38
J. L. Cie´ sli´ nski
a lot of time to structure preserving numerical methods (see, e.g., [15, 16]), which
is also close to Decio’s interests [17], but this subject evolved rather independently.
Spin-valued spectral problems seem to be quite natural in the geometric context
because Spin groups and Clifford algebras are very natural structures to deal with
orthogonal transformations, see the next section. Actually, even the standard su(2)valued Lax pairs for some well-known soliton equations (including sine-Gordon,
nonlinear Schrödinger and modKdV equations) can be rewritten in terms of the
group Spin(3) (which is isomorphic to SU (2), see, e.g., [18]). In particular, the
sine-Gordon equation φ, xy = sin φ arises as compatibility conditions for the Lax
pair
Ψ, x = UΨ , U =
iλ −
1
2 φ, x
1
2 φ, x −iλ
= λ σ 1 σ 2 +
1
2
φ, x σ 1 σ 3 ,
Ψ, t = UΨ , V =
1
4iλ
cos φ sin φ
sin φ − cos φ
=
σ 2 σ 1 cos φ + σ 3 σ 2 sin φ
4λ
,
(1)
where σ k denote Pauli matrices (and we replaced iσ 3 by σ 1 σ 2 , etc.). In order to
see the Lie algebra of a Spin group in the above formulas, one needs some basic
information on Clifford algebras and Spin groups, which will be provided in the
next section.
In the framework of Sym’s soliton surfaces approach su(2)-valued linear problems correspond to surfaces immersed in the Euclidean 3-space and other semisimple Lie algebras can be associated with surfaces in multi-dimensional (pseudo)Euclidean spaces [19]. Another natural possibility, followed in this paper, is to
extend this approach from SU (2) ∼ = Spin(3) on any Spin groups [20].
2 Clifford Algebras, Spin Groups, and Isometries of R p,q
Given linear space V and bilinear form ·| ·· (or, equivalently, the quadratic form
Q(v) := =v | v) we define the Clifford product by the following condition:
vw + wv = 2 | w ,
(v,w∈ V ) ,
(2)
where 1 is the unity (multiplicative identity) of the algebra. In other words, parallel
vectors commute and orthogonal vectors anti-commute. Hence, in particular,
v
−1
=
v
| v
,
(3)
i.e., vectors with Q(v) = 0 are invertible and the result is geometrically interpreted
as inversion with respect to the unit sphere.
J. L. Cie´ sli´ nski
a lot of time to structure preserving numerical methods (see, e.g., [15, 16]), which
is also close to Decio’s interests [17], but this subject evolved rather independently.
Spin-valued spectral problems seem to be quite natural in the geometric context
because Spin groups and Clifford algebras are very natural structures to deal with
orthogonal transformations, see the next section. Actually, even the standard su(2)valued Lax pairs for some well-known soliton equations (including sine-Gordon,
nonlinear Schrödinger and modKdV equations) can be rewritten in terms of the
group Spin(3) (which is isomorphic to SU (2), see, e.g., [18]). In particular, the
sine-Gordon equation φ, xy = sin φ arises as compatibility conditions for the Lax
pair
Ψ, x = UΨ , U =
iλ −
1
2 φ, x
1
2 φ, x −iλ
= λ σ 1 σ 2 +
1
2
φ, x σ 1 σ 3 ,
Ψ, t = UΨ , V =
1
4iλ
cos φ sin φ
sin φ − cos φ
=
σ 2 σ 1 cos φ + σ 3 σ 2 sin φ
4λ
,
(1)
where σ k denote Pauli matrices (and we replaced iσ 3 by σ 1 σ 2 , etc.). In order to
see the Lie algebra of a Spin group in the above formulas, one needs some basic
information on Clifford algebras and Spin groups, which will be provided in the
next section.
In the framework of Sym’s soliton surfaces approach su(2)-valued linear problems correspond to surfaces immersed in the Euclidean 3-space and other semisimple Lie algebras can be associated with surfaces in multi-dimensional (pseudo)Euclidean spaces [19]. Another natural possibility, followed in this paper, is to
extend this approach from SU (2) ∼ = Spin(3) on any Spin groups [20].
2 Clifford Algebras, Spin Groups, and Isometries of R p,q
Given linear space V and bilinear form ·| ·· (or, equivalently, the quadratic form
Q(v) := =v | v) we define the Clifford product by the following condition:
vw + wv = 2 | w ,
(v,w∈ V ) ,
(2)
where 1 is the unity (multiplicative identity) of the algebra. In other words, parallel
vectors commute and orthogonal vectors anti-commute. Hence, in particular,
v
−1
=
v
| v
,
(3)
i.e., vectors with Q(v) = 0 are invertible and the result is geometrically interpreted
as inversion with respect to the unit sphere.