Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation

Matrix solutions of a noncomm utativ e KP equation and a noncomm utativ e mKP equation C. R. Gilson, J. J. C. Nimmo and C. M. So oman Departmen t of Mathematics, Univ ersit y of Glasgow Glasgo w G12 8QW, UK Abstract Matrix solutions of a noncomm utative KP and a noncomm utative mKP equation whic h can be exp res sed as quasideterminan ts are discussed. In particular, w e in vestigate intera ction prop ertie s of tw o-soliton solutions. 1 In tro duction A considerable amount of literature exists concerning nonc o mm utativ e inte- grable sys tems. This includes noncommutativ e versions of the Burger s equa tion, the KdV equation, the KP equation, the mKP equation a nd the sine-Go rdon equation [6, 1 4, 15, 2, 19, 8, 5, 1 2 , 11, 1]. These equations can often be ob- tained by s imply r emo ving the assumption that the dep endent v ariables and their deriv ativ es in the La x pair commute. This pa per is co nc e r ned with a noncommutativ e KP e quation (ncKP) and a noncommutativ e mKP equa tion (ncmKP) [19, 13]. The Lax pair for ncKP is given by L KP = ∂ 2 x + v x − ∂ y , (1) M KP = 4 ∂ 3 x + 6 v x ∂ x + 3 v xx + 3 v y + ∂ t . (2) Both L KP and M KP are cov ariant under the Darb oux tra nsformation G θ = θ ∂ x θ − 1 , where θ is a n eigenfunction for L KP , M KP . F rom the compatibil- it y condition [ L KP , M KP ] = 0 we o bta in a noncommut ative version of the KP equation: ( v t + 3 v x v x + v xxx ) x + 3 v y y + 3[ v x , v y ] = 0 , (3) where u = v x . F or ncmKP , the La x pa ir [19] is given by L mKP = ∂ 2 x + 2 w∂ x − ∂ y , (4) M mKP = 4 ∂ 3 x + 12 w∂ 2 x + 6( w x + w 2 + W ) ∂ x + ∂ t . (5) 1 Both L mKP and M mKP are cov ariant under the Darb oux tra nsformation G θ = (( θ − 1 ) x ) − 1 ∂ x θ − 1 , wher e θ is an eigenfunction for L mKP , M mKP . The com- patibilit y condition [ L mKP , M mKP ] = 0 g iv es 0 = w t + w xxx − 6 ww x w + 3 W y + 3[ w x , W ] + − 3[ w xx , w ] − 3[ W, w 2 ] , (6) 0 = W x − w y + [ w, W ] . (7) Equations (6) and (7) form a noncommutativ e version of the mKP equation. Equation (7) is sa tisfied identically by applying the change of v aria bles [19] w = − f x f − 1 , W = − f y f − 1 . (8) The change of v ariables (8) has also b een used in [6, 2] to study a noncom- m utative mKP hierar chy . F or b oth ncK P and ncmKP equations, we co ns ider families of so lutions obta ined from iterating binar y Darb oux transforma tions. These