Modeling light bullets with the two-dimensional sine–Gordon equation

Abstract

Light bullets are spatially localized ultra-short optical pulses in more than one space dimensions. They contain only a few electromagnetic oscillations under their envelopes and propagate long distances without essentially changing shapes. Light bullets of femtosecond durations have been observed in recent numerical simulation of the full Maxwell systems. The sine–Gordon (SG) equation comes as an asymptotic reduction of the two level dissipationless Maxwell–Bloch system. We derive a new and complete nonlinear Schrödinger (NLS) equation in two space dimensions for the SG pulse envelopes so that it is globally well-posed and has all the relevant higher order terms to regularize the collapse of the standard critical NLS (CNLS). We perform a modulation analysis and found that SG pulse envelopes undergo focusing–defocusing cycles. Numerical results are in qualitative agreement with asymptotics and reveal the SG light bullets, similar to the Maxwell light bullets. We achieve the understanding that the light bullets are manifestations of the persistence and robustness of the complete NLS asymptotics.