an arb itrary elemen t C ∈ A and define a Lax op erator L = W C W − 1 , (2.4) the Lax equation of the matrix KP hierarch y is giv en b y L t A = [ M A , L ] . (2.5) The comm utativit y of ∂ t A and ∂ t B flo ws giv es r ise to the zero-curv ature equations of matrix KP hierarch y . M A,t B − M B ,t A + [ M A , M B ] = 0 . (2.6) It is kn o wn 3 that the evo lution of W giv en by W z = − N X i =1 Φ i ∂ − 1 Ψ T i W , (2.7a) Φ i,t A = M A (Φ i ) , (2.7b) Ψ i,t A = − M ∗ A (Ψ i ) (2.7c) is compatible with the matrix KP hierarch y (2.5) and reduces the matrix KP hierarc hy to the constrained matrix KP hierarc hy . If M A = P N i =0 u i ∂ i , the adj oint op erator M ∗ A is defined by M ∗ A = N X i =0 ( − 1) i ∂ i u T i . (2.8) 3 Based on this observ ation, we n o w introdu ce a new τ B flo w given b y W τ B = − P < 0 ( W B W − 1 ) W + N X i =1 Φ i ∂ − 1 Ψ T i W , (2.9a) Φ i,t A = M A (Φ i ) , (2.9b) Ψ i,t A = − M ∗ A (Ψ i ) . (2.9c) W e ha ve the follo wing lemma. Lemma 1 . P < 0 [ M A , Φ ∂ − 1 Ψ T ] = M A (Φ) ∂ − 1 Ψ T − Φ ∂ − 1 ( M ∗ A (Ψ)) T . Pro of . Without loss of generalit y , w e consider a monomial Q = u∂ k , u ∈ g 0 . Using (2.8), w e ha ve P < 0 ( ∂ − 1 Ψ T Q ) = P < 0 ( ∂ − 1 Ψ T u∂ k ) = P < 0 ( ∂ − 1 ∂ Ψ T u∂ k − 1 − ∂ − 1 ∂ (Ψ T u ) ∂ k − 1 ) = − P < 0 ( ∂ − 1 ∂ (Ψ T u ) ∂ k − 1 ) = · · · = ( − 1) k ∂ − 1 (Ψ T u ) ( k ) = ( − 1) k ∂ − 1 (( u T Ψ) ( k ) ) T = ∂ − 1 ( Q ∗ (Ψ)) T . So we h a v e P < 0 [ M A , Φ ∂ − 1 Ψ T ] = P < 0 ( M A Φ ∂ − 1 Ψ T ) − P < 0 (Φ ∂ − 1 Ψ T M A ) = M A (Φ) ∂ − 1 Ψ T − Φ ∂ − 1 ( M ∗ A (Ψ)) T . ✷ F ur ther more, we find that L τ B = W τ B C W − 1 − W C W − 1 W τ B W − 1 = M B L − LM B + Φ ∂ − 1 Ψ T L − L Φ ∂ − 1 Ψ T (2.10) = [ M B + Φ ∂ − 1 Ψ T , L ] . Lemma 2 . The τ B flo w giv en b y (2.9) and (2.10) is compatible w ith the matrix KP hierarc hy (2.5), namely , ( W t A ) τ B = ( W τ B ) t A , ( L t A ) τ B = ( L τ B ) t A . Pro of . F or con v enience, w e omit P . Notice th at W τ B = W t B + Φ ∂ − 1 Ψ T W , w e obtain ( W τ B ) t A = ( W t B ) t A + ( M A (Φ) ∂ − 1 Ψ T − Φ ∂ − 1 ( M ∗ A (Ψ)) T − Φ ∂ − 1 Ψ T P < 0 ( W AW − 1 )) W , ( W t A ) τ B = − P < 0 ( W t B AW − 1 + Φ ∂ − 1 Ψ T W AW − 1 − W AW − 1 W t B W − 1 − W AW − 1 Φ ∂ − 1 Ψ T ) W − P < 0 ( W AW − 1 ) W t B − P < 0 ( W AW − 1 )Φ ∂ − 1 Ψ T W , so ( W τ B ) t A − ( W t A ) τ B = ( W t B ) t A − ( W t A ) t B + ( M A (Φ) ∂ − 1 Ψ T − Φ ∂ − 1 ( M ∗ A (Ψ)) T − [Φ ∂ − 1 Ψ T , P < 0 ( W AW − 1 )] + P < 0 ([Φ ∂ − 1 Ψ T , W AW − 1 ])) W . 4 Using Lemma 1, w e find that P < 0 ([Φ ∂ − 1 Ψ T , W AW − 1 ]) = P < 0 (Φ ∂ − 1 Ψ T M A + Φ ∂ − 1 Ψ T P < 0 ( W AW − 1 ) − M A Φ ∂ − 1 Ψ T − P < 0 ( W AW − 1 )Φ ∂ − 1 Ψ T ) = [Φ ∂ − 1 Ψ T , P < 0 ( W AW − 1 )] + Φ ∂ − 1 ( M ∗ A (Ψ)) T − M A (Φ) ∂ − 1 Ψ T , so we hav e that ( W τ B ) t A − ( W t A ) τ B = 0. As ( L t A ) τ B = (( W C W − 1 ) t A ) τ B , ( L τ B ) t A = (( W C W − 1 ) τ B ) t A , w e find ( L t A ) τ B = ( L τ B ) t A from ( W t A ) τ B = ( W τ B ) t A . ✷ The comm utativit y of t A flo w and τ B flo w enables u s to obtain a new extended matrix KP hierarc hy as ∂ t A L = [ M A , L ] , (2.11a ) ∂ τ B L = [ M B + N X i =1 Φ i ∂ − 1 Ψ T i , L ] (2.11b) Φ i,t A = M A (Φ i ) , i = 1 , . . . , N , (2.11c ) Ψ i,t