solutio ns can b e expres s ed as qua sideterminan ts, which were intro duce d by Gelfand et al in [4 ]. An n × n matr ix Z = ( z ij ) n × n ov e r a r ing R (nonco m- m utative, in genera l) has n 2 quasideterminants, each o f which is deno ted | Z | ij for 1 ≤ i , j ≤ n . Let r j i denote the row vector obtaine d fro m the i th row of Z be deleting the j th entry , let c i j denote the column vector obtained from the j th row o f Z b y dele ting the i th en try , let Z ij be the matrix obtained from Z by deleting the i th r o w and j th column and assume tha t Z ij is invertible. Then | Z | ij exists and | Z | ij = z ij − r j i ( Z ij ) − 1 c i j . (9) F or notational conv enience, we b o x the leading element a bout whic h the e xpan- sion is made so that | Z | ij =      Z ij c i j r j i z ij      . (10) The qua sideterminan t solutio ns obtained from binary Dar boux transformations reduce to ratios of g r ammian determinants in the commut ative limit and w e call them quasigr ammians. In this pap er, we will consider the case where the dependent v ariables in ncKP and ncmKP are matrices and apply the metho ds used in [8, 9, 10, 7]. F r o m this platform, we inv estigate the interaction of the tw o- soliton s olution of the matrix versions o f ncKP a nd ncmKP . 2 Quasigrammian solutions of the ncKP equ a - tion In this section we r ecall the constructio n of the quasigr ammian solutions o f ncKP in [5]. The adjoint Lax pair is L † KP = ∂ 2 x + v † x + ∂ y , (11) M † KP = − 4 ∂ 3 x − 6 v † x ∂ x − 3 v † xx + 3 v † y − ∂ t . (12) 2 F ollowing the sta ndard constructio n of a binary Darb oux tr a nsformation (see [16]), one introduces a p oten tial Ω( φ, ψ ) satisfying Ω( φ, ψ ) x = ψ † φ, Ω( φ, ψ ) y = ψ † φ x − ψ † x φ, Ω( φ, ψ ) t = − 4( ψ † φ xx − ψ † x φ x + ψ † xx φ ) − 6 ψ † v x φ. The par ts of this definition are compatible when L KP [ φ ] = M KP [ φ ] = 0 a nd L † KP [ ψ ] = M † KP [ ψ ] = 0. Note that w e ca n define Ω(Φ , Ψ) for any row vectors Φ and Ψ such that L KP [Φ] = M KP [Φ] = 0 and L † KP [Ψ] = M † KP [Ψ] = 0. Conse- quently , if Φ is an m -vector and Ψ is an n -vector, then Ω is an m × n matrix . A binary Darb oux transformatio n is defined b y φ [ n +1] = φ [ n ] − θ [ n ] Ω( θ [ n ] , ρ [ n ] ) − 1 Ω( φ [ n ] , ρ [ n ] ) and ψ [ n +1] = ψ [ n ] − ρ [ n ] Ω( θ [ n ] , ρ [ n ] ) −† Ω( θ [ n ] , ψ [ n ] ) † , in which θ [ n ] = φ [ n ] | φ → θ n , ρ [ n ] = ψ [ n ] | ψ → ρ n . Using the notation Θ = ( θ 