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].

\[ Q(x,t)=\sum _{\alpha \in \nabla _{+}}(q_{\alpha }(x,t)E_{\alpha }+p_{\alpha }(x,t) E_{-\alpha })\in g/h, \] 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].

Reviewer: Iulia Hirică (Bucureşti)

##### 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 |