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On the reductions and Hamiltonian structures of $$N$$-wave type equations. (English) Zbl 1072.37050
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 156-170 (2001).
The $$N$$-wave equations $i[J, Q_t]-i[I, Q_x]+[[I, Q], [J, Q]]=0$ are solvable by the inverse scattering method (ISM) applied to the generalized system of Zakharov-Shabat type $L(\lambda )\Psi (x,t,\lambda )= \biggl(i{d\over dx} + [J, Q(x, t)]-\lambda J\biggr) \Psi (x,t,\lambda )=0, \quad J\in h,$
$Q(x,t)=\sum _{\alpha \in \nabla _{+}}(q_{\alpha }(x,t)E_{\alpha }+p_{\alpha }(x,t) E_{-\alpha })\in g/h,$ where $$h$$ is the Cartan subalgebra and $$E_{\alpha }$$ are the root vectors of the simple Lie algebra $$g.$$
In the present paper, it is shown how one can exibit new examples of integrable $$N$$-wave-type interactions some of which have applications to physics. The integrability of a rich family of $$N$$-wave equations and their importance as universal model of wave-wave interactions was proved by F. Calogero [J. Math. Phys. 30, 639–654 (1989; Zbl 0692.35098)].
The approach of this paper allows one to enrich this family still further. The authors paid special attention to the $${\mathbb Z}_2$$ and $${\mathbb Z}_2\times {\mathbb Z}_2$$-reductions including ones that can be imbedded also in the Weyl group of $$g.$$ The consequences of these restrictions on the properties of their Hamiltonian structure are analyzed on specific examples which find applications to nonlinear optics.
For the entire collection see [Zbl 0957.00038].
##### MSC:
 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems 35Q58 Other completely integrable PDE (MSC2000) 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 35L05 Wave equation 78A60 Lasers, masers, optical bistability, nonlinear optics
Zbl 0692.35098