44
J. L. Cie´ sli´ nski
The full multi-soliton case is still under construction but basic results are already
obtained. In the general case, when D(λ) ∈ Pin(V ⊕ W ) is a polynomial in λ, we
multiply (16) by the spinor norm N(D), defined by (9), and obtain:
D, μ β β β(D) + DU μ β β β(D) = ˜
U μ N(D)
(26)
Note that the special case D ∈ V ⊕ W reduces to (17), because then β β β(D) = D and
N(D) = D 2 . In the general case the right-hand side of (26) vanishes if N(D(λ)) =
0. This is a polynomial in λ. Its roots, denoted by λ k , are constant by virtue of (20).
Similarly as in the case (17) we define
C k := Ψ
−1 (λ k )D(λ k )Ψ (λ k ) .
(27)
Substituting it to (26) we obtain
C k β β β(C k ) = 0 ,
C k , μ β β β(C k ) = 0 .
(28)
Following [34], we can solve this system:
C k = ˆ
C k d k , d k = const ∈ V , d
2
k = 0 , ˆ
C k ∈ Pin(V ⊕ W ) .
(29)
Therefore, D(λ k ) = ˆ
D k Ψ (λ k )d k Ψ −1 (λ k ), where ˆ
D k ∈ Pin(V ⊕ W ). It is natural to
conjecture that D(λ) can be uniquely (up to a normalization) defined by eigenvalues
λ k and null vectors d k and can be expressed in an algebraic way by elements
Ψ (λ k )d k Ψ −1 (λ k ). Reduction groups will be expressed in terms of involutions in
the Clifford algebra.
5 Conclusions and Open Problems
Extending results of [31] we presented a method of construction of the Darboux
matrix for Spin-valued linear problems. The construction is fully done only in
the one-soliton case. The iteration of one-soliton transformations is not difficult,
and two-soliton Darboux matrix is shown. However, the multi-soliton case needs
further elaboration. Then, our approach should work also for multi-dimensional
Lobachevsky spaces [20, 35] and, possibly, for all problems described in [35, 36].
References
1. D. Levi, A. Sym, Integrable systems describing surfaces of non-constant curvature. Phys. Lett.
A 149, 381–387 (1990)
2. D. Levi, A. Sym, G.-Z. Tu, A working Algorithm to Isolate Integrable Surfaces in E 3 . Dip.
Fis. INFN N.761, 10/10/1990. Preprint (1990)
J. L. Cie´ sli´ nski
The full multi-soliton case is still under construction but basic results are already
obtained. In the general case, when D(λ) ∈ Pin(V ⊕ W ) is a polynomial in λ, we
multiply (16) by the spinor norm N(D), defined by (9), and obtain:
D, μ β β β(D) + DU μ β β β(D) = ˜
U μ N(D)
(26)
Note that the special case D ∈ V ⊕ W reduces to (17), because then β β β(D) = D and
N(D) = D 2 . In the general case the right-hand side of (26) vanishes if N(D(λ)) =
0. This is a polynomial in λ. Its roots, denoted by λ k , are constant by virtue of (20).
Similarly as in the case (17) we define
C k := Ψ
−1 (λ k )D(λ k )Ψ (λ k ) .
(27)
Substituting it to (26) we obtain
C k β β β(C k ) = 0 ,
C k , μ β β β(C k ) = 0 .
(28)
Following [34], we can solve this system:
C k = ˆ
C k d k , d k = const ∈ V , d
2
k = 0 , ˆ
C k ∈ Pin(V ⊕ W ) .
(29)
Therefore, D(λ k ) = ˆ
D k Ψ (λ k )d k Ψ −1 (λ k ), where ˆ
D k ∈ Pin(V ⊕ W ). It is natural to
conjecture that D(λ) can be uniquely (up to a normalization) defined by eigenvalues
λ k and null vectors d k and can be expressed in an algebraic way by elements
Ψ (λ k )d k Ψ −1 (λ k ). Reduction groups will be expressed in terms of involutions in
the Clifford algebra.
5 Conclusions and Open Problems
Extending results of [31] we presented a method of construction of the Darboux
matrix for Spin-valued linear problems. The construction is fully done only in
the one-soliton case. The iteration of one-soliton transformations is not difficult,
and two-soliton Darboux matrix is shown. However, the multi-soliton case needs
further elaboration. Then, our approach should work also for multi-dimensional
Lobachevsky spaces [20, 35] and, possibly, for all problems described in [35, 36].
References
1. D. Levi, A. Sym, Integrable systems describing surfaces of non-constant curvature. Phys. Lett.
A 149, 381–387 (1990)
2. D. Levi, A. Sym, G.-Z. Tu, A working Algorithm to Isolate Integrable Surfaces in E 3 . Dip.
Fis. INFN N.761, 10/10/1990. Preprint (1990)