The light filament as a new nonlinear polarization state

We present an analytical approach to the theory of nonlinear propagation in gases of femtosecond optical pulses with broad-band spectrum . The vector character of the nonlinear third-order polarization of the electrical field in air is investigated in details. A new polarization state is presented by using left-hand and right-hand circular components of the electrical field . The corresponding system of vector amplitude equations is derived in the rotating basis. We found that this system of nonlinear equations has $3D+1$ vector soliton solutions with Lorentz shape. The solution presents a relatively stable propagation and rotation with GHz frequency of the vector of the electrical field in a plane orthogonal to the direction of propagation. The evolution of the intensity profile demonstrates a weak self-compression and a week spherical wave in the first milliseconds of propagation.