516
P. Adamopoulou and G. Papamikos
3 Soliton Solutions for the Vector mKdV Hierarchy
Employing the method of rational dressing (see for example [8, 12]), we construct
soliton and breather solutions to the vmKdV hierarchy. Given that Ψ (x, t 2n+1 , λ)
is the fundamental solution to the overdetermined compatible system of linear
equations
L(λ)Ψ = 0 , A 2n+1 (λ)Ψ = 0 , n = 1, 2, . . . ,
(8)
then a Darboux transformation
Ψ → Φ = M(λ)Ψ , with det M(λ) = 0 ,
is a gauge transformation that leaves the linear equations form invariant
L(λ)Φ = 0 ,
A 2n+1 (λ)Φ = 0 ,
(9)
where
L(λ) = D x −
U (λ) with
U(λ) = U ( u, λ), and similarly for
A 2n+1 (λ). Here u
is a new exact solution of the vmKdV hierarchy. The Darboux matrix M(λ) can be a
constant matrix with respect to the independent variables and it will result to the O N -
invariance of the vmKdV hierarchy, or it can be a non-constant matrix which leads to
Darboux-dressing relations or to a dressing chain (Bäcklund transformation). From
the compatibility condition of (8) and (9) it follows that the Darboux matrix M(λ)
satisfies the Lax–Darboux equations
D x M =
UM − MU , D t 2n+1 M =
V 2n+1 M − MV 2n+1 ,
(10)
or equivalently,
L = MLM
−1 ,
A 2n+1 = MA 2n+1 M
−1 .
(11)
We further require that the Darboux matrix also satisfies the relations
M(λ)M(λ)
T
= 1 , M(λ) = M(λ
∗ )
∗ , M(λ) = QM(−λ)Q
−1 ,
(12)
which follow from the symmetries (4) of the Lax operators. The Darboux matrices
that lead to the one-soliton and one-breather solutions of the vmKdV hierarchy are
parameterised by a complex number and an element of the complex Grassmannian
Gr(s, C
N +2 ), with s = 1, 2, . . . , N + 1 in the case of a breather solution, and s = 1
for a soliton solution. Further details are presented in [1], see also [8] in relation to
the dressing for the vector sine-Gordon equation.