Periodic solutions of the short pulse model equation

Abstract

We develop a systematic procedure for constructing periodic solutions of the short pulse (SP) model equation that describes the propagation of ultrashort pulses in nonlinear media. We first summarize a novel exact method of solution that consists of a hodograph transformation to convert the SP equation into the sine-Gordon (sG) equation. We then exemplify some one- and two-phase periodic solutions of the sG equation for which the system of linear partial differential equations governing the inverse mapping can be integrated analytically to obtain periodic solutions of the SP equation in the form of the parametric representation. We obtain a new class of two-phase periodic solutions and investigate their properties in detail. Of particular interest is a nonsingular periodic solution that reduces to the breather solution in the long-wave limit. It may play an important role in studying the propagation of ultrashort pulses in a finite-length optical system.