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

0.2 MB·English

by  C. R. Gilson

Matrix solutions of a noncommutative KP
equation and a noncommutative mKP equation
9
0 C. R. Gilson, J. J. C. Nimmo and C. M. Sooman
0 Department of Mathematics,
2
University of Glasgow

n
Glasgow G12 8QW, UK
a
J

8
Abstract

]
I Matrix solutions of a noncommutative KP and a noncommutative
S
mKPequationwhichcanbeexpressedasquasideterminantsarediscussed.
.
n Inparticular,weinvestigateinteractionpropertiesoftwo-solitonsolutions.
i
l
n
1 Introduction
[

2
A considerable amount of literature exists concerning noncommutative inte-
v
1 grablesystems. ThisincludesnoncommutativeversionsoftheBurgersequation,
9 the KdV equation, the KP equation, the mKP equation and the sine-Gordon
8 equation [6, 14, 15, 2, 19, 8, 5, 12, 11, 1]. These equations can often be ob-
1
tained by simply removing the assumption that the dependent variables and
.
0 their derivatives in the Lax pair commute.
1 This paper is concerned with a noncommutative KP equation (ncKP) and
8
a noncommutative mKP equation (ncmKP) [19, 13]. The Lax pair for ncKP is
0
given by
:
v
Xi LKP =∂x2+vx−∂y, (1)
ar MKP =4∂x3+6vx∂x+3vxx+3vy+∂t. (2)
Both L and M are covariant under the Darboux transformation
KP KP
G = θ∂ θ−1, where θ is an eigenfunction for L ,M . From the compatibil-
θ x KP KP
ity condition [L ,M ] = 0 we obtain a noncommutative version of the KP
KP KP
equation:
(v +3v v +v ) +3v +3[v ,v ]=0, (3)
t x x xxx x yy x y
where u=v . For ncmKP, the Lax pair [19] is given by
x
L =∂2+2w∂ −∂ , (4)
mKP x x y
M =4∂3+12w∂2+6(w +w2+W)∂ +∂ . (5)
mKP x x x x t
1
Both L and M are covariant under the Darboux transformation
mKP mKP
G =((θ−1) )−1∂ θ−1, where θ is an eigenfunction for L ,M . The com-
θ x x mKP mKP
patibility condition [L ,M ]=0 gives
mKP mKP
0=w +w −6ww w+3W +3[w ,W] −3[w ,w]−3[W,w2], (6)
t xxx x y x + xx
0=W −w +[w,W]. (7)
x y
Equations (6) and (7) form a noncommutative version of the mKP equation.
Equation (7) is satisfied identically by applying the change of variables [19]
w =−f f−1, W =−f f−1. (8)
x y
The change of variables (8) has also been used in [6, 2] to study a noncom-
mutative mKP hierarchy. For both ncKP and ncmKP equations, we consider
families of solutions obtained from iterating binary Darboux transformations.
These solutions can be expressed as quasideterminants, which were introduced
by Gelfand et al in [4]. An n×n matrix Z =(z ) over a ring R (noncom-
ij n×n
mutative, in general) has n2 quasideterminants, each of which is denoted |Z|
ij
for 1 ≤ i,j ≤ n. Let rj denote the row vector obtained from the ith row of Z
i
bedeletingthejthentry,letci denotethecolumnvectorobtainedfromthejth
j
row of Z by deleting the ith entry, let Zij be the matrix obtained from Z by
deleting the ith row and jth column and assume that Zij is invertible. Then
|Z| exists and
ij
|Z| =z −rj(Zij)−1ci. (9)
ij ij i j
Fornotationalconvenience,weboxthe leadingelementaboutwhichthe expan-
sion is made so that
Zij ci
j
|Z|ij = rj z . (10)
(cid:12) i ij (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Thequasideterminantsolutionsobtai(cid:12)nedfromb(cid:12)inaryDarbouxtransformations
(cid:12) (cid:12)
reducetoratiosofgrammiandeterminantsinthecommutativelimitandwecall
them quasigrammians.
In this paper, we will consider the case where the dependent variables in
ncKP and ncmKP are matrices and apply the methods used in [8, 9, 10, 7].
Fromthis platform, we investigate the interactionof the two-solitonsolutionof
the matrix versions of ncKP and ncmKP.
