An integrable (classical and quantum) four-wave mixing Hamiltonian system

A four-wave mixing Hamiltonian system on the classical as well as on the quantum level is investigated. In the classical case, if one assumes the frequency resonance condition of the form $\omega_0 -\omega_1 +\omega_2 -\omega_3=0$, this Hamiltonian system is integrated in quadratures and the explicit formulas of solutions are presented. Under the same condition the spectral decomposition of quantum Hamiltonian is found and thus, the Heisenberg equation for this system is solved. Some applications of the obtained results in non-linear optics are disscused.