A = − M ∗ A (Ψ i ) , i = 1 , . . . , N . (2.11d) Prop osition 1 . Th e comm u tativit y of (2.11a ) and (2.11b) un der (2.11c) and (2.11d) giv es rise to the zero-curv ature equation for extended matrix KP h ierarc hy (2.11) M A,τ B − ( M B + N X i =1 Φ i ∂ − 1 Ψ T i ) t A + [ M A , M B + N X i =1 Φ i ∂ − 1 Ψ T i ] = 0 , (2.12a ) Φ i,t A = M A (Φ i ) , (2.12b) Ψ i,t A = − M ∗ A (Ψ i ) , i = 1 , . . . , N , (2.12c ) or equiv alen tly M A,τ B − M B ,t A + [ M A , M B ] − N X i =1 P ≥ 0 ([Φ i ∂ − 1 Ψ T i , M A ]) = 0 , (2.13a ) Φ i,t A = M A (Φ i ) , (2.13b) Ψ i,t A = − M ∗ A (Ψ i ) , i = 1 , . . . , N , (2.13c ) with the Lax represent ation giv en b y ψ t A = M A ( ψ ) , (2.14a ) ψ τ B = ( M B + N X i =1 Φ i ∂ − 1 Ψ T i )( ψ ) . (2.14b) Pro of . T he commutat ivity of (2.11a) and (2.11b) u n der (2.11c) and (2.11d) immediately giv es rise to (2.12 ). No w we p ro ve (2.13). Using L emma 1, we ha v e − (Φ ∂ − 1 Ψ T ) t A + [ M A , Φ ∂ − 1 Ψ T ] = − M A (Φ) ∂ − 1 Ψ T + Φ ∂ − 1 M ∗ A (Ψ) T + [ M A , Φ ∂ − 1 Ψ T ] = − P < 0 [ M A , Φ ∂ − 1 Ψ T ] + [ M A , Φ ∂ − 1 Ψ T ] (2.15) = P ≥ 0 [ M A , Φ ∂ − 1 Ψ T ] . ✷ 5 Remark . The extended matrix KP hierarch y extends the m atrix KP h ierarch y by con taining t wo time series t A and τ B and more comp onen ts Φ i and Ψ i , i = 1 , . . . , N . In the follo wing w e restrict w i to 2 × 2 matrices and consider the extended matrix KP hierarch y with A = σ 3 ∂ , B = σ 3 ∂ 2 and C = σ 3 ∂ , where σ 3 =  1 0 0 − 1  . The asso ciated times are t A = y , t B = t . W e in tro d uce U =  0 r q 0  = [ ω 1 , σ 3 ] = − 2 σ 3 ω of f 1 , D =  a 0 0 b  = − 2 ω diag 1 . The zero-curv ature equation (2.6) leads to the ev olution equation U t = 1 2 σ 3 ( U xx + U y y ) + σ 3 U 3 + [ D y , U ] , (2.16a ) σ 3 D y − D x + U 2 = 0 , (2.16b) i.e. r t = 1 2 ( r xx + r y y ) + r 2 q + ( a y − b y ) r , a y − a x + r q = 0 , (2.17a ) q t = − 1 2 ( q xx + q y y ) − q 2 r − ( a y − b y ) q , b y + b x − r q = 0 , (2.17b) As a x − a y = b y + b x , we ma y assu me a + b = φ x , a − b = φ y , which leads to φ xx − φ y y = 2 r q . Denote v = φ y , then w e hav e r t = 1 2 ( r xx + r y y ) + r 2 q + v y r , (2.18a ) q t = − 1 2 ( q xx + q y y ) − q 2 r − v y q , (2.18b) v xx − v y y = 2( r q ) y , (2.1 8c) whic h is 2 + 1 dimen sional AKNS equation 3 . If we replace U t b y iU t and assu me that q = ¯ r , φ = ¯ φ , then the ab o ve s y s tem redu ces to the Da vey- S tew artson I (DSI) equation ir t = 1 2 ( r xx + r y y ) + | r | 2 r + v y r , v xx − v y y = 2( | r | 2 ) y . (2.19) No w we derive the (2+1) dim en sional AKNS equation with self-consistent s ou r ces and Dav ey- Stew artson equation with s elf-consistent sour ces from (2.13). Example 1 . If w e tak e t A = y and τ B = t , we obtain th e fi rst t yp e of 2 + 1 d imensional AK NS equation with s elf-consistent sour ces from (2.13) U t = 1 2 σ 3 ( U xx + U y y ) + σ 3 U 3 + [ D y , U ] + [ N X i =1 Φ i Ψ T i , σ 3 ] , (2.20a ) σ 3 D y − D x + U 2 = 0 , (2.20b) Φ i,y = σ 3 Φ i,x + U Φ i , (2.20c ) Ψ i,y = σ 3 Ψ i,x − U T Ψ i , (2.20d) 6 or w e can rewr ite it as r t = 1 2 ( r xx + r y y ) + r 2 q + v y r − N X i =1 (2( φ i 11 ψ i 21 + φ i 12 ψ i 22 )) , (2.21a ) q t = − 1 2 ( q xx + q