Integrability of Nonlinear Systems

Nonlinear Waves, Solitons, and IST 25
There has been significant interest in the SDYM equations as a “master” integrable
system. Ward [19] has conjectured that perhaps all “integrable” equations,
e.g. soliton equations, may be obtained as a reduction of the SDYM equations.
The reduction process has three aspects (see [7]).
i) Employ the gauge freedom (5.6) of the equations. Frequently the choice of
gauge can simplify the analysis and make the search for integrable reductions
considerably easier.
ii) Reduction of independent variables, i.e., γ a (α, ᾱ, β, ¯β) can be functions of
α, orα, β, etc.
iii) Choice of the underlying gauge group (algebra) in which one carries out the
analysis. Sometimes it is a matrix algebra, e.g., su(n),gl(n); but in many
interesting cases the gauge algebra is infinite dimensional, e.g., sdiff(S 3 ).
It is often easiest to make identifications via the linear pair of SDYM. For
example, suppose γ a ∈ gl(N), γ a = γ a (α, β), γ¯β = iJ = diagonal matrix,
γᾱ = iA 0 = diagonal matrix. Then, calling γ α = Q, γ β = A 1 (5.1a–5.1b) reduce
to
∂Ψ
=(Q + iζJ)Ψ
∂α (5.7a)
∂Ψ
∂β =(A 1 − iζA 0 )Ψ.
(5.7b)
In fact, (5.7) is the linear pair associated with the N wave system (when N =3
it is the 3 wave system). We need go no further in writing the equations (cf. [7])
except to point out that once the spatial part of the linear system is known, then
actually the entire hierarchy can be ascertained. Other special cases include KdV,
NLS, sine Gordon etc.
It is also worth remarking that the well-known 2+1 dimensional soliton system
can be obtained from SDYM if we assume that the gauge potentials are
elements of the infinite dimensional gauge algebra of differential polynomials.
For example, suppose γᾱ = γ ¯β = 0, Q = Q(α, y, β), A 1 = A 1 (α, y, β), J, A 0
are diagonal matrices and
Then (5.1) reduces to
γ α = Q + J ∂ ∂y
γ β = A 1 + A 0
∂
∂y . (5.8)
∂Ψ
(Q
∂α = + J ∂ )
Ψ
∂y
(
∂Ψ
∂β = ∂
A 1 + A 0
∂y
)
Ψ.
(5.9)
Compatibility of (5.9) yields the N wave equations in 2+1 dimensions (here
the independent variables are α, y, β; i.e., β plays the role of time). Again the