Quasideterminant solutions of the generalized Heisenberg magnet model

Quasideterminan t solutions of the generalized Heisen b erg magnet mo del U. Saleem 1 and M. Hassan 2 Dep artment of Physics, University of the Punjab, Quaid-e-Azam Campus, L ahor e-54 590, P akistan. Abstract In this pap er w e present Darb oux transformatio n for the genera lized Heisenber g mag net (GHM) mo del bas ed on genera l linear Lie group GL ( n ) and co nstruct multi-soliton solutions in terms of quasidetermina n ts. F urther we relate the quas ideterminant m ulti-soliton s o lutions obtained b y the means of Darb oux tra nsformation with those of o bta ined by dressing method. W e also discuss the mo del based on the Lie group S U ( n ) a nd obtain explicit s o liton solutions of the mo de l ba sed on S U (2). P A CS : 11.10.N x, 02.30. Ik Keyw ords: In tegrable systems, Heisen b erg mo del, Darb oux transformation, quasid etermin ants 1 T el No: +92-42-99231243, F ax No: +92-42-35856892 e-mail:usaleem@ph ysics.pu.edu.pk, usman physics@y aho o.com 2 mhassan@physics.pu.edu.pk 1 1 In t ro duction During the past decades, there has b een an increasing int erest in the study of classical and quan tum in tegrabilit y of Heisen b erg f er r omagnet (HM) mo d el [1]-[15]. The Heisenberg ferromagnet (HM) mo del based on Hermitian symmetric spaces h as b een studied in [11]-[14]. The integ rabilit y of the HM mo d el based on S U (2) via inv erse scattering metho d is presented in [2]-[3] and its S U ( n ) generalizat ion is studied in [4 ]. The in tegrabilit y of the GHM mod el based on the ge neral linear Lie group GL ( n ) via L ax formalism has b een in ve stigated in [1]. In this pap er w e pr esen t the Darb oux transformation of the GHM mo d el based on general linear group GL ( n ) with Lie algebra g l(n) and calculate multi-solit on solutions in term of quasideterminants. W e also establish the r elation b et w een the Darb oux transformation and the well- kno w n dressing metho d [16]. In the last secti on, w e discuss the mo d el based S U ( n ) and calc ulate an exp licit expression of th e single-soliton s olution of the HM m o del b ased on the Lie group S U (2) u sing Darb oux transformation. The Hamiltonian of th e GHM mo d el is defin ed by [1] H = 1 2 T r  ( ∂ x U ) T ( ∂ x U )  , (1.1) with ” T ” is transp ose and U ( x, t ) is a m atrix-v alued function whic h tak es v alues in the Lie algebra gl(n) of the general linear group GL ( n ). The corresp onding equation of motion can b e expr essed as ∂ t U = {H , ∂ x U } . (1.2) The ab o ve equation (1.2) can b e w ritten as ∂ t U =  U, ∂ 2 x U  , (1.3) where ∂ x = ∂ ∂ x and ∂ t = ∂ ∂ t . Let u s assume that U ( x, t ) is diagonizable, i.e., U = g T g − 1 , (1.4) where g ∈ GL ( n ) is matrix fun ction of ( x, t ) and T is a n × n constant m atrix T =                  c 1 0 · · · 