Appl Phys B (2011) 103:591–596 DOI 10.1007/s00340-010-4282-5 Nonlinear light bullets in purely lossy, self-focusing media M.A. Porras Received: 4 August 2010 / Revised version: 13 September 2010 / Published online: 30 October 2010 © Springer-Verlag 2010 Abstract We report on light bullet propagation supported by a dynamic equilibrium between self-focusing and nonlin- ear losses. Unlike solitons, self-focusing is not eliminated, but only frozen. Losses are not compensated by a coni- cal energy flux, but only by that arising from self-focusing. These light bullets rebuild after obstacles, a property that was thought to be exclusive of conical waves. 1 Introduction In this Letter we call attention to a family of purely nonlin- ear waves that are localized in multiple dimensions, prop- agate without distortion, are resistant to nonlinear losses (NLLs), and self-reconstruct after obstacles. This property set is believed belong exclusively to light bullets (LBs) of conical type [1–7], characterized by asymptotic linear be- havior. However, it is also verified by these fully nonlinear waves with no conical geometry as a result of a balance be- tween two nonlinearities: self-focusing (SF) and NLLs. These LBs present a central peak with the approximate form of a soliton, but with converging wave fronts, that it is continuously being absorbed and replenished by an in- ward energy flux from its surrounding and created by SF alone. No conical geometry is necessary for replenishment. At the same time, NLLs prevent SF to progress toward col- lapse, resulting in a frozen SF state. Unlike the well-known dissipative solitons and dissipative bullets, [8–10] these LBs M.A. Porras ( ) Departamento de Física Aplicada a los Recursos Naturales, Universidad Politécnica de Madrid, Rios Rosas 21, 28003 Madrid, Spain e-mail: miguelangel.porras@upm.es Fax: 34-91-3366952 are permanently losing energy without a compensating gain, and therefore will be called lossy LBs (LLBs). Stationarity with permanent losses requires a reservoir with infinite en- ergy, which in turn implies weak localization, exactly the same way as conical LBs [3, 4]. After describing the (rather) complex structure of LLBs, we show that their finite energy versions (truncated LLBs) preserve the stationarity prop- erty for hundreds of the diffraction (dispersion) lengths as- sociated to the width (duration) of the central peak. Also, blocking the central portion of the LLB results in a self- reconstructed peak after a propagation distance that is quan- titatively characterized. The truncated, three-dimensional LLB that maximizes transfer of energy into the medium has recently been shown to be spontaneously formed upon SF of Gaussian wave packets halted by NLLs [11], and this fact can explain basic features of filamentation in media with anomalous disper- sion [12, 13]. In fact, unless specific non-zero angles are favored by nonlinear phase matching conditions [14], no conical structure can be efficiently created from an input Gaussian-like wave packet, and LLBs seems more suited than conical LBs to interpret experiments of spontaneous wave localization. In addition, it follows from this work that the observed self-reconstruction and self-healing properties of light filaments [7, 15] do not require a conical structure either, but these properties can also result from a balance between phase and amplitude nonlinearities, as is given in these LLBs. 2 Non-solitary, non-conical light-bullet solutions of the nonlinear Schrödinger equation with nonlinear losses We consider a wave packet E = A exp(−iω 0 t + ik 0 z) of carrier frequency ω 0 and propagation constant k 0 that self-