The applied method of slowly varying amplitudes gives us the possibility to reduce the nonlinear vector integrodifferential wave equation of the electrical and magnetic vector fields to the amplitude vector nonlinear differential equations. Using this approximation, different orders of dispersion of the linear and nonlinear susceptibility can be estimated. Critical values of parameters to observe different linear and nonlinear effects are determined. The obtained amplitude equations are a vector version of 3D + 1 nonlinear Schrödinger equation (VNSE) describing the evolution of slowly varying amplitudes of electrical and magnetic fields in dispersive nonlinear Kerr-type media. We show that VNSE admits exact vortex solutions with classical orbital momentum = 1 and finite energy. Dispersion region and medium parameters necessary for experimental observation of these vortices are determined.2000 Mathematics Subject Classification: 35Q55, 45G15, 90C30.
Introduction.At the present time, there are no difficulties for experimentalist in laser physics and nonlinear optics to obtain picosecond or femtosecond optical pulses with equal durations in x, y, and z directions. The problems with the so-generated light bullets arise in the process of their propagation in a nonlinear media with dispersion. In the transparency region of a dispersive Kerr-type media, as it was established in [6,14], the scalar paraxial approximation (no dispersion in the direction of propagation) is in a very good accordance with the experimental results. The paraxial approximation, used in the derivation of the scalar 2D + 1 nonlinear Schrödinger equation (NLS), does not include (in first-or second-order of magnitude) a small second derivative of the amplitude function in the direction of propagation. The inclusion of this term does not change the main result dramatically in the case of a linear propagation (pulses with small intensity), as generated optical bullets at short distance are transformed in optical disks, with large transverse and small lengthwise dimension. Some of the experimental possibilities for this small term to become important were discussed in [21]. In [2] it was shown that this second derivative in the direction of propagation term is in the same order and with the same sign as the others only in some special cases: optical pulses near the Langmuir frequency or near some of the electronic resonances. In these regions, the sign of the dispersion is negative and the scalar 2D+1 NLS becomes a 3D+1 NLS one. Propagation of optical bullets under the dynamics of 2D+1 and 3D+1 NLS is investigated also in relation to different kinds of nonlinearity [9,10,19,23,24]. A generation of a new kind of 2D and 3D optical pulses, the so-called optical vortices, has recently become a topic of considerable interest. Generally, the optical vortices