0 0 0 · · · 0 0 0 c 1 · · · 0 0 0 · · · 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · c 1 0 0 · · · 0 0 0 0 · · · 0 c 2 0 · · · 0 0 0 0 · · · 0 0 c 2 · · · 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 0 · · · 0 c 2                  , (1.5) 2 where 1 ≤ p ≤ n and c 1 , c 2 ∈ R (or C ). F rom equations (1.4) and (1.5), we ha v e [ U, [ U, [ U, χ ]]] = c 2 [ U, χ ] , (1.6) for an arb itrary matrix function χ and c = c 1 − c 2 6 = 0. Since ∂ x U ≡ U x =  ∂ x g g − 1 , U  , (1.7) implies [ U, [ U, U x ]] = c 2 U x , (1.8) The equation of motion (1.3) can also b e w ritten as th e zero-curv ature condition i.e.,  ∂ x − 1 (1 − λ ) U, ∂ t − c 2 (1 − λ ) 2 U − 1 (1 − λ ) [ U, U x ]  = 0 . (1.9) The ab o ve zero-curv ature condition (1.9 ) is equiv alen t to the compatibilit y condition of the follo wing Lax pair ∂ x Ψ( x, t ; λ ) = 1 (1 − λ ) U ( x, t )Ψ( x, t ; λ ) (1.10) ∂ t Ψ( x, t ; λ ) =  c 2 (1 − λ ) 2 U + 1 (1 − λ ) [ U, U x ]  Ψ( x, t ; λ ) (1.11) where λ is a real (or complex) parameter and Ψ is an in vertible n × n matrix-v alued function b elonging to GL ( n ). In the next section, w e defin e the Darb oux transformation on matrix solutions Ψ of the Lax p air (1.10)-(1.11). T o write d o wn the explicit expressions for matrix solutions of the GHM m o del, we will use the n otion of quasideterminan t in tro duced by Gelfand and Retakh [17]-[21 ]. Let X b e an n × n matrix o ver a ring R (noncommutat iv e, in general). F or an y 1 ≤ i , j ≤ n , let r i b e the i th row and c j b e the j th column of X . Th ere exist n 2 quasideterminan ts d enoted by | X | ij for i, j = 1 , . . . , n and are defined by | X | ij =      X ij c i j r j i x ij      = x ij − r j i  X ij  − 1 c i j , (1.12) where x ij is the ij th entry of X , r j i represent s the i th row of X without the j th en try , c i j represent s the j th column of X without the i th en try and X ij is the submatrix of X obtained by remo ving from X the i th row and the j th column. T he qu asideterminats are also d enoted b y th e follo w in g notation. If the ring R is comm u tative i.e. the en tr ies of the m atrix X all commute, then | X | ij = ( − 1) i + j det X det X ij . (1.13) 3 F or a detailed account of qu asideterminan ts and their p rop erties see e.g. [17]-[21 ]. In this pap er, w e will consider only quasideterminan ts that are expanded ab out an n × n matrix o ver a commuta tiv e ring. Let A B C D ! , b e a blo c k decomp osition of any K × K matrix wh ere the matrix D is n × n and A is inv ertible. The ring R in this case is the (noncomm u tativ e) ring of n × n matrices ov er another comm utativ e ring. The quasideterminan t of K × K matrix expanded ab out the n × n m atrix D is d efined by      A B C D      = D − C A − 1 B . (1.14) The quasideterminants h av e foun d v arious applications in the theory of inte grable sy s tems, where the m ultisoliton solutions of v arious noncommutat iv e integ rable systems are expr essed in terms of quisideterminant s (see e.g. [22]-[30]). 