Reductions and Real Forms of Hamiltonian Systems Related to N-wave Type Equations

Reductions of N -wave type equations related to simple Lie algebras and the hierarchy of their Hamiltonian structures are studied. The reduction group GR is realized as a subgroup of the Weyl group of the corresponding algebra. Some of the reduced equations are of physical interest. 1. Preliminary. The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method (ISM) is interesting and still open problem. We start with the well known form for the N -wave equations [1] : i[J,Qt]− i[I, Qx] + [[I, Q], [J,Q]] = 0 (1) which is solvable by the ISM applied to the generalized system of Zakharov-Shabat type: L(λ) = idx + [J,Q(x, t)]− λJ, J ∈ h. (2) The potential matrix Q(x, t) = ∑ α∈∆+ (qα(x, t)Eα + pα(x, t)E−α) ∈ g\h (3) takes values in the simple Lie algebra g with Cartan subalgebra h, ∆+ is the set of positive roots of g, and Eα, E−α and Hk form the Cartan-Weyl basis of g. Indeed the N -wave equation (1) is the compatibility condition [L(λ),M(λ)] = 0, where M(λ) = idt + [I, Q(x, t)]− λI, I ∈ h (4) 2. The reduction group. Our basic tool is the reduction group GR introduced by A. V. Mikhailov [2]. GR is a finite group which preserves the Lax representation (3), i.e. it ensures that the reduction constraints are automatically compatible with the evolution. Therefore GR must have two realizations: 1) GR ∈ Aut g and 2) GR ∈ ConfC. To each gk ∈ GR we relate a reduction condition for the Lax pair as follows [2]: Ck(L(Γk(λ))) = L(λ), Ck(M(Γk(λ))) = M(λ), (5) where Ck ∈ Aut g and Γk(λ) ∈ ConfC are the images of gk . It is well known that Aut g = V ⊗ Aut0 g where V is the group of outer automorphisms (the symmetry group of the Dynkin diagram) and Aut0 g is the group of inner automorphisms. We consider only those groups of inner automorphisms that preserve the form of L and M ; this means that GR must preserve the Cartan subalgebra. Then