Variational Principles, Lie Point Symmetries, and Similarity Solutions of the Vector Maxwell Equations in Non-linear Optics

The vector Maxwell equations of nonlinear optics coupled to a single Lorentz oscillator and with instantaneous Kerr nonlinearity are investigated by using Lie symmetry group methods. Lagrangian and Hamiltonian formulations of the equations are obtained. The aim of the analysis is to explore the properties of Maxwell's equations in nonlinear optics, without resorting to the commonly used nonlinear Schr\"odinger (NLS) equation approximation in which a high frequency carrier wave is modulated on long length and time scales due to nonlinear sideband wave interactions. This is important in femto-second pulse propagation in which the NLS approximation is expected to break down. The canonical Hamiltonian description of the equations involves the solution of a polynomial equation for the electric field $E$, in terms of the the canonical variables, with possible multiple real roots for $E$. In order to circumvent this problem, non-canonical Poisson bracket formulations of the equations are obtained in which the electric field is one of the non-canonical variables. Noether's theorem, and the Lie point symmetries admitted by the equations are used to obtain four conservation laws, including the electromagnetic momentum and energy conservation laws, corresponding to the space and time translation invariance symmetries. The symmetries are used to obtain classical similarity solutions of the equations. The traveling wave similarity solutions for the case of a cubic Kerr nonlinearity, are shown to reduce to a single ordinary differential equation for the variable $y=E^2$, where $E$ is the electric field intensity. The differential equation has solutions $y=y(\xi)$, where $\xi=z-st$ is the traveling wave variable and $s$ is the velocity of the wave. These solutions exhibit new phenomena not obtainable by the NLS approximation. The characteristics of the solutions depends on the values of the wave velocity $s$ and the energy integration constant $\epsilon$. Both smooth periodic traveling waves and non-smooth solutions in which the electric field gradient diverges (i.e. solutions in which $|E_\xi|\to\infty$ at specific values of $E$, but where $|E|$ is bounded) are obtained. The traveling wave solutions also include a kink-type solution, with possible important applications in femto-second technology.