2 Quasigrammian solutions of the ncKP equa-
tion
In this section we recall the construction of the quasigrammian solutions of
ncKP in [5]. The adjoint Lax pair is
L† =∂2+v† +∂ , (11)
KP x x y
M† =−4∂3−6v†∂ −3v† +3v†−∂ . (12)
KP x x x xx y t
2
Following the standard construction of a binary Darboux transformation (see
[16]), one introduces a potential Ω(φ,ψ) satisfying
Ω(φ,ψ) =ψ†φ, Ω(φ,ψ) =ψ†φ −ψ†φ,
x y x x
Ω(φ,ψ) =−4(ψ†φ −ψ†φ +ψ† φ)−6ψ†v φ.
t xx x x xx x
The parts of this definition are compatible when L [φ] = M [φ] = 0 and
KP KP
L† [ψ] = M† [ψ] = 0. Note that we can define Ω(Φ,Ψ) for any row vectors
KP KP
Φ and Ψ such that L [Φ] = M [Φ] = 0 and L† [Ψ] = M† [Ψ] = 0. Conse-
KP KP KP KP
quently, if Φ is an m-vector and Ψ is an n-vector, then Ω is an m×n matrix.
A binary Darboux transformation is defined by
φ =φ −θ Ω(θ ,ρ )−1Ω(φ ,ρ )
[n+1] [n] [n] [n] [n] [n] [n]
and
ψ =ψ −ρ Ω(θ ,ρ )−†Ω(θ ,ψ )†,
[n+1] [n] [n] [n] [n] [n] [n]
in which
θ =φ | , ρ =ψ | .
[n] [n] φ→θn [n] [n] ψ→ρn
Using the notation Θ=(θ ,...θ ) and P=(ρ ,...,ρ ) we have, for n≥1
1 n 1 n
Ω(Θ,P) Ω(φ,P) Ω(Θ,P)† Ω(Θ,ψ)†
φ = , ψ =
[n+1] Θ φ [n+1] P ψ
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
and (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Ω(Θ,P) Ω(φ,P)
Ω(φ ,ψ )= .
[n+1] [n+1] Ω(Θ,ψ) Ω(φ,ψ)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The effect of the binary Darboux trans(cid:12)formation (cid:12)
(cid:12) (cid:12)
L =G L G−1, M =G M G−1
KP θ,φ KP θ,φ KP θ,φ KP θ,φ
is that b c
vˆ=v +2θΩ(θ,ρ)−1ρ†.
0
After n binary Darboux transformations we have
n
v =v +2 θ Ω(θ ,ρ )−1ρ†
[n+1] 0 [k] [k] [k] [k]
k=1
X
Ω(Θ,P) P†
=v −2 . (13)
0 Θ 0
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3
2.1 Two-soliton matrix solution
We now derive matrix solutions of ncKP using the methods applied in [8]. The
trivial vacuum solution v =O gives
0
Ω(Θ,P) P†
v =−2 . (14)
Θ 0
(cid:12) (cid:12)
(cid:12) (cid:12)
The eigenfunctions θ and the adjoin(cid:12)t eigenfuncti(cid:12)ons ρ satisfy
i (cid:12) (cid:12) i
θ =θ , θ =−4θ , (15)
i,xx i,y i,t i,xxx
and
ρ =−ρ , ρ =−4ρ , (16)
i,xx i,y i,t i,xxx
respectively. We choose the simplest nontrivial solutions of (15) and (16):
θj =Ajeηj, ρi =Bie−γi, (17)
where η = p (x+p y −4p2t),γ = q (x+q y −4q2t) and A ,B are d×m
j j j j i i i i j i
matrices. With this, we have
BTA
Ω(θ ,ρ )= i j e(ηj−γi)+δ I.
j i i,j
(p −q )
j i
We takeA =r P ,wherer isascalarandP isaprojectionmatrix. With A
j j j j j j
chosen in this way, we must have m=d. We choose B =I and the solution u
i
will be a d×d matrix.
