42
J. L. Cie´ sli´ nski
˜
Ψ = DΨ,
˜
Ψ , μ = ˜
U μ ˜
Ψ ,
˜
U μ = D, μ D
−1
+ DU μ D
−1 ,
(16)
provided that ˜
U μ has the same dependence on dependent variables as U μ .
Instead of caring about dependent variables one can try to describe the structure
of the Lax pair. The considered nonlinear system follows uniquely from compatibility conditions [8, 9, 13]. Then the Darboux transformation has to preserve this
structure.
The structure is characterized primarily by the dependence on λ (e.g., divisor of
poles) [24, 25], reduction group (loop group) [26], and other invariants of Darboux
transformations, like linear and multilinear constraints on coefficients of the Laurent
expansion around poles [13].
Different methods of constructing the Darboux matrix need different form of λdependence of D (these forms are equivalent up to a λ-dependent scalar factor).
In particular, one can assume D as polynomial in λ (eigenvalues, corresponding to
solitons, are zeros of det D) [27, 28], sum of simple fractions (eigenvalues: poles of
D and D −1 ) [24, 26], or a “realization” (D = N + F (λ − A) −1 G) [29, 30].
The motivation for the case of Spin groups came from yet another approach [31].
Multiplying (16) by D 2 (λ) we get
D, μ D + DU μ D = ˜
U μ D
2 .
(17)
It is crucial point that the right-hand side vanishes for λ + and λ − such that
D 2 (λ ± ) = 0. Then, we obtain a solution of the remaining equation: D(λ ± ) =
ϕ ± Ψ (λ ± )d ± Ψ (λ ± ) −1 , where d ± = const, (d ± ) 2 = 0 and ϕ ± are two scalar
functions. Finally, D(λ) is given as a linear combination of D(λ + ) and D(λ − ) with
coefficients linear in λ [31], which yields one-soliton Darboux matrix.
This approach was extended on the multi-soliton case for 2 × 2 matrix problems
[32], and analogous generalization for Spin-valued linear problems is introduced
below.
4.1 The Darboux-Bäcklund Transformation in the Case of
Spin Groups
In this section we focus on the linear problem (12) and assume Ψ ∈ Spin(V ⊕ W ).
Using commutation relations (4) one can easily verify that
β β β(U μ ) = −U μ ,
e U μ (λ) = U μ (−λ)e ,
U, μ (λ) = U μ ( ¯
λ) ,
(18)
where e = e r+1 e r+2 . . . e r+q . In a way analogous to the matrix case one can obtain
corresponding formulas for Ψ :
N(Ψ ) ≡ Ψ β β β(Ψ ) = const ,
e Ψ (λ) = Ψ (−λ)e ,
Ψ (λ) = Ψ ( ¯
λ) .
(19)
J. L. Cie´ sli´ nski
˜
Ψ = DΨ,
˜
Ψ , μ = ˜
U μ ˜
Ψ ,
˜
U μ = D, μ D
−1
+ DU μ D
−1 ,
(16)
provided that ˜
U μ has the same dependence on dependent variables as U μ .
Instead of caring about dependent variables one can try to describe the structure
of the Lax pair. The considered nonlinear system follows uniquely from compatibility conditions [8, 9, 13]. Then the Darboux transformation has to preserve this
structure.
The structure is characterized primarily by the dependence on λ (e.g., divisor of
poles) [24, 25], reduction group (loop group) [26], and other invariants of Darboux
transformations, like linear and multilinear constraints on coefficients of the Laurent
expansion around poles [13].
Different methods of constructing the Darboux matrix need different form of λdependence of D (these forms are equivalent up to a λ-dependent scalar factor).
In particular, one can assume D as polynomial in λ (eigenvalues, corresponding to
solitons, are zeros of det D) [27, 28], sum of simple fractions (eigenvalues: poles of
D and D −1 ) [24, 26], or a “realization” (D = N + F (λ − A) −1 G) [29, 30].
The motivation for the case of Spin groups came from yet another approach [31].
Multiplying (16) by D 2 (λ) we get
D, μ D + DU μ D = ˜
U μ D
2 .
(17)
It is crucial point that the right-hand side vanishes for λ + and λ − such that
D 2 (λ ± ) = 0. Then, we obtain a solution of the remaining equation: D(λ ± ) =
ϕ ± Ψ (λ ± )d ± Ψ (λ ± ) −1 , where d ± = const, (d ± ) 2 = 0 and ϕ ± are two scalar
functions. Finally, D(λ) is given as a linear combination of D(λ + ) and D(λ − ) with
coefficients linear in λ [31], which yields one-soliton Darboux matrix.
This approach was extended on the multi-soliton case for 2 × 2 matrix problems
[32], and analogous generalization for Spin-valued linear problems is introduced
below.
4.1 The Darboux-Bäcklund Transformation in the Case of
Spin Groups
In this section we focus on the linear problem (12) and assume Ψ ∈ Spin(V ⊕ W ).
Using commutation relations (4) one can easily verify that
β β β(U μ ) = −U μ ,
e U μ (λ) = U μ (−λ)e ,
U, μ (λ) = U μ ( ¯
λ) ,
(18)
where e = e r+1 e r+2 . . . e r+q . In a way analogous to the matrix case one can obtain
corresponding formulas for Ψ :
N(Ψ ) ≡ Ψ β β β(Ψ ) = const ,
e Ψ (λ) = Ψ (−λ)e ,
Ψ (λ) = Ψ ( ¯
λ) .
(19)