2 Darb oux transformation The Darb oux transformation is one of the well-kno wn metho d of ob taining multi -soliton solutions of many integ rable mo dels [31]-[33]. W e define the Darb oux transformation on the matrix solutions of the Lax pair (1.10)-(1.11), in terms of an n × n matrix D ( x, t, λ ), called the Da rb oux matrix. F or a general d iscussion on Darb oux matrix approac h see e.g. [34]-[39]. The Darb oux m atrix r elates the t w o matrix solutions of the Lax pair (1.10)-(1.11), in such a w ay that the Lax pair is co v arian t under the Darb oux transformation. Th e one-fold Darb oux trans formation on the matrix solution of the L ax p air (1.10 )-(1.11) is defi ned b y Ψ [1] ( x, t ; λ ) = D ( x, t, λ )Ψ( x, t ; λ ) , (2.1) where D ( x, t, λ ) is the Darb oux matrix. F or our case, we can mak e the follo wing ansatz D ( x, t, λ ) = λI − M ( x, t ) , (2.2) where M ( x, t ) is an n × n matrix function and I is an n × n identit y matrix. Th e new solution Ψ [1] ( x, t ; λ ) satisfies the follo wing Lax p air, i.e. ∂ x Ψ [1] ( x, t ; λ ) = 1 1 − λ U [1] Ψ [1] ( x, t ; λ ) , (2.3) ∂ t Ψ [1] ( x, t ; λ ) =  c 2 (1 − λ ) 2 U [1] + 1 1 − λ [ U [1] , U x [1]]  Ψ [1] ( x, t ; λ ) , (2.4) 4 where U [1] satisfies the equation of motion (1.3). By op erating ∂ x and ∂ t on equation (2.1) and equating th e co efficients of different p ow ers of λ , w e get the follo w in g transformation on the matrix field U U [1] = U + M x , (2.5 ) and the follo wing conditions w hic h M is required to satisfy M x ( I − M ) = [ U, M ] , (2.6) M t ( I − M ) 2 =  c 2 U + [ U, U x ] , M  + M [ U, U x ] M − [ U, U x ] M 2 . (2.7) One can solv e equ ations (2.6)-(2.7) to obtain an explicit expression for the matrix function M ( x, t ). An explicit expression for M ( x, t ) can b e found as follo w s. Let us tak e n distinct real (or complex) constan t parameters λ 1 , · · · , λ n ( 6 = 1). Also tak e n constan t column v ectors e 1 , e 2 , · · · , e n and construct an inv ertible n on-degenerate n × n m atrix function Θ( x, t ) Θ( x, t ) = (Ψ( λ 1 ) e 1 , · · · , Ψ ( λ n ) e n ) = ( θ 1 , · · · , θ n ) . (2.8) Eac h column θ i = Ψ( λ i ) e i in the matrix Θ is a column solution of th e Lax pair (1.10)-(1.11 ) wh en λ = λ i and i = 1 , 2 , . . . , n i.e. ∂ x θ i = 1 1 − λ i U θ i , (2.9) ∂ t θ i =  c 2 (1 − λ i ) 2 U + 1 1 − λ i [ U, U x ]  θ i . (2.10) Let us tak e an n × n in vertible diagonal m atrix with entries b eing eigen v alues λ i corresp ondin g to the eigen ve ctors θ i Λ = diag( λ 1 , . . . , λ n ) . (2.11) The n × n matrix generalizatio n of the Lax pair (2.9)-(2.1 0 ) will b e ∂ x Θ = U Θ ( I − Λ) − 1 , (2.12) ∂ t Θ i = c 2 U Θ ( I − Λ) − 2 + [ U, U x ] Θ ( I − Λ) − 1 . (2.13) The