In the case n=1, we obtain a one-soliton matrix solution. Expanding (14)
gives
2rP
v = .
e(γ−η)+ r
(p−q)
The above calculation and others that that follow use the formula
(I −aP)−1 =I +aP(1−a)−1 where a6=1 is a scalar and P is any projection
matrix. We now have
1 1
u=v = (p−q)2Psech2 (η−γ+ξ) , (18)
x
2 2
(cid:18) (cid:19)
where ξ =log r .
(p−q)
In the case(cid:16)n = d(cid:17)= 2, we obtain a two-soliton 2×2 matrix solution. By
expanding (14) we get
v =2 A1eη1 A2eη2 Aje((pηjj−−qγii)) +δi,jI −2×12 IIee−−γγ21
(cid:0) (cid:1)(cid:16) Ie−γ1 (cid:17) (cid:18) (cid:19)
=2 K1eγ1 K2eγ2 Ie−γ2 =2(K1+K2).
(cid:18) (cid:19)
(cid:0) (cid:1)
4
Therefore
K I+ r1e(η1−γ1)P =e(η1−γ1)A − e(η1−γ1) K A ,
1 (p −q ) 1 1 (p −q ) 2 1
(cid:18) 1 1 (cid:19) 1 2
K I+ r2e(η2−γ2)P =e(η2−γ2)A − e(η2−γ2) K A .
2 (p −q ) 2 2 (p −q ) 1 2
(cid:18) 2 2 (cid:19) 2 1
We assume that the P are the rank-1 projection matrices
j
P = µj ⊗νj = µjνjT,
j (µ ,ν ) µTν
j j j j
wherethe 2-vectorsµ ,ν satisfythe condition(µ ,ν )6=0. Solving forK and
j j j j 1
K gives
2
(p −q )
K = 2 1 (g (p −q )I −A )A ,
1 g 2 1 2 2 1
(p −q )
K = 1 2 (g (p −q )I −A )A ,
2 g 1 2 1 1 2
where gi =e(γi−ηi)+ (pir−iqi), g =g1g2(p1−q2)(p2−q1)−αr1r2 and
α= (µ1,ν2)(µ2,ν1).
(µ1,ν1)(µ2,ν2)
We now investigatethe behaviour of v as t→±∞. This will demonstrate
[3]
that each soliton emerges from interaction undergoing a phase shift and that
the amplitude of each soliton may also change due to the interaction. We first
fix γ −η and assume without loss of generality that 0 > p > q > p > q .
1 1 2 2 1 1
As t→−∞
r P
v ∼2 1 1
g
1
and therefore
1 1
u=v ∼ (p −q )2P sech2 η −γ +ξ− , (19)
x 2 1 1 1 2 1 1 1
(cid:18) (cid:19)
(cid:0) (cid:1)
where ξ− =log r1 .
1 (p1−q1)
Note that u(cid:16)=vx is i(cid:17)nvariantunder the transformationv →v+C, where C
is a constant matrix. As t→+∞, we get
(r (p −q )−(p −q )A )(p −q )A −((p −q )A −αr (p −q ))(p −q )A
v∼2 2 1 2 2 2 2 2 1 1 1 2 1 1 2 2 2 2 2
r (p −q )(p −q ) g − αr1(p2−q2)
2 1 2 2 1 1 (p1−q2)(p2−q1)
rˆPˆ (cid:16) (cid:17)
∼2 1 1 ,
eγ1−η1 + p1rˆ1
(p1−q1)
5
where rˆ = r 1− α(p1−q1)(p2−q2) = r1(µˆ1,νˆ1), µˆ = µ − (p2−q2)(µ1,ν2)µ2,
1 1 (p1−q2)(p2−q1) (µ1,ν1) 1 1 (p1−q2)(µ2,ν2)
νˆ =ν − (p2−q2(cid:16))(µ2,ν1)ν2 and P = (cid:17)µˆ1⊗νˆ1. Therefore
1 1 (p2−q1)(µ2,ν2) 1 (µˆ1,νˆ1)
1 b 1
u=v ∼ (p −q )2P sech2 η −γ +ξ+ (20)
x 2 1 1 1 2 1 1 1
(cid:18) (cid:19)
(cid:0) (cid:1)
b
where ξ+ =log rˆ1 .
