Canonical Hamiltonian formulation of the nonlinear Schrödinger equation in a one-dimensional, periodic Kerr medium.

A canonical Hamiltonian formulation of the nonlinear Schrödinger equation has been derived in this paper. This formulation governs the dynamics of pulse propagation in a one-dimensional, periodic Kerr medium when the frequency content of the pulse is sufficiently narrow relative to a carrier frequency, and sufficiently far removed from a photonic band gap of the medium. Our Hamiltonian is numerically equal to the energy, and our fields obey canonical commutation relations, so the theory can easily be quantized. We clarify the nature of the conserved quantities associated with simple symmetries.