1 , . . . θ n ) and P = ( ρ 1 , . . . , ρ n ) we have, for n ≥ 1 φ [ n +1] =      Ω(Θ , P) Ω( φ, P) Θ φ      , ψ [ n +1] =      Ω(Θ , P) † Ω(Θ , ψ ) † P ψ      and Ω( φ [ n +1] , ψ [ n +1] ) =      Ω(Θ , P) Ω( φ, P) Ω(Θ , ψ ) Ω( φ, ψ )      . The effect of the binary Darb oux transforma tion b L KP = G θ ,φ L KP G − 1 θ ,φ , c M KP = G θ ,φ M KP G − 1 θ ,φ is tha t ˆ v = v 0 + 2 θ Ω( θ , ρ ) − 1 ρ † . After n binary Darb oux tr ansformations we have v [ n +1] = v 0 + 2 n X k =1 θ [ k ] Ω( θ [ k ] , ρ [ k ] ) − 1 ρ † [ k ] = v 0 − 2     Ω(Θ , P) P † Θ 0     . (13) 3 2.1 Tw o-soliton matrix solution W e now derive matrix solutions of ncKP using the metho ds applied in [8]. The trivial v acuum solutio n v 0 = O gives v = − 2     Ω(Θ , P) P † Θ 0     . (14) The eig enfunctions θ i and the adjoint eige nf unctions ρ i satisfy θ i,xx = θ i,y , θ i,t = − 4 θ i,xxx , (15) and ρ i,xx = − ρ i,y , ρ i,t = − 4 ρ i,xxx , (16) resp ectiv ely . W e choo se the simplest nontrivial solutio ns of (15) and (16): θ j = A j e η j , ρ i = B i e − γ i , (17) where η j = p j ( x + p j y − 4 p 2 j t ) , γ i = q i ( x + q i y − 4 q 2 i t ) and A j , B i are d × m matrices. With this, we have Ω( θ j , ρ i ) = B T i A j ( p j − q i ) e ( η j − γ i ) + δ i,j I . W e take A j = r j P j , where r j is a sca la r and P j is a pro jection matrix. With A j chosen in this way , we must hav e m = d . W e choose B i = I and the solutio n u will b e a d × d matrix. In the c a se n = 1, we o btain a one-soliton matrix solution. Expanding (14 ) gives v = 2 rP e ( γ − η ) + r ( p − q ) . The a bov e calculation and others that that follow use the for m ula ( I − aP ) − 1 = I + a P (1 − a ) − 1 where a 6 = 1 is a scala r and P is any pr o jection matrix. W e now hav e u = v x = 1 2 ( p − q ) 2 P sech 2  1 2 ( η − γ + ξ )  , (18) where ξ = log  r ( p − q )  . In the case n = d = 2, we obtain a tw o-solito n 2 × 2 matrix solution. By expanding (14) we get v = 2  A 1 e η 1 A 2 e η 2   A j e ( η j − γ i ) ( p j − q i ) + δ i,j I  − 1 2 × 2  I e − γ 1 I e − γ 2  = 2  K 1 e γ 1 K 2 e γ 2   I e − γ 1 I e − γ 2  = 2( K 1 + K 2 ) . 