Evolution of higer-order light bullets in saturable Kerr-type medium

In the paper we consider propagation of (3+1)D ultra-short spatio-temporal light pulses (bullets) in saturable nonlinear medium of Kerr type. The envelope of the complex amplitude describing such pulse is a solution of (3+1)D Nonlinear Schrodinger Equation. In order to obtain possibility of stationary and stable transmission along the z-axis we assume dielectric permittivity with square graded index linear profile and saturable nonlinearity of cubic-quintic type. The corresponding Higher Order Nonlinear Schrodinger Equation (HONSE) can be solved approximately by means of variational method. Following after the solutions in linear medium we predict existence of the whole series of solution in a form of higher-order modes. Assuming the trial function with appropriate spatial profile we obtain corresponding Euler-Lagrange equations. We solve these equations analytically in stationary case obtaining parameters of stationary light pulses. Assuming small deviations from stationary pulse parameters we obtain and solve equations of oscillatory type that describe non-stationary propagation of bullets. We compare the exact numerical and approximate analytical solutions. We analyze coupling between oscillations with different frequencies in the evolution of higher-order light bullets.