Generalized Hamiltonian Formalism in Nonlinear Optics

The basic mathematical apparatus of nonlinear optics consists of an array of nonlinear PDEs for the complex amplitudes of an envelope of interacting wave trains. In the general case, these equations include linear and nonlinear dissipative terms. However, in many important cases, they are small and can be neglected: therefore the equations are conservative, and the medium is transparent. According to the Kramers–Kronig relations, stemming from the principle of causality, the transparency can be realized at most in a limited spectral band, and even in this case some dissipation inevitably exists. Nevertheless, such fundamental nonlinear effects as the generation of high harmonics, induced Raman scattering, and self-focusing can be described by the conservative equations, preserving energy. It is remarkable that these equations are not just conservative. It is an experimental fact that in all known cases, the conservative equations of nonlinear optics are also Hamiltonian systems. The nonlinear Schrödinger equation is a perfect example of that sort. Actually, it is not astonishing. All macroscopic equations describing real media can be derived, at least in principle, from the microscopic quantum equations, which are Hamiltonian by definition. The original Hamiltonian