n × n matrix Θ is a particular matrix solution of the Lax pair (2.9)-(2.10) with Λ b eing a matrix of particular eigen v alues. In terms of particular matrix solution Θ of the Lax pair (2.9)-(2.10), we mak e the f ollo wing choic e of the matrix M ( x, t ) M ( x, t ) = ΘΛΘ − 1 . (2.14) 5 Our next step is to chec k that equation (2.14) is a solution of equations (2.6)-(2.7). In order to sho w this, w e fi rst op erate ∂ x on equ ation (2.14 ) to get ∂ x M = ∂ x (ΘΛΘ − 1 ) , = ( ∂ x Θ) ΛΘ − 1 + ΘΛ ∂ x (Θ − 1 ) , = U Θ( I − Λ) − 1 ΛΘ − 1 − ΘΛΘ − 1 U Θ( I − Λ) − 1 Θ − 1 , = − U + Θ( I − Λ)Θ − 1 j + Θ( I − Λ) − 1 Θ − 1 , = − U + ( I − M ) U ( I − M ) − 1 , (2.15) whic h is the equ ation (2.6 ). Similarly op erate ∂ t on (2.14 ), we get ∂ t M = ∂ t  ΘΛΘ − 1  = ( ∂ t Θ) ΛΘ − 1 + ΘΘΛ ∂ t (Θ − 1 ) =  c 2 U Θ ( I − Λ) − 2 + [ U, U x ] Θ ( I − Λ) − 1  ΛΘ − 1 − ΘΛΘ − 1  c 2 U Θ ( I − Λ) − 2 + [ U, U x ] Θ ( I − Λ) − 1  Θ − 1 , (2.16) whic h is equation (2.7). This sh o ws that the c hoice (2.14) of the m atrix M satisfies the equations (2.6)-(2.7). In other wo rds w e can say that if the collectio n (Ψ , U ) is a solution of the Lax p air (1.10)-(1.11) and the matrix M is defined by (2.14), then (Ψ[1] , U [1]) defined by (2.1) and (2.5) resp ectiv ely , is also a solution of the same Lax pair. T herefore w e say that Ψ[1] =  λI − ΘΛΘ − 1  Ψ , U [1] =  I − ΘΛΘ − 1  U  I − ΘΛΘ − 1  − 1 , is the requir ed Darb oux transf ormation on the solution Ψ to the Lax pair (1.10)-(1.11) and U to the equation of motion (1.3 ) resp ectiv ely . 3 Quasideterminan t solutions W e hav e sho wn that the matrix M = Θ ΛΘ − 1 satisfies the conditions (2.6)-(2.7). Therefore, the one-fold Darb oux transformation (2.1) can also b e w ritten in terms of quasidetermen ts as Ψ[1] ≡ D ( x, t ; λ )Ψ =  λI − Θ 1 Λ 1 Θ − 1 1  Ψ , =      Θ 1 Ψ Θ 1 Λ 1 λ Ψ      . (3.1) 6 The ab ov e equation d efines the Darb oux trans f ormation on the matrix solution Ψ of the Lax p air (1.10)-(1.11). Th e corresp ondin g one-fold Darb oux transf orm ation on the matrix field U is U [1] =  I − Θ 1 Λ 1 Θ − 1 1  U  I − Θ 1 Λ 1 Θ − 1 1  − 1 , =      Θ 1 I Θ 1 ( I − Λ 1 ) 0      U      Θ 1 I Θ 1 ( I − Λ 1 ) 0      − 1 . (3.2) W e write t wo- fold Darb oux transformation on Ψ as Ψ[2] ≡ D ( x, t ; λ )Ψ[1] = λ Ψ [1] − Θ 2 [1]Λ 2 Θ − 1 2 [1]Ψ[1] = λ  λI − Θ 1 Λ 1 Θ − 1 1  Ψ −  Θ 2 Λ 2 − Θ 1 Λ 1 Θ − 1 1 Θ 2  Λ 2  Θ 2 Λ 2 − Θ 1 Λ 1 Θ − 1 1 Θ 2  − 1  λI − Θ 1 Λ 1 Θ − 1 1  Ψ , =         Θ 1 Θ 2 Ψ Θ 1 Λ 1 Θ 2 Λ 2 λ Ψ Θ 1 Λ 2 1 Θ 2 Λ 2 2 λ 2 Ψ         . (3.3) Similarly the expression for tw o-fold Darb oux transformation on the matrix fi eld U as U [2] = Θ 2 [1] ( I − Λ 2 ) Θ − 1 2 [1] U [1]  Θ 2 [1] ( I − Λ 2 ) Θ − 1 2 [1]  − 1 , =  Θ 2 Λ 2 − Θ 1 Λ 1 Θ − 1 1 Θ 2  ( I − Λ 2 )  Θ 2 Λ 2 − Θ 1 Λ 1 Θ − 1 1 Θ 2  − 1 ×  I − Θ 1 Λ 1 Θ − 1 1  U  I − Θ 1 Λ 1 Θ − 1 1  − 1 ×   Θ 2 Λ 2 − Θ 1 Λ 1 Θ − 