1 (p1−q1)
Similarly, fix(cid:16)ing γ2−(cid:17)η2 gives
1 1
u∼ (p −q )2P sech2 η −γ +ξ− , t→−∞ (21)
2 2 2 2 2 2 2 2
(cid:18) (cid:19)
1 1(cid:0) (cid:1)
u∼ (p −q )2Pb sech2 η −γ +ξ+ , t→+∞, (22)
2 2 2 2 2 2 2 2
(cid:18) (cid:19)
(cid:0) (cid:1)
where rˆ =r 1− α(p1−q1)(p2−q2) = r2(µˆ2,νˆ2), µˆ =µ − (p1−q1)(µ2,ν1)µ1,
2 2 (p1−q2)(p2−q1) (µ2,ν2) 2 2 (p2−q1)(µ1,ν1)
νˆ =ν −(p1−q(cid:16)1)(µ1,ν2)ν1,P = µˆ2⊗(cid:17)νˆ2,ξ− =log rˆ2 andξ+ =log r2 .
2 2 (p1−q2)(µ1,ν1) 2 (µˆ2,νˆ2) 2 (p2−q2) 2 (p2−q2)
The soliton phase shifts ∆ =ξ+−ξ− are (cid:16) (cid:17) (cid:16) (cid:17)
j j j
b
rˆ r
∆ =log 1 =logβ, ∆ =log 2 =−logβ,
1 r 2 rˆ
(cid:18) 1(cid:19) (cid:18) 2(cid:19)
where β =1− α(p1−q1)(p2−q2).
(p1−q2)(p2−q1)
The matrix amplitude of the first soliton changes from 1(p − q )2P to
2 1 1 1
1(p − q )2P and the matrix amplitude of the second soliton changes from
2 1 1 1
1(p −q )2P to 1(p −q )2P as t changes from −∞ to +∞. If
2 2 2 2 2 2 2 2
(µ ,ν ) = 0b(P P = 0) or (µ ,ν ) = 0(P P = 0), then α = 0 and therefore
1 2 2 1 2 1 1 2
β =1, so thbere is no phase shift but the matrix amplitudes may still change. If
(µ ,ν )=0and(µ ,ν )=0(givingP P =P P =0),thereisnophaseshiftor
1 2 2 1 1 2 2 1
changeinamplitudeandsothe solitonshavetrivialinteraction. Figure1shows
1 −2 96 −16
a plot of the interaction with P = and P = 1 .
1 0 0 2 121 −150 −25
(cid:18) (cid:19) (cid:18) (cid:19)
3 Quasigrammian solutions of the ncmKP equa-
tion
The construction of this particular binary Darboux transformation is given in
[18] and also in [17] (for Lax operators with matrix coefficients). The adjoint
Lax pair is
L† =∂2−2w† −2w†∂ +∂ ,
mKP x x x y
M† =−4∂3+12w†∂2+6(3w†−w†2−W†)∂ +6(w† −[w†,w†] −W†)−∂ .
mKP x x x x xx x + x t
6
Figure1: Plotofu=(u ) witht=0,p =−1,q =−39,p = 19,q = 1,
i,j 2×2 1 4 1 4 2 2 2 2
r =2 and r =1.
1 2
Fornotationalconvenience,wedenoteanelementofkerL† ∩kerM† byφ .
mKP mKP x
One introduces a potential Ω(φ,ψ) satisfying
Ω(φ,ψ) =ψ†φ , Ω(φ,ψ) =2ψ†wφ +ψ†φ −ψ†φ ,
x x y x xx x x
Ω(φ,ψ) =2(−2ψ† φ −2ψ†φ +2ψ†φ −3ψ†w2φ −3ψ†Wφ −3ψ†w φ
t xx x xxx x xx x x x x
+6ψ†wφ −6ψ†wφ ).
x x xx
A binary Darboux transformationis defined by
φ =φ −θ Ω(ρ ,θ )−1Ω(ρ ,φ )
[n+1] [n] [n] [n] [n] [n] [n]
and
ψ =ψ −ρ Ω(ρ ,θ )†−1Ω(ψ ,θ )†,
[n+1] [n] [n] [n] [n] [n] [n]
in which
θ =φ | , ρ =ψ | .
[n] [n] φ→θn [n] [n] ψ→ρn
7
Using the notation Θ=(θ ,...θ ) and P=(ρ ,...,ρ ), we have, for n≥1
1 n 1 n
Ω(Θ,P) Ω(φ,P) Ω(Θ,P)† Ω(Θ,ψ)†
φ = , ψ = .
