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Nonlinear losses accompanying self-focusing substantially impacts the dynamic balance of diffraction and nonlinearity, permitting the existence of localized and stationary solutions of the 2D+1 nonlinear Schrödinger equation which are stable against radial collapse. These are featured by linear, conical tails that continually refill the nonlinear, central spot. An experiment shows that the discovered solution behaves as strong attractor for the self-focusing dynamics in Kerr media.PACS numbers: 42.65. Re, 42.65.Tg One of the main goals of modern nonlinear wave physics is the achievement of wave localization, stationarity and stability. While in a one-dimensional geometry (e.g., in optical fibers), nonlinearity suitably balances linear wave dispersion, leading to the soliton regime, in the multidimensional case, nonlinearity drives waves either to collapse or instability. In self-focusing of optical beams, for instance, many stabilizing mechanisms, such as Kerr saturation, plasma-induced defocusing, or stimulated Raman scattering, have been explored, and are being the subject of intense debate, mainly in the context of light filamentation in air or condensed matter [1]. These mechanisms, however, are either intrinsically lossy, or, due to the huge intensities involved, are accompanied by losses, which lead ultimately to the termination of any soliton regime. Similar pictures can be traced in all phenomena commonly discussed in the context of the nonlinear Schrödinger equation (NLSE), as Bose-Einstein condensates (BEC) or Langmuir waves in plasma [2]. Nonlinear losses (NLL) arise in BEC from two-and three-body inelastic recombination, and as the natural mechanism for energy dissipation in Langmuir turbulence [3].The question then arises of whether any stationary and localized (SL) wave propagation is possible in the presence of NLL. The response, as shown in this Letter, is affirmative. These SL waves cannot be ascribed to the class of solitary waves, but are instead nonlinear conical waves (as the non-linear X waves [4]) of dissipative type, whose stationarity is sustained by a continuous refilling of the nonlinearly absorbed central spot with the energy supplied by linear, conical tails. These waves are not only robust against NLL, but find their stabilizing mechanism against perturbations in NLL themselves.Among the linear conical waves [5], the simplest one is the monochromatic Bessel beam (BB) [6], made of a superposition of plane waves whose wave vectors are evenly distributed over the surface of a cone, resulting in a nondiffracting transversal Bessel profile. Despite the ideal nature of BBs (they carry infinite power), they not only have revealed to be a paradigm for understanding wave phenomena, but also have found applications as diverse as in frequency conversion, or in atom trapping and alignment [7]. Of particular interest for us is the finding [8] that the BB is describable in terms of the interference of two conical Hankel beams [5], carrying equal amounts of energy towards and outwards the beam ...

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The precise observation of the angle-frequency spectrum of light filaments in water reveals a scenario incompatible with current models of conical emission (CE). Its description in terms of linear X-wave modes leads us to understand filamentation dynamics requiring a phase-and groupmatched, Kerr-driven four-wave-mixing process that involves two highly localized pumps and two X-waves. CE and temporal splitting arise naturally as two manifestations of this process. PACS numbers: 190.5940, 320.2250 Filamentation of intense light pulses in nonlinear media has attracted much interest ever since first experimental evidences in the early '60's ([1] and references therein). Owing to the very high intensities reached during the process, several nonlinear phenomena, e.g., multiphoton absorption, plasma formation, saturable nonlinear response, stimulated Raman scattering etc., occur in addition to the optical Kerr effect. Indeed, the filament regime is enriched by peculiar phenomena like pulse splitting, self-steepening, shock-wave formation, supercontinuum generation, and conical emission (CE) [2]. In media with normal group velocity dispersion (GVD), no matter if of solid, liquid or gaseous nature, CE accompanies filamentation, producing radiation at angles that increase with increasing detuning from the carrier frequency [3,4]. In spite of the generality of the process, a clear understanding of the interplay between CE and filament dynamics is still missing. Only recently, Kolesik et al. have proposed an interpretation of filamentation dynamics in water on the basis of pulse splitting and dynamic nonlinear X waves at the far field [5], in which the double X-like structure observed in simulated anglefrequency spectra arises from the scattering of an incident field at the two main peaks of the split material response wave.Originally, CE in light filaments was interpreted in terms of the modulation instability (MI) angle-frequency gain pattern of the plane and monochromatic (PM) modes of the nonlinear Schrödinger equation (NSE) [6,7]. Measurements at large angles and detunings from the carrier frequency gave in fact results fairly compatible with this interpretation [8,9]. In the present work, owing to the use of a novel imaging spectrograph technique [3], we have been able to observe for the first time the CE in the region of small angles and detunings. The results clearly indicate a scenario not compatible with the MI analysis of PM modes. Our description by means of the spectra of the stationary linear X-waves supported by the medium, indicates that the strong localization of the self-focused field plays a crucial role in the substantial modification experienced by the MI pattern. We propose a simple picture in which the latter results from the parametric amplification of two weak X-waves by the strong, highly localized pump. Supporting this interpretation, we are able to derive, from the matching condition among the interacting waves, a simple analytical expression [Eq. (4)] that accurately determines the overall CE...

We investigate the formation of X waves during filamentation in Kerr media. From the standard model developed for femtosecond filamentation in liquids, solids, and gases, the influence of several physical effects and parameters is numerically studied in the strongly nonlinear regime where group velocity dispersion alone is insufficient to arrest collapse. The collapse is shown to be arrested by multiphoton absorption and plasma defocusing, but not by dispersion. The postcollapse dynamics takes the form of a pulse splitting, which induces large gradients in the near field and seeds the formation of X waves, appearing both in the near and far fields. We discuss the universal features of the X-wave patterns, among which the long arms in the far field that follow the linear dispersive properties of the medium [Conti, Phys. Rev. Lett. 90, 170406 (2003); Kolesik, Phys. Rev. Lett. 92, 253901 (2004)] and are accompanied by a strong modulated axial emission.

Excitation of unbalanced-Bessel beams by a gradual increase of nonlinearity in a water sample outlines the achievement of the first ever observed quasimonochromatic wave packet that propagates stably for hundreds of Rayleigh lengths in a focusing and dispersive Kerr medium, i.e., in the absence of spectral broadening and conical emission. A modulational instability analysis reveals the key role of nonlinear dissipation in quenching the growth of spatiotemporal unstable modes.

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