1 1 Θ 2  ( I − Λ 2 )  Θ 2 Λ 2 − Θ 1 Λ 1 Θ − 1 1 Θ 2  − 1  − 1 , =         Θ 1 Θ 2 I Θ 1 ( I − Λ 1 ) Θ 2 ( I − Λ 2 ) 0 Θ 1 ( I − Λ 1 ) 2 Θ 2 ( I − Λ 2 ) 2 0         × U × ×         Θ 1 Θ 2 I Θ 1 ( I − Λ 1 ) Θ 2 ( I − Λ 2 ) 0 Θ 1 ( I − Λ 1 ) 2 Θ 2 ( I − Λ 2 ) 2 0         − 1 . (3.4) The result can b e generalized to obtain N -fold Darb oux transf ormation on matrix solution Ψ as Ψ[ N ] =              Θ 1 Θ 2 · · · Θ N Ψ Θ 1 Λ 1 Θ 2 Λ 2 · · · Θ N Λ N λ Ψ Θ 1 Λ 2 1 Θ 2 Λ 2 2 · · · Θ N Λ 2 N λ 2 Ψ . . . . . . . . . . . . . . . Θ 1 Λ N 1 Θ 2 Λ N 2 · · · Θ N Λ N N λ N Ψ              . (3.5) 7 Similarly the expression for U [ N ] is U [ N ] =              Θ 1 Θ 2 · · · Θ N I Θ 1 ( I − Λ 1 ) Θ 2 ( I − Λ 2 ) · · · Θ N ( I − Λ N ) 0 Θ 1 ( I − Λ 1 ) 2 Θ 2 ( I − Λ 2 ) 2 · · · Θ N ( I − Λ N ) 2 0 . . . . . . . . . . . . . . . Θ 1 ( I − Λ 1 ) N Θ 2 ( I − Λ 2 ) N · · · Θ N ( I − Λ N ) N 0              × U × ×              Θ 1 Θ 2 · · · Θ N I Θ 1 ( I − Λ 1 ) Θ 2 ( I − Λ 2 ) · · · Θ N ( I − Λ N ) 0 Θ 1 ( I − Λ 1 ) 2 Θ 2 ( I − Λ 2 ) 2 · · · Θ N ( I − Λ N ) 2 0 . . . . . . . . . . . . . . . Θ 1 ( I − Λ 1 ) N Θ 2 ( I − Λ 2 ) N · · · Θ N ( I − Λ N ) N 0              − 1 . (3.6) W e n o w relate the q u asideterminan t solutions of GHM with the solutions obtained by d ressing metho d an d the in v erse scattering m etho d. F or this pur p ose, w e pro ceed as follo ws. F rom the definition of the m atrix M , w e hav e M Θ = ΘΛ . (3.7) Let θ i and θ j b e the column solutions of the Lax pair (1.10)-(1.1 1 ) wh en λ = λ i and λ = λ j resp ectiv ely i.e. M θ i = λ i θ i , i = 1 , 2 , . . . , p M θ j = λ j θ j . j = p + 1 , p + 2 , . . . , n (3.8) No w we take λ i = µ and λ j = ¯ µ , we m ay wr ite the matrix M as M = µP + ¯ µP ⊥ , (3.9) where P is the hermitian pr o jector i.e. P † = P . Th e pr o jector P satisfies P 2 = P and P ⊥ = 1 − P . The pro jector P is hermitian pr o jection on a complex sp ace and P ⊥ as pro jection on orthogonal space. No w equ ation (3.9) can also written as M = ( µ − ¯ µ ) P + ¯ µI , (3.10) where the h ermitian pro jector can b e expressed as P = θ i  θ † i , θ i  − 1 θ † i . (3.11) 8 The one-fold Darb oux transf ormation (3.1) on the matrix solution Ψ can also b e exp r essed in terms of pro jector P as Ψ[1] ≡ D ( x, t ; λ )Ψ =  I − µ − ¯ µ λ − ¯ µ P  Ψ , (3.12) where D ( x, t ; λ ) is the rescaled Darb oux-dressing function i.e. D ( x, t ; λ ) = ( λ − µ ) − 1 D ( x, t ; λ ). Similarly the N -fold Darb oux transformation (3.5) on the matrix solution Ψ can also b e w ritten as (tak e P [1] = P ) Ψ[ N ] = N − 1 Y k =0  I − µ N − k − ¯ µ N − k λ − ¯ µ N − k P [ N − k ]  Ψ . (3.13) No w we can expr ess the N -fold Darb oux transformation (3.6) on the matrix field U can b e written as U [ N ] = N − 1 Y k =0  I − µ N − k − ¯ µ N − k 1 − ¯ µ N − k P [ N − k ]  U N − 1 Y l =1  I − ¯ µ l − µ l 1 − ¯ µ l P [ l ]  , (3.14) and