y y ) − q 2 r − v y q + N X i =1 (2( φ i 21 ψ i 11 + φ i 22 ψ i 12 )) , (2.21b) v xx − v y y = 2( r q ) y , (2.21c ) Φ i,y = σ 3 Φ i,x + U Φ i , (2.21d) Ψ i,y = σ 3 Ψ i,x − U T Ψ i . (2.21e ) Here and afterw ard Φ i =  φ i 11 φ i 12 φ i 21 φ i 22  , Ψ i =  ψ i 11 ψ i 12 ψ i 21 ψ i 22  , i = 1 , . . . , N are 2 × 2 matrices. Its Lax r epresen tation is ψ y = ( σ 3 ∂ + U ) ψ , (2.22a ) ψ t = ( σ 3 ∂ 2 + U ∂ + 1 2 U x + 1 2 σ 3 U 2 + 1 2 σ 3 U y + D y + N X i =1 Φ i ∂ − 1 Ψ T i ) ψ . (2.22b) Example 2 . When we tak e τ A = y and t B = t , we get the second t yp e of 2 + 1 dimen s ional AKNS equation w ith self-consisten t sources from (2.13) U t = 1 2 σ 3 ( U xx + U y y ) + σ 3 U 3 + [ D y , U ] − N X i =1  [ U, (Φ i Ψ T i ) diag ] + 2 σ 3 (Φ i,x Ψ T i ) of f  , (2.23a ) ( σ 3 D y − D x + U 2 ) y − N X i =1  [ U, (Φ i Ψ T i ) of f ] + 2 σ 3 (Φ i Ψ T i ) diag x  = 0 , (2.23b) Φ i,t = σ 3 Φ i,xx + U Φ i,x + 1 2 ( U x + σ 3 U 2 + σ 3 U y + D y )Φ i , i = 1 , . . . , N , (2.23c ) Ψ i,t = − σ 3 Ψ i,xx + U T Ψ i,x − 1 2 ( U x + σ 3 U 2 + σ 3 U y + D y ) T Ψ i , i = 1 , . . . , N , (2.23d) 7 or r t = 1 2 ( r xx + r y y ) + r 2 q + v y r + N X i =1 ( r ( φ i 11 ψ i 11 + φ i 12 ψ i 12 − φ i 21 ψ i 21 − φ i 22 ψ i 22 ) − 2( φ i 11 ,x ψ i 21 + φ i 12 ,x ψ i 22 )) , (2.24a ) q t = − 1 2 ( q xx + q y y ) − q 2 r − v y q − N X i =1 ( q ( φ i 11 ψ i 11 + φ i 12 ψ i 12 − φ i 21 ψ i 21 − φ i 22 ψ i 22 ) − 2( φ i 21 ,x ψ i 11 + φ i 22 ,x ψ i 12 )) , (2.24b) v y y − v xx = 2( N X i =1 ( r ( φ i 21 ψ i 11 + φ i 22 ψ i 12 − q ( φ i 11 ψ i 21 + φ i 12 ψ i 22 ) +2( φ i 11 ψ i 11 + φ i 12 ψ i 12 ) x ))) , (2.24c ) Φ i,t = σ 3 Φ i,xx + U Φ i,x + 1 2 ( U x + σ 3 U 2 + σ 3 U y + D y )Φ i , i = 1 , . . . , N , (2.24d) Ψ i,t = − σ 3 Ψ i,xx + U T Ψ i,x − 1 2 ( U x + σ 3 U 2 + σ 3 U y + D y ) T Ψ i , i = 1 , . . . , N , (2.24e ) Its Lax r epresen tation is ψ y = ( σ 3 ∂ + U + N X i =1 Φ i ∂ − 1 Ψ T i ) ψ , (2.25a ) ψ t = ( σ 3 ∂ 2 + U ∂ + 1 2 U x + 1 2 σ 3 U 2 + 1 2 σ 3 U y + D y ) ψ . (2.25b) Example 3 . Defin e ˜ U =  0 r ¯ r 0  , the first t yp e of DSI equation with self-consistent sources is ir t = 1 2 ( r xx + r y y ) + | r | 2 r + v y r − 2 N X j =1 ( φ j 11 ψ j 21 + φ j 12 ψ j 22 ) , (2.26a ) v xx − v y y = 2( | r | 2 ) y , (2.26b) Φ j,y = σ 3 Φ j,x + ˜ U Φ j , j = 1 , . . . , N , (2.26c ) Ψ j,y = σ 3 Ψ j,x − ˜ U T Ψ j , j = 1 , . . . , N . (2.26d) Its Lax r epresen taion is ψ y = ( σ 3 ∂ + ˜ U ) ψ , (2.27a ) ψ t = − i ( σ 3 ∂ 2 + ˜ U ∂ + 1 2 ˜ U x + 1 2 σ 3 ˜ U 2 + 1 2 σ 3 ˜ U y + D y + N X j =1 Φ j ∂ − 1 Ψ T j ) ψ . (2.27b) 8 Example 4 . T he second type of DSI equation with self-consistent sources is ir t = 1 2 ( r xx + r y y ) + | r | 2 r + v y r + N X j =1 ( r ( φ j 11 ψ j 11 + φ j 12 ψ j 12 − φ j 21 ψ j 21 − φ j 22 ψ j 22 ) − 2( φ j 11 ,x ψ j 21 + φ j 12 ,x ψ j 22 )) , (2.28a ) v y y − v xx = 2( N X j =1 ( r ( φ j 21 ψ j 11 + φ j 22 ψ j 12 − ¯ r ( φ j 11 ψ j 21 + φ j 12 ψ j 22 ) +2( φ j 11 ψ j 11 + φ j 12 ψ j 12 ) x ))) , (2.28 b ) Φ j,t = σ 3 Φ j,xx + ˜ U Φ j,x + 1 2 ( ˜ U x + σ 3 ˜ U 2 + σ 3 ˜ U y + D y )Φ j , j = 1 , . . . , N , (2.28c ) Ψ j,t = − σ 3 Ψ j,xx + ˜ U T Ψ j,x − 1 2 ( ˜ U x + σ 3 ˜ U 2 + σ 3 ˜ U y + D y ) T Ψ j , j = 1 , . . . , N . (2 .28d) Its Lax r epresen tation is ψ y = ( σ 3 ∂ + ˜ U + N X j =1 Φ j ∂ − 1 Ψ T j ) ψ , (2 .29a) ψ t = − i ( σ 3 ∂ 2 + ˜ U ∂ + 1 2 ˜ U x + 1 2 σ 3 ˜ U 2 + 1 2 σ 3 ˜ U y + D y ) ψ . (2.29b) Remark . The extended matrix KP hierarc hy (2.11 ) p ro vides a un ified w ay to construct the first t yp e and second t yp e of the (2+1)-dimensional AKNS equation (and DSI equation) with self-consisten t sources and their Lax repr esen tation. 