Hierarchy of integrable Hamiltonians describing the nonlinearn-wave interaction

Abstract

In the paper we construct an hierarchy of integrable Hamiltonian systems which describe the variation of n-wave envelopes in nonlinear dielectric medium. The exact solutions for some special Hamiltonians are given in terms of elliptic functions of the first kind.

Integrable Systems of Neumann Type

“…In this paper we study a family of systems that interpolate between harmonic oscillators and Neumann systems in arbitrary dimensions on a specific symplectic leaf in the dual of the Lie algebra of skew-symmetric matrices viewed as upper triangular matrices endowed with the "constant coefficient" Poisson bracket. The idea of considering these systems comes from [14] who studied similar systems in the complex setting and for matrices with a different internal structure. It is remarkable that for the systems we consider, even though they are induced from Lie-Poisson systems, the general known involution theorems do not apply to our knowledge, so we give a direct proof of involution.…”

Integrability and correspondence of classical and quantum non-linear three-mode systems

“…and is called a three-wave system [3,4]. Other non-linear n-wave systems were integrated in [14]. Although the integrable Hamiltonian structure of the three-wave equations is well known, we will follow the general construction presented in [15] to reduce the considered system to a system of one degree of freedom.…”

Integrable Systems Related to Deformed so(5)

“…We refer to [4] and [9] for the treatment of Hamiltonian formulation of propagation of optical traveling wave pulses. Also one can find this type of nonlinear Hamiltonian optical system integrated by quadratures in [16].…”