Physica D 135 (2000) 345–368 Modeling light bullets with the two-dimensional sine–Gordon equation J.X. Xin Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA Received 16 November 1998; received in revised form 22 February 1999; accepted 26 April 1999 Communicated by C.K.R.T. Jones 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. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Light bullets; Sine–Gordon equation; Maxwell–Bloch system 1. Introduction Ultra short optical pulses of femtosecond duration are of tremendous technological and fundamental interest, and have been the subject of many recent studies in nonlinear optics [4,10,15,16,20,21], among others. Such short light pulses have potential applications in time-domain spectroscopy of dielectrics, semiconductors and transient chemical processes, probing high-intensity plasmas, imaging and medical infrared tomography [20], light propagation through atmosphere, and near material interfaces [4]. Conventional nonlinear optics usually operate with almost harmonic EM (electromagnetic) oscillations modulated by an envelope much longer than a single cycle of the oscillation so that there are on the order of 100–1000 oscillations under the envelope. These pulses are hence referred to as long pulses. The time honored approach for long pulses is the slowly varying envelope approximation and the derivation of the nonlinear Schrödinger (NLS) equation for the envelope, see e.g [27]. In clear contrast, short pulses typically have only a few EM oscillation cycles under their envelopes. So there is a lack of separation of scales between the envelope and the underlying EM oscillations, which seems to make ∗ E-mail address: xin@math.arizona.edu (J.X. Xin) 0167-2789/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII:S0167-2789(99)00128-1