3 Reductions of the extended matrix KP hierarc h y The extended KP hierarc hy dep ends on t w o time series t A and τ B . It is n atur al to consid er its t A -reduction and τ B -reduction. 3.1 The t A -reduction The t A -reduction of th e extended m atrix KP hierarch y is giv en b y W AW − 1 = M A , (3.1) where A = C k ∂ k . The wa ve f unction and the adjoin t w a ve function are giv en b y Φ( t, z ) = W exp ( ξ ( t, z )) , Φ ∗ ( t, z ) = ( W ∗ ) − 1 exp ( − ξ ( t, z )) , (3.2) where ξ ( t, z ) = P i> 0 t i z i , t 1 = x . T hen w e ha ve M A (Φ) = W AW − 1 Φ = z k Φ C k , (3.3) M ∗ A (Φ ∗ ) = ( W AW − 1 ) ∗ Φ ∗ = − z k Φ ∗ C k . (3.4) L t A = [ M A , L ] = [ W AW − 1 , W C W − 1 ] = W [ A, C ] W − 1 = 0 . (3.5) 9 So L is indep en d en t of t A and w e can dr op t A dep end ence from (2.12) ( M A ) τ B = [ M B + N X i =1 Φ i ∂ − 1 Ψ T i , M A ] , (3.6a) M A (Φ i ) = λ k i Φ i C k , (3.6b) M ∗ A (Ψ i ) = λ k i Ψ i C k . (3.6c) When A = σ 3 ∂ and B = σ 3 ∂ 2 w e ha ve M A (Φ i ) = W AW − 1 (Φ i ) = λ i Φ i σ 3 . Th en the first typ e of the 2 + 1-dimensional AKNS equation with self-consisten t sources reduces to the first typ e of 1 + 1-dimensional AKNS equation with self-consisten t sources U t = 1 2 σ 3 U xx + σ 3 U 3 + [ N X i =1 Φ i Ψ T i , σ 3 ] , (3.7a) U 2 = D x , (3.7b) σ 3 Φ i,x + U Φ i = λ i Φ i σ 3 , (3.7c) σ 3 Ψ i,x − U T Ψ i = λ i Ψ i σ 3 . (3.7d) 3.2 The τ B -reduction The τ B -reduction is giv en b y 3 W B W − 1 = M B + N X i =1 Φ i ∂ − 1 Ψ T i . (3.8) Then we can drop τ B dep end ence fr om (2.12) ( M B + N X i =1 Φ i ∂ − 1 Ψ T i ) t A = [ M A , M B + N X i =1 Φ i ∂ − 1 Ψ T i ] , (3.9a) Φ i,t A = M A (Φ i ) , (3.9b) Ψ i,t A = − M ∗ A (Ψ i ) . (3.9c) whic h is jus t th e constrained matrix KP hierarch y giv en in 3 . When B = σ 3 ∂ and A = σ 3 ∂ 2 , we get the constrained (2+1)-dimensional AKNS equation or second t yp e of AKNS equation with self-consisten t sources. U t = 1 2 σ 3 U xx + σ 3 U 3 + [ m X j =1 (Φ j Ψ T j ) diag , U ] + σ 3 m X j =1 (Φ j x Ψ T j − Φ j Ψ T j x ) of f , (3.10a ) Φ it = σ 3 Φ ixx + U Φ ix + 1 2 U x Φ i + 1 2 σ 3 U 2 Φ i + m X j =1 Φ j Ψ T j Φ i + m X j =1 (Φ j Ψ T j ) diag Φ i , (3.10b) Ψ it = − σ 3 Ψ ixx + U T Ψ ix + 1 2 U T x Ψ i − 1 2 σ 3 U 2 Ψ i − m X j =1 Ψ j Φ T j Ψ i − m X j =1 (Ψ j Φ T j ) diag Ψ i . (3.10c ) 10 Remark . Th e extended matrix K P hierarch y and its t A -reduction and τ B -reduction pro vide a simple and unified wa y to obtain the t wo t yp es of (2+1)-dimensional and (1+1)-dimensional AKNS equation with s elf-consistent sour ces. 4 Generalized dr essing approac h for extended matrix KP hierar- c h y In the follo wing, we restrict g 0 to b e the matrix alge b ra of dimen s ional 2 × 2. No w we prop ose a generalized d r essing approac h for the extended matrix KP hierarc hy . F