4 Therefore K 1  I + r 1 e ( η 1 − γ 1 ) ( p 1 − q 1 ) P 1  = e ( η 1 − γ 1 ) A 1 − e ( η 1 − γ 1 ) ( p 1 − q 2 ) K 2 A 1 , K 2  I + r 2 e ( η 2 − γ 2 ) ( p 2 − q 2 ) P 2  = e ( η 2 − γ 2 ) A 2 − e ( η 2 − γ 2 ) ( p 2 − q 1 ) K 1 A 2 . W e assume tha t the P j are the ra nk-1 pro jectio n ma trices P j = µ j ⊗ ν j ( µ j , ν j ) = µ j ν T j µ T j ν j , where the 2-vectors µ j , ν j satisfy the condition ( µ j , ν j ) 6 = 0 . Solving for K 1 and K 2 gives K 1 = ( p 2 − q 1 ) g ( g 2 ( p 1 − q 2 ) I − A 2 ) A 1 , K 2 = ( p 1 − q 2 ) g ( g 1 ( p 2 − q 1 ) I − A 1 ) A 2 , where g i = e ( γ i − η i ) + r i ( p i − q i ) , g = g 1 g 2 ( p 1 − q 2 )( p 2 − q 1 ) − αr 1 r 2 and α = ( µ 1 ,ν 2 )( µ 2 ,ν 1 ) ( µ 1 ,ν 1 )( µ 2 ,ν 2 ) . W e now in vestigate the be haviour of v [3] as t → ± ∞ . This will demons trate that each soliton emerges fro m interaction undergoing a phas e shift and that the amplitude of each so liton may also c hange due to the in teractio n. W e first fix γ 1 − η 1 and ass ume without loss of genera lit y that 0 > p 2 > q 2 > p 1 > q 1 . As t → −∞ v ∼ 2 r 1 P 1 g 1 and therefore u = v x ∼ 1 2 ( p 1 − q 1 ) 2 P 1 sech 2  1 2  η 1 − γ 1 + ξ − 1   , (19) where ξ − 1 = log  r 1 ( p 1 − q 1 )  . Note that u = v x is inv a rian t under the tr ansformation v → v + C , where C is a constant matrix. As t → + ∞ , we get v ∼ 2 ( r 2 ( p 1 − q 2 ) − ( p 2 − q 2 ) A 2 )( p 2 − q 1 ) A 1 − (( p 1 − q 2 ) A 1 − αr 1 ( p 2 − q 2 ))( p 2 − q 2 ) A 2 r 2 ( p 1 − q 2 )( p 2 − q 1 )  g 1 − αr 1 ( p 2 − q 2 ) ( p 1 − q 2 )( p 2 − q 1 )  ∼ 2 ˆ r 1 ˆ P 1 e γ 1 − η 1 + p 1 ˆ r 1 ( p 1 − q 1 ) , 5 where ˆ r 1 = r 1  1 − α ( p 1 − q 1 )( p 2 − q 2 ) ( p 1 − q 2 )( p 2 − q 1 )  = r 1 ( ˆ µ 1 , ˆ ν 1 ) ( µ 1 ,ν 1 ) , ˆ µ 1 = µ 1 − ( p 2 − q 2 )( µ 1 ,ν 2 ) µ 2 ( p 1 − q 2 )( µ 2 ,ν 2 ) , ˆ ν 1 = ν 1 − ( p 2 − q 2 )( µ 2 ,ν 1 ) ν 2 ( p 2 − q 1 )( µ 2 ,ν 2 ) and b P 1 = ˆ µ 1 ⊗ ˆ ν 1 ( ˆ µ 1 , ˆ ν 1 ) . Therefo r e u = v x ∼ 1 2 ( p 1 − q 1 ) 2 b P 1 sech 2  1 2  η 1 − γ 1 + ξ + 1   (20) where ξ + 1 = log  ˆ r 1 ( p 1 − q 1 )  . Similarly , fixing γ 2 − η 2 gives u ∼ 1 2 ( p 2 − q 2 ) 2 b P 2 sech 2  1 2  η 2 − γ 2 + ξ − 2   , t → −∞ (21) u ∼ 1 2 ( p 2 − q 2 ) 2 P 2 sech 2  1 2  η 2 − γ 2 + ξ + 2   , t → + ∞ , (22) where ˆ r 2 = r 2  1 − α ( p 1 − q 1 )( p 2 − q 2 ) ( p 1 − q 2 )( p 2 − q 1 )  = r 2 ( ˆ µ 2 , ˆ ν 2 ) ( µ 2 ,ν 2 ) , ˆ µ 2 = µ 2 − ( p 1 − q 1 )( µ 2 ,ν 1 ) µ 1 ( p 2 − q 1 )( µ 1 ,ν 1 ) , ˆ ν 2 = ν 2 − ( p 1 − q 1 )( µ 1 ,ν 2 ) ν 1 ( p 1 − q 2 )( µ 1 ,ν 1 ) , b P 2 = ˆ µ 2 ⊗ ˆ ν 2 ( ˆ µ 2 , ˆ ν 2 ) , ξ − 2 = log  ˆ r 2 ( p 2 − q 2 )  and ξ + 2 = log  r 2 ( p 2 − q 2 )  . The soliton phase shifts ∆ j = ξ + j − ξ − j are ∆ 1 = log  ˆ r 1 r 1  = log β , ∆ 2 = log  r 2 ˆ r 2  = − log β , where β = 1 − α ( p 1 − q 1 )( p 2 − q 2 ) ( p 1 − q 2 )( p 2 − q 1 ) . The matrix