Discrete energy levels of bright solitons in lithium niobate ferroelectrics

In the framework of a simple model of a one-dimensional array of a ferroelectric slab of domains, the polarization with time and space is explained by the Klein-Gordon equation. A perturbation of the K-G equation makes it possible to use this continuum model through a progressive-wave nonlinear Schrodinger equation (NLSE). The latter gives rise to a bright soliton that moves with a certain velocity. The bright soliton controls itself so that both bright and dark solitons appear at the same time with discrete energy levels that are estimated from a hypergeometric function, but the dark soliton is not visible as it is part of the complex solution that indicates absorption of energy, i.e., the presence of an energy gap. The pulse width for switching of ferroelectrics can be estimated from the analysis of the NLSE and matches that calculated from this model.