On a coupled system of equations describing pulse propagation in quadratic media

Abstract

In the slowly varying envelope approximation we derive the basic equations that describe the propagation of ultrashort pulses in quadratically nonlinear media in which a wave at a fundamental frequency interacts with its second harmonic. In the governing equations we keep linear terms that account for both second- and third-order dispersion and nonlinear terms describing both nonlinear dispersion and self-steepening of the pulse edge. We then perform the Painleve singularity structure analysis of the most general system of coupled partial differential equations we derived. In a specific case, when third-order dispersion is negligible, by using a Hirota-like method, we found zero- and one-parameter families of bright (fundamental frequency) and dark (second harmonic) solitary waves which travel at a locked group velocity.