a mplitude of the firs t soliton changes fro m 1 2 ( p 1 − q 1 ) 2 P 1 to 1 2 ( p 1 − q 1 ) 2 b P 1 and the matrix amplitude of the se c ond soliton changes from 1 2 ( p 2 − q 2 ) 2 b P 2 to 1 2 ( p 2 − q 2 ) 2 P 2 as t changes from −∞ to + ∞ . If ( µ 1 , ν 2 ) = 0 ( P 2 P 1 = 0) o r ( µ 2 , ν 1 ) = 0 ( P 1 P 2 = 0), then α = 0 and therefor e β = 1, so there is no phase shift but the matr ix amplitudes may still change. If ( µ 1 , ν 2 ) = 0 a nd ( µ 2 , ν 1 ) = 0 (g iving P 1 P 2 = P 2 P 1 = 0), there is no phase shift or change in amplitude and so the solitons ha ve trivial int era c tio n. Fig ure 1 shows a plot of the interaction with P 1 =  1 − 2 0 0  and P 2 = 1 121  96 − 16 − 150 − 25  . 3 Quasigrammian solutions of the ncmKP equa- tion The constructio n o f this particular binar y Darb o ux transformation is given in [18] a nd also in [17] (for Lax o perator s with matrix co efficien ts). The adjo int Lax pair is L † mKP = ∂ 2 x − 2 w † x − 2 w † ∂ x + ∂ y , M † mKP = − 4 ∂ 3 x + 12 w † ∂ 2 x + 6(3 w † x − w † 2 − W † ) ∂ x + 6( w † xx − [ w † x , w † ] + − W † x ) − ∂ t . 6 Figure 1: P lot of u = ( u i,j ) 2 × 2 with t = 0, p 1 = − 1 4 , q 1 = − 39 4 , p 2 = 19 2 , q 2 = 1 2 , r 1 = 2 and r 2 = 1. F or notational convenience, w e denote an element of ker L † mKP ∩ k er M † mKP by φ x . One int ro duces a p oten tia l Ω( φ, ψ ) sa tis fying Ω( φ, ψ ) x = ψ † φ x , Ω( φ, ψ ) y = 2 ψ † wφ x + ψ † φ xx − ψ † x φ x , Ω( φ, ψ ) t = 2( − 2 ψ † xx φ x − 2 ψ † φ xxx + 2 ψ † x φ xx − 3 ψ † w 2 φ x − 3 ψ † W φ x − 3 ψ † w x φ x + 6 ψ † x wφ x − 6 ψ † wφ xx ) . A binary Darb oux transformatio n is defined b y φ [ n +1] = φ [ n ] − θ [ n ] Ω( ρ [ n ] , θ [ n ] ) − 1 Ω( ρ [ n ] , φ [ n ] ) and ψ [ n +1] = ψ [ n ] − ρ [ n ] Ω( ρ [ n ] , θ [ n ] ) † − 1 Ω( ψ [ n ] , θ [ n ] ) † , in which θ [ n ] = φ [ n ] | φ → θ n , ρ [ n ] = ψ [ n ] | ψ → ρ n . 7 Using the notation Θ = ( θ 1 , . . . θ n ) and P = ( ρ 1 , . . . , ρ n ), we have, for n ≥ 1 φ [ n +1] =      Ω(Θ , P) Ω( φ, P) Θ φ      , ψ [ n +1] =      Ω(Θ , P) † Ω(Θ , ψ ) † P ψ      . The effect of the binary Darb oux transforma tion b L mKP = G θ ,φ x L mKP G − 1 θ ,φ x , c M mKP = G θ ,φ x M mKP G − 1 θ ,φ x is tha t ˆ f =     Ω ρ † θ 1     f , where f is g iv en in (8). After n Darb oux transforma tions we hav e f [ n +1] =     Ω(P , Θ) P † Θ I     f . 