or the dr essing form of L giv en b y (2.4) L = W C W − 1 , (4.1) usually the W has fin ite terms, so it is equiv alen t to assu me that th e dressin g op erator W is a p ure differen tial op erator of order N as follo ws W = ∂ N + w 1 ∂ N − 1 + w 2 ∂ N − 2 + · · · + w N . (4.2) Let 2 × 2 matrices f i , g i satisfy f i,t A = A ( f i ) , f i,τ B = B ( f i ) , g i,t A = A ( g i ) , g i,τ B = B ( g i ) , i = 1 , · · · , N . (4.3) By means of the metho d of v ariation of constan ts, let 2 × 2 matrices h i b e the linear com bination of f i and g i as h i = f i + α i ( τ B ) g i , i = 1 , · · · , N (4.4) with α i b eing a fun ction of τ B . W e assume that h i and its deriv ativ es are in ve r tible matrices an d the 2 N × 2 N matrix       h 1 h 2 · · · h N h (1) 1 h (1) 2 · · · h (1) N . . . . . . . . . . . . h ( N − 1) 1 h ( N − 1) 2 · · · h ( N − 1) N       is inv ertible. Theorem 1 3 : Let a 0 , . . . , a N − 1 b e the 2 × 2 m atrix fu nctions determined as the solution of the linear algebraic system N X i =0 a i h ( i ) j = 0 , j = 1 , . . . , N , (4.5) with a N = 1. Then W = P N i =0 a i ∂ i satisfies (2.2) and L = W C W − 1 satisfies the matrix K P hierarc hy (2.5). W e ha ve W ( h i ) = 0 , i = 1 , · · · , N (4 .6) Theorem 2 : Let b 1 , . . . , b N b e the 2 × 2 m atrix functions satisfy N X j =1 h ( i ) j b j = δ i,N − 1 , i = 0 , . . . , N − 1 . (4.7) 11 Define Φ i = − ˙ α i W ( g i ), and Ψ T i = b i , where ˙ α i = dα i /dτ B , then W = P N i =0 a i ∂ i , Φ i , Ψ i and L = W C W − 1 satisfy extended matrix KP h ierarc hy (2.11) . T o pro of T h eorem 2, we need sev eral lemmas und er the ab o ve assump tions. Lemma 3 : W − 1 = P N i =1 h i ∂ − 1 Ψ T i . Pro of: Using(4.6) and (4.7), we ha ve P ≥ 0 ( W N X i =1 h i ∂ − 1 Ψ T i ) = P ≥ 0 ( W N X i =1 h i ∂ − 1 b i ) = P ≥ 0 ( W ∞ X k =0 ∂ − k − 1 N X i =1 h ( k ) i b i ) = P ≥ 0 ( W ∂ − N + ∞ X k = N ∂ − k − 1 N X i =1 h ( k ) i b i ) = P ≥ 0 ( W ∂ − N (1 + ∞ X k =0 ∂ − k − 1 N X i =1 h ( N + k ) i b i )) = 1 . P ≤ 0 ( W N X i =1 h i ∂ − 1 Ψ T i ) = N X i =1 ( W ( h i )) ∂ − 1 Ψ T i = 0 . (4.8) So we kn o w that W − 1 = P N i =1 h i ∂ − 1 Ψ T i . Lemma 4 : W ∗ (Ψ i ) = 0. Pro of: F rom the relation W ∗ ( W − 1 ) ∗ ∂ j = ∂ j , we kn ow that 0 = Res ∂ W ∗ ( N X i =1 h i ∂ − 1 Ψ T i ) ∗ ∂ j = − Res ∂ W ∗ N X i =1 Ψ i ∂ − 1 h T i ∂ j = ( − 1) j +1 N X i =1 W ∗ (Ψ i ) h T ( j ) i . As the 2 N × 2 N m atrix       h 1 h 2 · · · h N h (1) 1 h (1) 2 · · · h (1) N . . . . . . . . . . . . h ( N − 1) 1 h ( N − 1) 2 · · · h ( N − 1) N       is inv ertible, w e fin d that W ∗ (Ψ i ) = 0. Lemma 5 : The op erator ∂ − 1 Ψ T i W is a pur e differen tial op erator f or eac h i . F urth er more, for 1 ≤ i, j ≤ N , ( ∂ − 1 Ψ T i W )( h j ) = δ ij I . Pro of: As P < 0 ( ∂ − 1 Ψ T i W ) = ∂ − 1 ( W ∗ (Ψ i )) T = 0, we kno w that ∂ − 1 Ψ T i W is a p ure differen tial op erator. Let c ij = ( ∂ − 1 Ψ T i W )( h j ). W e find that ∂ ( c ij ) = Ψ T i W ( h j ) = 0 and N X i =1 h ( k ) i c ij = ∂ k ( X i h i c ij ) = ∂ k ( X i ( h i ∂ − 1 Ψ T i W )( h j )) = h ( k ) j . So we h a v e c ij = δ ij I . Prop osition 2 : W satisfies W τ B = − P < 0 ( W B W − 1 ) W + P N i =1 Φ i ∂ − 1 Ψ T i W . 