hermitian pro jector is defined as P [ k ] = θ i [ k ]  θ † i [ k ] , θ i [ k ]  − 1 θ † i [ k ] . (3.15) The expressions (3.13) and (3.14) can also b e written as sum of K terms [27] Ψ[ N ] = N − 1 X k =0  I − 1 λ − ¯ µ k R k  Ψ , (3.16) and U [ N ] = N − 1 X k =0  I − 1 1 − ¯ µ k R k  U N − 1 X l =0  I − 1 1 − ¯ µ l R l  − 1 , (3.17) where R k = N − 1 X l =0 ( µ l − ¯ µ k ) θ ( k ) i  θ ( k ) † i , θ ( l ) i  − 1 θ ( l ) † i . (3.18) 4 The explicit solutions of t he GHM mo del In this section we calculate explicit expr ession of soliton s olution. First of all w e will stu dy GHM mo del based on S U ( n ). In this case the sp in fu nction U takes v alues in the Lie algebra su (n) so 9 that one can d ecomp ose the spin fu n ction in to comp onent s U = U a T a , and T a , a = 1 , 2 , . . . , n 2 are an ti-hermitian n × n matrices with normalization T r  T a T b  = 1 2 δ ab and are the generators of the S U ( n ) in the fundamental represent ation satisfying the algebra h T a , T b i = f abc T c , (4.1) where f abc are the structure constan ts of the Lie algebra su(n) . F or an y X ∈ su (n) , we wr ite X = X a T a and U a = − 2T r( U T a ). The matrix-field U b elongs to the Lie algebra su(n) of the Lie group S U ( n ) th er efore U † = − U, T r( U ) = 0 . (4.2) The equations (2.1)-(2.2) and (2.5) define a Darb oux transformation for the GHM mo del based on the Lie group S U ( n ). The new solution of the equation of motion (1.3) U [1] m ust b e su(n) v alued i.e. U † [1] = − U [1] , T r( U [1]) = 0 , (4.3) therefore, w e ha ve th e follo wing conditions on the matrix M M † = − M , T r( M ) = 0 . (4.4) In other words we wa n t to m ake sp ecific M to satisfy the (4.4). This can b e ac hiev ed if we c h o ose the particular solutions θ i at λ = λ i , let us fi rst calculat e ∂ x  θ † i θ j  =  ∂ x θ † i  θ j + θ † i ( ∂ x θ j ) =  1 − ¯ λ i  − 1 θ † i U † θ j + (1 − λ j ) − 1 θ † i U θ j , (4.5) using equation (4.2) th e ab o ve equation (4.5) b ecomes ∂ x  θ † i θ j  = 0 , (4.6) when λ i 6 = λ j (i.e. ¯ λ i = λ j ). Similarly w e can chec k ∂ t  θ † i θ j  = 0 . (4.7) F rom the d efinition of the matrix M , w e h a ve θ † i  M † + M  θ j =  ¯ λ i + λ j  θ † i θ j , (4.8) 10 when λ i 6 = λ j then the ab o ve exp ression (4.8) imp lies θ † i θ j = 0 . (4.9) The column v ectors θ i are linearly indep end en t and the equ ation (4.9) holds ev eryw here. F or the HM mo d el b ased on S U ( n ), the constan t matrix (1.5) b ecomes T =                  2 − 2 n 0 · · · 0 0 0 · · · 0 0 0 − 2 n · · · 0 0 0 · · · 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · − 2 n 0 0 · · · 0 0 0 0 · · · 0 − 2 n 0 · · · 0 0 0 0 · · · 0 0 − 2 n · · · 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 0 · · · 0 − 2 n                  . (4.10) Then U 2 b ecomes U 2 = 4 ( n − 1) n 2 I + 2 ( n − 2) n U. (4.11) These are the constrain