[n+1] Θ φ [n+1] P ψ
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
The effect of the(cid:12)binary Darboux tr(cid:12)ansformation(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
L =G L G−1 , M =G M G−1
mKP θ,φx mKP θ,φx mKP θ,φx mKP θ,φx
is that b c
Ω ρ†
fˆ= f,
θ 1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where f is given in (8). After n Darb(cid:12)oux tra(cid:12)nsformations we have
Ω(P,Θ) P†
f = f.
[n+1] Θ I
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3.1 Two-soliton matrix so(cid:12)lution (cid:12)
The trivial vacuum solution f =I (giving w =W =O) gives
Ω(P,Θ) P†
F = . (23)
Θ I
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The eigenfunctions θi and the adj(cid:12)oint eigenfun(cid:12)ctions ρi satisfy (15) and (16)
respectively. We again choose the eigenfunction solutions of the form (17).
With this, we have
p BTA
Ω(θj,ρi)=δi,jI − j i j e(ηj−γi).
q (p −q )
i j i
As in the previous section, we take A = r P and B = I. So the solutions w
j j j i
and W will be d×d matrices.
In the case n=1, we obtain a one-soliton matrix solution. Expanding (23)
gives
rP
F =I + q .
e(γ−η)− rp
q(p−q)
If r >0 and either q >p>0 or 0> q >p, or alternatively, if r <0 and either
p>q >0 or 0>p>q then
1 1 1
w =−FxF−1 =− (pq)−21(p−q)2Psech (η−γ+ϕ) sech (η−γ+χ) ,
4 2 2
(cid:18) (cid:19) (cid:18) (cid:19)
W =−F F−1 =(p+q)w,
y
8
where ϕ = log( −pr ) and χ = log( −r ). Both w and W have a unique
q(p−q) (p−q)
maximum where
−(pq−1)12r
η−γ =−log =λ.
(p−q)
!
In the case n = d = 2, we obtain a two-soliton 2 × 2 matrix solution.
Expanding (23) gives
−1 Ie−γ1
F =I+ A1eη1 A2eη2 δi,jI− qi(ppjjA−jqi)e(ηj−γi) 2×2 Ie−q1γ2!
(cid:0) (cid:1)(cid:16) (cid:17) q2
Ie−γ1 1 1
=I+(cid:0)L1eγ1 L2eγ2(cid:1) Ie−qq12γ2!=I+ q1L1+ q2L2.
Solving for L and L gives
1 2
(p −q )q
L = 2 1 1((p −q )q h I +p A )A ,
1 h 1 2 2 2 1 2 1
(p −q )q
L = 1 2 2((p −q )q h I +p A )A ,
2 h 2 1 1 1 2 1 2
where hi =e(γi−ηi)− (pip−iqrii)qi, h=h1h2q1q2(p1−q2)(p2−q1)−αp1p2r1r2 and
α is as defined in the previous section.
We now investigate the behaviour of F as t → ±∞. We first fix η −γ
1 1
and assume without loss of generality that 0 > p > q > p > q . Then, as
2 2 1 1
t→−∞,
r1P
F ∼I+ q1 1
h
1
and therefore
1 1
w =−FxF−1 ∼−4(p1q1)−12(p1−q1)2P1sech 2 η1−γ1+ϕ−1
(cid:18) (cid:19)
1 (cid:0) (cid:1)
×sech η −γ +χ− , (24)
2 1 1 1
(cid:18) (cid:19)
(cid:0) (cid:1)
where ϕ− =log( −p1r1 ) and χ− =log( −r1 ).