12 Pro of: T aking ∂ τ B to the identit y W ( h i ) = 0 and using (4.3), Lemma 3 and Lemma 5, w e ha v e 0 = W τ B ( h i ) + W B ( h i ) + ˙ α i W ( g i ) = ( ∂ τ B W )( h i ) + ( W B W − 1 W )( h i ) − N X j =1 Φ j δ j i = ( ∂ τ B W + P < 0 ( W B W − 1 ) W − N X j =1 Φ j ∂ − 1 Ψ T j W )( h i ) . (4.9) Since the non-negativ e op erator acti n g on h i ab o ve h as degree lo w er than N and h i are N inde- p endent fun ctions, the op erator itself must b e zero. Hence the p r op osition is pro ve n . Pro of of the Theorem 2: The p ro of of (2.11a) is similar as w e ha ve in Th eorem 1. By u sing (2.10) , w e get (2.11 b ). By a d irect calculation, w e get (2.11c ) and (2.11d). 5 Solutions for extended matrix KP hierarc hy By using the generalized d r essing approac h giv en b y Theorem 2, we can constru ct the explicit solutions to the extended matrix KP hierarc hy . F or the (2+1)-dimensional AK NS equation with self-consisten t sources, we c ho ose prop er 2 × 2 matrices f i and g i . The solution of f y = σ 3 f x , f t = σ 3 f xx is f = c 11 λ + d 11 e λx + λy + λ 2 t c 12 µ + d 12 e µx + µy + µ 2 t c 21 λ + d 21 e λx − λy − λ 2 t c 22 µ + d 22 e µx − µy − µ 2 t ! , (5.1) where c ij , d ij are arbitrary constan ts, bu t they should b e c h osen such that f is in vertible for any x, y , t . W e can defin e f i = 1 λ i + e λ i x + λ i y + λ 2 i t 0 0 1 µ i + e µ i x − µ i y − µ 2 i t ! , (5.2) g i = 0 − e µ i x + µ i y + µ 2 i t e λ i x − λ i y − λ 2 i t 0 ! . (5.3) where λ i > 0, µ i > 0. In this wa y we ha v e a i and b i and the N-soliton solution of the (2+1)- dimensional AKNS equ ation with s elf-consistent sources is U = − 2 σ 3 a of f n − 1 , D = a diag n − 1 , Φ i = − ˙ α i W ( g i ) , Ψ i = b i , i = 1 , . . . , n . (5.4) The one-soliton solution of the fi rst t yp e of (2 + 1)-AKNS equation with a self-consiten t source (2.21) can b e constructed from h = f + α ( t ) g = 1 λ + e λx + λy + λ 2 t − α ( t ) e µx + µy + µ 2 t α ( t ) e λx − λy − λ 2 t 1 µ + e µx − µy − µ 2 t ! :=  1 λ + e ξ 11 − α ( t ) e ξ 12 α ( t ) e ξ 21 1 µ + e ξ 22  13 F rom Theorem 1 and Theorem 2 we ha ve a 0 = − h x h − 1 , a 1 = 1 , b 0 = h − 1 , wh ic h giv es rise to W = ∂ − h x h − 1 , U = 2 σ 3 ( h x h − 1 ) of f , D = 2( h x h − 1 ) diag , Φ = − ˙ α ( t )( g x − h x h − 1 g ) , Ψ T = h − 1 . (5.5) So the one-soliton of the fir st t yp e of (2+1)-AKNS equation with a self-consisten t source (2.21) is q = − 2 α ( t ) λ µ e ξ 21 + ( λ − µ ) e ξ 21 + ξ 22 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 ) + α ( t ) 2 e ξ 12 + ξ 21 , (5.6) r = 2 α ( t ) − µ λ e ξ 12 + ( λ − µ ) e ξ 11 + ξ 12 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 ) + α ( t ) 2 e ξ 12 + ξ 21 , (5.7) v = 2 λ µ e ξ 11 − µ λ e ξ 22 + ( λ − µ )( e ξ 11 + ξ 22 − α ( t ) 2 e ξ 12 + ξ 21 ) ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 ) + α ( t ) 2 e ξ 12 + ξ 21 , (5 .8) Φ = − ˙ α ( t )    α ( t ) 2 µ λ e ξ 12 + ξ 21 +( µ − λ ) e ξ 11 + ξ 12 + ξ 21 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21 − α ( t ) e ξ 12 ( µ − λ ) e ξ 11 + ξ 22 +(1 − λ µ ) e ξ 11 + µ λ e ξ 22 + 1 λ ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21 α ( t ) e ξ 21 ( λ − µ ) e ξ 11 + ξ 22 +(1 − λ µ ) e ξ 22 + λ µ e ξ 11 + 1 