ts giv en in ref. [4]. F o r th e construction of explicit soliton solution f or the S U ( n ) HM mo del, w e constru ct the matrix M by defining a Hermitian p ro jector P . F or this case, w e take th e seed solution to b e U 0 ≡ U = i     a 1 . . . a n     , (4.12) where a i are r eal constants and P n i =1 a i = 0. The corresp onding solution of the Lax pair is expressed in blo c k diagonal matrix Ψ( x, t ; λ ) = W p ( λ ) O O W n − p ( λ ) ! , (4.13) where W p ( λ ) =     e i ω 1 ( λ ) . . . e i ω p ( λ )     , (4.14) and W n − p ( λ ) =     e i ω p +1 ( λ ) . . . e i ω n ( λ )     , (4.15) 11 are p × p and ( n − p ) × ( n − p ) matrices resp ectiv ely and ω i ( λ ) = a i  1 1 − λ x + 4 (1 − λ ) 2 t  . (4.16) No w d efine a particular matrix solution Θ of the Lax pair as Θ = ( Ψ( µ ) L 1 , Ψ( ¯ µ ) L 2 ) , (4.17) where L 1 is an n × p constan t matrix of p column v ectors and L 2 is the orthogonal complementa ry n × ( n − p ) matrix of ( n − p ) column ve ctors. Th e columns of L 1 span a p -dimensional sub space U of C n , and those of L 2 span the orthogonal subs p ace V . The pro jector P is completely charac terized b y the tw o su bspaces U = Im P an d V = K er P giv en by the condition P ⊥ U = 0 and P V = 0. Let us write L 1 = A B ! and L 2 = C D ! , where A , B , C and D are constan t p × p , ( n − p ) × n , p × ( n − p ) and ( n − p ) × ( n − p ) constant matrices resp ectiv ely . Give n this, the n × n matric Θ is giv en by Θ = W p ( µ ) A W p ( ¯ µ ) C W n − p ( µ ) B W n − p ( ¯ µ ) D ! . (4.18) W e no w d efine the p ro jector P in terms of the matrix Φ = Ψ( µ ) L 1 = ( θ 1 , · · · θ p ) give n by Φ = ( θ 1 , · · · , θ p ) = W p ( µ ) A W n − p ( µ ) B ! . The pro jector is th u s giv en b y P = W p ( µ ) A ∆ A † W † p ( ¯ µ ) W p ( µ ) A ∆ B † W † n − p ( ¯ µ ) W n − p ( µ ) B ∆ A † W † p ( ¯ µ ) W n − p ( µ ) B ∆ B † W † n − p ( ¯ µ ) ! , (4.19) where ∆ − 1 = A † W † p ( ¯ µ ) W p ( µ ) A + B † W † n − p ( ¯ µ ) W n − p ( µ ) A . The Darb ou x matrix D ( λ ) can now b e constructed to giv e explicit soliton solution of the S U ( n ) HM mo del. T o elab orate the resu lt more explicitly , we pro ceed with the example of S U (2) HM mo del. F or the S U (2) mo del, the equations (4.10) and (4.11) b ecome T = 1 0 0 − 1 ! . (4.20) Then U 2 b ecomes U 2 = I . (4.21) 12 The Lax p air (1.10)-(1.11) for the S U (2) mo del can b e written as ∂ x Ψ( x, t ; λ ) = 1 (1 − λ ) U ( x, t )Ψ( x, t ; λ ) , (4.22 ) ∂ t Ψ( x, t ; λ ) =  4 (1 − λ ) 2 U + 2 (1 − λ ) U U x  Ψ( x, t ; λ ) . (4.23) If we tak e trivial solution (as seed solution), single soliton and multi-solito n solutions can b e ob- tained b y Darb oux transformation as explained ab o v e. W e take th e s eed solution to b e U 0 ≡ U = i 0 0 − i ! . ( 4.24) The corresp onding solution of the lin ear system (4.22)-(4.23) can b e written as Ψ( x, t ; λ ) =   