1 q1(p1−q1) 1 (p1−q1)
Note that w = −F F−1 and W = −F F−1 are invariant under the trans-
x y
formation F → FC where C is a non-singular constant matrix. As t → +∞,
we get
(r p (p −q )−p (p −q )A )(p −q )A −(p (p −q )A −αp r (p −q ))
F∼I+ 2 2 1 2 1 2 2 2 2 1 1 2 1 2 1 1 1 2 2
h r p q (p −q )(p −q )+αp p r r (p −q )
1 2 2 1 1 2 2 1 1 2 1 2 2 2
(p −q )A
×(p −q )A I + 2 2 2
2 2 2 q r
(cid:18) 2 2 (cid:19)
r˜1P
∼I + q1 1 ,
eγ1−η1 − p1r˜1
eq1(p1−q1)
9
where r˜ = r 1− α(p1−q1)(p2−q2) = r1(µ˜1,ν˜1), µ˜ = µ − p1(p2−q2)(µ1,ν2)µ2,
1 1 (p1−q2)(p2−q1) (µ1,ν1) 1 1 p2(p1−q2)(µ2,ν2)
ν˜ =ν − q1(p2−(cid:16)q2)(µ2,ν1)ν2 and P =(cid:17) µ˜1⊗ν˜1. Therefore
1 1 q2(p2−q1)(µ2,ν2) 1 (µ˜1,ν˜1)
1 1
w =−FxF−1 ∼−4(p1q1)−e12(p1−q1)2P1sech 2 η1−γ1+ϕ+1
(cid:18) (cid:19)
1 (cid:0) (cid:1)
×sech η −γ +eχ+ , (25)
2 1 1 1
(cid:18) (cid:19)
(cid:0) (cid:1)
where ϕ+ =log( −p1r˜1 ) and χ+ =log( −r˜1 ).
1 q1(p1−q1) 1 (p1−q1)
Similarly, fixing γ −η gives
2 2
1 1
w ∼−4(p2q2)−12(p2−q2)2P2sech 2 η2−γ2+ϕ−2
(cid:18) (cid:19)
1 (cid:0) (cid:1)
×sech η −γ +eχ− , t→−∞, (26)
2 2 2 2
(cid:18) (cid:19)
1 (cid:0) (cid:1) 1
w ∼−4(p2q2)−12(p2−q2)2P2sech 2 η2−γ2+ϕ+2
(cid:18) (cid:19)
1 (cid:0) (cid:1)
×sech η −γ +χ+ , t→+∞, (27)
2 2 2 2
(cid:18) (cid:19)
(cid:0) (cid:1)
where r˜ = r 1− α(p1−q1)(p2−q2) = r2(µ˜2,ν˜2), µ˜ = µ − p2(p1−q1)(µ2,ν1)µ1,
2 2 (p1−q2)(p2−q1) (µ2,ν2) 2 2 p1(p2−q1)(µ1,ν1)
ν˜ =ν −q2(p1−(cid:16)q1)(µ1,ν2)ν1,P = µ˜2⊗(cid:17)ν˜2,ϕ− =log( −p2r˜2 ),χ− =log( −r˜2 ),
2 2 q1(p1−q2)(µ1,ν1) 2 (µ˜2,ν˜2) 2 q2(p2−q2) 2 (p2−q2)
ϕ+ =log( −p2r2 ) and ϕ− =log( −p2r2 ).
2 The soqli2t(opn2−pq2h)ase shift2seΛ =λ+q2(−p2λ−−q2)are
i i i
r r˜
Λ =log 1 =−logβ, Λ =log 2 =logβ.
1 r˜ 2 r
(cid:18) 1(cid:19) (cid:18) 2(cid:19)
The matrix amplitude of the first soliton changes from
−41(p1q1)−12(p1−q1)2P1 to −41(p1q1)−12(p1−q1)2P1 and the matrix amplitude
of the second soliton changes from −41(p2q2)−21(p2−q2)2P2 to
−41(p2q2)−12(p2−q2)2P2 as t changes from −∞ toe+∞. Figure 2 shows a plot
1 −1 16 e−6
of the interaction with P = and P = 1 .
1 0 0 2 13 8 −3
(cid:18) (cid:19) (cid:18) (cid:19)
4 Conclusions
Inthispaper,wehaveconsideredanoncommutativeKPandanoncommutative
mKP equation. It was shown that solutions of ncmKP obtained from a binary
Darboux transformation could be expressed as a single quasideterminant. In
addition, we have used methods similar to those employed in [8] and obtained
matrix versions of both ncKP and ncmKP. Finally, we investigated the inter-
action properties of the two-soliton solution of both ncKP and ncmKP. This
showed that as well as undergoing a phase-shift, the amplitude of each soliton
can also change, giving a more elegant picture than the commutative case.
10