3.1 Tw o-soliton matrix solution The tr ivial v acuum solution f = I (giving w = W = O) gives F =     Ω(P , Θ) P † Θ I     . ( 23) The eigenfunctions θ i and the adjoint eigenfunctions ρ i satisfy (15) and (16) resp ectiv ely . W e a gain choos e the eig enfunction solutions of the form (17). With this, we hav e Ω( θ j , ρ i ) = δ i,j I − p j B T i A j q i ( p j − q i ) e ( η j − γ i ) . As in the previo us section, we take A j = r j P j and B i = I . So the solutions w and W will b e d × d matrices. In the c a se n = 1, we o btain a one-soliton matrix solution. Expanding (23 ) gives F = I + r q P e ( γ − η ) − r p q ( p − q ) . If r > 0 and either q > p > 0 or 0 > q > p , or alterna tiv ely , if r < 0 a nd either p > q > 0 or 0 > p > q then w = − F x F − 1 = − 1 4 ( pq ) − 1 2 ( p − q ) 2 P sech  1 2 ( η − γ + ϕ )  sech  1 2 ( η − γ + χ )  , W = − F y F − 1 = ( p + q ) w, 8 where ϕ = log( − pr q ( p − q ) ) and χ = lo g( − r ( p − q ) ). Both w and W ha ve a unique maximum where η − γ = − log − ( pq − 1 ) 1 2 r ( p − q ) ! = λ. In the cas e n = d = 2, we obtain a tw o-so liton 2 × 2 matrix solution. Expanding (23) gives F = I +  A 1 e η 1 A 2 e η 2   δ i,j I − p j A j q i ( p j − q i ) e ( η j − γ i )  − 1 2 × 2 I e − γ 1 q 1 I e − γ 2 q 2 ! = I +  L 1 e γ 1 L 2 e γ 2  I e − γ 1 q 1 I e − γ 2 q 2 ! = I + 1 q 1 L 1 + 1 q 2 L 2 . Solving for L 1 and L 2 gives L 1 = ( p 2 − q 1 ) q 1 h (( p 1 − q 2 ) q 2 h 2 I + p 1 A 2 ) A 1 , L 2 = ( p 1 − q 2 ) q 2 h (( p 2 − q 1 ) q 1 h 1 I + p 2 A 1 ) A 2 , where h i = e ( γ i − η i ) − p i r i ( p i − q i ) q i , h = h 1 h 2 q 1 q 2 ( p 1 − q 2 )( p 2 − q 1 ) − αp 1 p 2 r 1 r 2 and α is as defined in the previous section. W e now inv estiga te the b eha viour of F a s t → ±∞ . W e first fix η 1 − γ 1 and a ssume without loss of gener alit y that 0 > p 2 > q 2 > p 1 > q 1 . The n, as t → −∞ , F ∼ I + r 1 q 1 P 1 h 1 and therefore w = − F x F − 1 ∼ − 1 4 ( p 1 q 1 ) − 1 2 ( p 1 − q 1 ) 2 P 1 sech  1 2  η 1 − γ 1 + ϕ − 1   × sech  1 2  η 1 − γ 1 + χ − 1   , (24) where ϕ − 1 = log ( − p 1 r 1 q 1 ( p 1 − q 1 ) ) and χ − 1 = log ( − r 1 ( p 1 − q 1 ) ). Note that w = − F x F − 1 and W = − F y F − 1 are inv ariant under the trans- formation F → F C where C is a non-singula r co ns tan t matrix. As t → + ∞ , we get F ∼ I + ( r 2 p 2 ( p 1 − q 2 ) − p 1 ( p 2 − q 2 ) A 2 )( p 2 − q 1 ) A 1 − ( p 2 ( p 1 − q 2 ) A 1 − αp 1 r 1 ( p 2 − q 2 )) h 1 r 2 p 2 q 1 ( p 1 − q 2 )( p 2 − q 1 ) + αp 1 p 2 r 1 r 2 ( p 2 − q 2 ) × ( p 2 − q 2 ) A 2  I + ( p 2 − q 2 ) A 2 q 2 r 2  ∼ I + ˜ r 1 q 1 e P 1 e γ 1 − η 1 − p 1 ˜ r 1 q 1 ( p 1 − q 1 ) , 9 where ˜ r 1 = r 1  1 − α ( p 1 − q 1 )( p 2 − q 2 ) ( p 1 − q 2 )( p 2 − q 1 )  = r 1 ( ˜ µ 1 , ˜ ν 1 ) ( µ 1 ,ν 1 ) , ˜ µ 1 = µ 1 − p 1 ( p 2 − q 2 )( µ 1 ,ν 2 ) µ 2 p 2 ( p 1 − q 2 )( µ 2 ,ν 2 ) , ˜ ν 1 = ν 1 − q 1 ( p 2 − q 2 )( µ 2 ,ν 1 ) ν 2 q 2 ( p 2 − q 1 )( µ 2 ,ν 2 ) and e P 1 = ˜ µ 1 ⊗ ˜ ν 1 ( ˜ µ 1 , ˜ ν 1 ) . Therefo r e w = − F x F − 1 ∼ − 1 4 ( p 1 q 1 ) − 1 2 ( p 1 − q 1 ) 2 e P 1 sech  1 2  η 1 − γ 1 + ϕ + 1   × sech  1 2  η 1 − γ 1 + χ + 1   , (25) where ϕ + 1 = log ( − p 1 ˜ r 1 q 1 ( p 1 − q 1 ) ) and χ + 1 = log ( − ˜ r 1 ( p 1 − q 1 ) ). Similarly , fixing γ 2 − η 2 gives w ∼ − 1 4 ( p 2 q 2 ) − 1 2 ( p 2 − q 2 ) 2 e P 2 sech  1 2  η 2 − γ 2 + ϕ − 2   × sech  1 2  η 2 − γ 2 + χ − 2   , t → −∞ , (26) w ∼ − 1 4 ( p 2 q 2 ) − 1 2 ( p 2 − q 2 ) 2 P 2 sech  1 2  η 2 − γ 2 + ϕ + 2   × sech  1 2  η 2 − γ 2 + χ + 2   , t → + ∞ , (27) where ˜ r 2 = r 2  1 − α ( p 1 − q 1 )( p 2 − q 2 ) ( p 1 − q 2 )( p 2 − q 1 )  = r 2 ( ˜ µ 2 , ˜ ν 2 ) ( µ 2 ,ν 2 ) , ˜ µ 2 = µ 2 − p 2 ( p 1 − q 1 )( µ 2 ,ν 1 ) µ 1 p 1 ( p 2 − q 1 )( µ 1 ,ν 1 ) , ˜ ν 2 = ν 2 − q 2 ( p 1 − q 1 )( µ 1 ,ν 2 ) ν 1 q 1 ( p 1 − q 2 )( µ 1 ,ν 1 ) , e P 2 = ˜ µ 2 ⊗ ˜ ν 2 ( ˜ µ 2 , ˜ ν 2 ) , ϕ − 2 = log ( − p 2 ˜ r 2 q 2 ( p 2 − q 2 ) ), χ − 2 = log ( − ˜ r 2 ( p 2 − q 2 ) ), ϕ + 2 = log ( − p 2 r 2 q 2 ( p 2 − q 2 ) ) and ϕ − 2 = log ( − p 2 r 2 q 2 ( p 2 − q 2 ) ). The soliton phase shifts Λ i = λ + i − λ − i are Λ 1 = log  r 1 ˜ r 1  = − log β , Λ 2 = log  ˜ r 2 r 2  = log β . The matrix amplitude o f the first so liton changes from − 1 4 ( p 1 q 1 ) − 1 2 ( p 1 − q 1 ) 2 P 1 to − 1 4 ( p 1 q 1 ) − 1 2 ( p 1 − q 1 ) 2 e P 1 and the matrix amplitude of the second soliton changes from − 1 4 ( p 2 q 2 ) − 1 2 ( p 2 − q 2 ) 2 e P 2 to − 1 4 ( p 2 q 2 ) − 1 2 ( p 2 − q 2 ) 2 P 2 as t changes from −∞ to + ∞ . Figure 2 shows a plot of the in teraction with P 1 =  1 − 1 0 0  and P 2 = 1 13  16 − 6 8 − 3  . 4 Conclusions In this pap er, we hav e considered a noncommutativ e KP and a noncomm utative mKP equation. It was s hown that solutions of ncmKP obtained from a binar y Darb oux transfor mation co uld b e expre s sed as a s ingle quasideterminant. In addition, we hav e used methods s imilar to thos e employ e d in [8] and obtained matrix versions of b oth ncKP and ncmKP . Finally , we inv estigated the inter- action prop erties of the tw o - soliton solution of b o th ncK P and ncmKP . 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