e i “ 1 (1 − λ ) x + 4 (1 − λ ) 2 t ” 0 0 e − i “ 1 (1 − λ ) x + 4 (1 − λ ) 2 t ”   . (4.25) T ak e λ 1 = µ and λ 2 = ¯ µ , th e constan t matrix Λ is giv en by Λ = µ 0 0 ¯ µ ! , (4.26) and corresp ondin g 2 × 2 matrix solution Θ b ecomes Θ ≡ ( θ 1 , θ 2 ) =   e i “ 1 (1 − µ ) x + 4 (1 − µ ) 2 t ” e i “ 1 (1 − ¯ µ ) x + 4 (1 − ¯ µ ) 2 t ” − e − i “ 1 (1 − µ ) x + 4 (1 − µ ) 2 t ” e − i “ 1 (1 − ¯ µ ) x + 4 (1 − ¯ µ ) 2 t ”   . (4.27) The matrix M is giv en b y M = Θ ΛΘ − 1 , = 1 e u + e − u µe u + ¯ µe − u ( ¯ µ − µ ) e iv ( ¯ µ − µ ) e − iv ¯ µe u + µe − u ! , (4.28) where the fu nctions u ( x, t ) and v ( x, t ) are defined b y u ( x, t ) = i  1 (1 − µ ) − 1 (1 − ¯ µ )  x + 4i  1 (1 − µ ) 2 − 1 (1 − ¯ µ ) 2  t, v ( x, t ) =  1 (1 − µ ) + 1 (1 − ¯ µ )  x + 4  1 (1 − µ ) 2 + 1 (1 − ¯ µ ) 2  t. (4.29) 13 Let us tak e th e eigen v alue to b e µ = e i θ . Th e exp r ession (4.28 ) then b ecomes M = cos θ + i sin θ tanh u − i (sin θ sec h u ) e i v − i (sin θ sec h u ) e − i v cos θ − i sin θ tanh u ! , (4.30) and the corresp onding Darb oux matrix D ( λ ) in this case is D ( λ ) = λ − cos θ − i s in θ tanh u i (sin θ sec h u ) e i v i (sin θ sec h u ) e − i v λ − cos θ + i sin θ tanh u ! . (4.31) Comparing the ab o v e equation with (3.12), we fin d the follo wing expression for the pro jector P = 2 e u sec h u − 2 e i v sec h u − 2 e − i v sec h u 2 e − u sec h u ! . (4.32) Using (3.2) and (4.24), we get U [1] = i U 3 U + − U − − i U 3 ! , (4.33) where U 3 = 1 − (1 + cos θ )sec h 2 u, U + ≡ U − = − i e i v [(1 + cos θ )tanh u + i sin θ ] s ech u. (4.34) F rom equ ation (4.34), we see that U † [1] = − U [1] and T r( U [1]) = 0. Therefore equation (4.34) is an explicit expression of the single-soliton solution of the HM mo d el based on S U (2) obtained b y using Darb oux transformation. Similarly one can calculate explicit expression for the m ulti-soliton solution of the mo d el. The expression (4.34) is similar to the exp r ession of the single soliton giv en in [2]. 5 Concluding remarks In this pap er, w e ha ve stu died GHM mo d el based on general linear Lie group GL ( n ) and expressed the m ulti-soliton solutions in terms of the quasideterminan t using the Darb oux transformation defined on the solution of the Lax p air. W e ha ve also established equiv alence b et ween the Darb ou x matrix approac h and the Zakharov-M ikhailo v’s dressin g metho d. In last section we hav e redu ced the GHM m o del int o the HM mo d el based on S U ( n ) and calculated an explicit expression for the single-soliton solution. It wo uld b e interesti ng to stu dy the GHM mo dels based on Hermitian symmetric spaces. W e shall address this problem in a sep arate w ork. 14 References [1] I. Ch erednik, Basic metho ds of soliton the ory , Adv. Ser. Math. Phys. 25 (1996) 1-250. [2] L. 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