µ ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21 α ( t ) 2 λ µ e ξ 21 + ξ 12 +( λ − µ ) e ξ 21 + ξ 22 + ξ 12 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21    , (5.9) Ψ =    1 µ + e ξ 22 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21 − α ( t ) e ξ 21 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21 α ( t ) e ξ 12 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21 1 λ + e ξ 11 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( t ) 2 e ξ 12 + ξ 21    . (5.10) The one-soliton solution of the second t yp e of (2 + 1)-AKNS equation with a self-consiten t source (2.24 ) can b e constru cted from h = f + α ( y ) g = 1 λ + e λx + λy + λ 2 t − α ( y ) e µx + µy + µ 2 t α ( y ) e λx − λy − λ 2 t 1 µ + e µx − µy − µ 2 t ! :=  1 λ + e ξ 11 − α ( y ) e ξ 12 α ( y ) e ξ 21 1 µ + e ξ 22  . F rom Theorem 1 and Theorem 2 we ha ve a 0 = − h x h − 1 , a 1 = 1 , b 0 = h − 1 , wh ic h giv es rise to W = ∂ − h x h − 1 , U = 2 σ 3 ( h x h − 1 ) of f , D = 2( h x h − 1 ) diag , Φ = − ˙ α ( y )( g x − h x h − 1 g ) , Ψ T = h − 1 . (5.11) So th e one-soliton of the second t yp e of (2+1)-AKNS equation with a self-consistent sour ce 14 (2.24) is q = − 2 α ( y ) λ µ e ξ 21 + ( λ − µ ) e ξ 21 + ξ 22 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 ) + α ( y ) 2 e ξ 12 + ξ 21 , (5.12) r = 2 α ( y ) − µ λ e ξ 12 + ( λ − µ ) e ξ 11 + ξ 12 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 ) + α ( y ) 2 e ξ 12 + ξ 21 , (5.13) v = 2 λ µ e ξ 11 − µ λ e ξ 22 + ( λ − µ )( e ξ 11 + ξ 22 − α ( y ) 2 e ξ 12 + ξ 21 ) ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 ) + α ( y ) 2 e ξ 12 + ξ 21 , (5.14) Φ = − ˙ α ( y )    α ( t ) 2 µ λ e ξ 12 + ξ 21 +( µ − λ ) e ξ 11 + ξ 12 + ξ 21 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21 − α ( y ) e ξ 12 ( µ − λ ) e ξ 11 + ξ 22 +(1 − λ µ ) e ξ 11 + µ λ e ξ 22 + 1 λ ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21 α ( y ) e ξ 21 ( λ − µ ) e ξ 11 + ξ 22 +(1 − λ µ ) e ξ 22 + λ µ e ξ 11 + 1 µ ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21 α ( y ) 2 λ µ e ξ 21 + ξ 12 +( λ − µ ) e ξ 21 + ξ 22 + ξ 12 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21    , (5.15) Ψ =    1 µ + e ξ 22 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21 − α ( y ) e ξ 21 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21 α ( y ) e ξ 12 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21 1 λ + e ξ 11 ( 1 λ + e ξ 11 )( 1 µ + e ξ 22 )+ α ( y ) 2 e ξ 12 + ξ 21    . (5.16) The one-solito n solution of the first t yp e of the (2 + 1)-DS equation with a self-consisten t source (2.26) can b e constructed from h = f + α ( t ) g = 1 λ + e λx + λy + iλ 2 t − α ( t ) e − λx − λy + iλ 2 t α ( t ) e λx − λy − iλ 2 t 1 λ + e − λx + λy − iλ 2 t ! :=  1 λ + e η 11 − α ( t ) e η 12 α ( t ) e η 21 1 λ + e η 22  . where λ > 0. W e find th at the one-solito n solution of the fir st t yp e of the (2 + 1)-DS equation with a self- consisten t source is r = 2 α ( t ) e η 12 + 2 λe η 11 + η 12 ( 1 λ + e η 11 )( 1 λ + e η 22 ) + α ( t ) 2 e η 12 + η 21 , (5.17) v = 2 e η 22 + e η 11 + 2 λ ( e η 11 + η 22 − α ( t ) 2 e η 12 + η 21 ) ( 1 λ + e η 11 )( 1 λ + e η 22 ) + α ( t ) 2 e η 12 + η 21 , (5.18) Φ = − ˙ α ( t )   − α ( t ) e η 12 + η 21 +2 λe η 11 + η 12 + η 21 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21 α ( t ) e η 12 2 λe η 11 + η 22 +2 e η 11 + e η 22 + 1 λ ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21 α ( t ) e η 21 2 λe η 11 + η 22 +2 e η 22 + e η 11 + 1 λ ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21 α ( t ) e η 21 + η 12 +2 λe η 21 + η 22 + η 12 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21   , (5.19) Ψ =   1 λ + e η 22 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21 − α ( t ) e η 21 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21 α ( t ) e η 12 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21 1 λ + e η 11 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( t ) 2 e η 12 + η 21   . (5.20) The one-solit on solution of the second t yp e of the (2 + 1)-DS equation with a self-consisten t source (2.28) can b e constru cted from h = f + α ( y ) g = 1 λ + e λx + λy + iλ 2 t − α ( y ) e − λx − λy + iλ 2 t α ( y ) e λx − λy − iλ 2 t 1 λ + e − λx + λy − iλ 2 t ! :=  1 λ + e η 11 − α ( y ) e η 12 α ( y ) e η 21 1 λ + e η 22  . 15 where λ > 0. W e find that the one-soliton solution of the second t yp e of the (2 + 1)-DS equation with a self-consisten t source is r = 2 α ( y ) e η 12 + 2 λe η 11 + η 12 ( 1 λ + e η 11 )( 1 λ + e η 22 ) + α ( y ) 2 e η 12 + η 21 , (5.21) v = 2 e η 22 + e η 11 + 2 λ ( e η 11 + η 22 − α ( y ) 2 e η 12 + η 21 ) ( 1 λ + e η 11 )( 1 λ + e η 22 ) + α ( y ) 2 e η 12 + η 21 , (5.22) Φ = − ˙ α ( y )   − α ( y ) e η 12 + η 21 +2 λe η 11 + η 12 + η 21 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21 α ( y ) e η 12 2 λe η 11 + η 22 +2 e η 11 + e η 22 + 1 λ ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21 α ( y ) e η 21 2 λe η 11 + η 22 +2 e η 22 + e η 11 + 1 λ ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21 α ( y ) e η 21 + η 12 +2 λe η 21 + η 22 + η 12 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21   , (5.23) Ψ =   1 λ + e η 22 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21 − α ( y ) e η 21 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21 α ( y ) e η 12 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21 1 λ + e η 11 ( 1 λ + e η 11 )( 1 λ + e η 22 )+ α ( y ) 2 e η 12 + η 21   . (5.24) 6 Conclusion W e extend the matrix K P h ierarc hy by introd ucing a new τ B flo w and adding eigenfunctions and adjoin t eig enf unctions as new comp onents. The zero curv ature equatio n and Lax repr esen tation for the extended matrix KP h ierarc hy and its t A -reduction and τ B -reduction are presen ted. The extended matrix KP hierarch y and its t wo r eductions pro vide an u nified w ay to fi nd t wo t yp es of (2+1)-dimensional and (1+1)-dimensional AKNS equation (and DS equation) with s elf-consisten t sources. With the com bination of d ressing metho d and the metho d of v ariation of constan ts, w e prop ose a generalized dressing metho d to solve the extended matrix K P hierarch y and obtain some of its solutions. Th e soliton solution of t w o types of 2+1 AKNS equation with self-consisten t sources and t wo t yp es of DS equation w ith self-consisten t sources are stu d ied. 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