Two-component few-cycle light bullets in a gradient waveguide with quadratic nonlinearity

In this work we present results of our study of light bullets in inhomogeneous media with quadratic nonlinearity. We consider the second harmonics generation by few-cycle pulses having about 3 - 5 oscillations under the envelope. We give reasons to apply "slowly varying envelope approximation" in this case. The self-consistent system of nonlinear equations for the envelopes of both harmonics is substantially modified in comparison with the case of quasi-monochromatic signals. This system is supplemented by a third order group dispersion and by a dispersion of nonlinearity. The diffraction terms are also modified. The appropriate system of parabolic equations for the envelopes of both harmonics is obtained. To solve an arising 2D+1 system numerically we construct an original nonlinear finite-difference scheme based on the Crank-Nicolson and pseudo-spectral methods preserving the integrals of motion. We discuss different regimes of pulse propagation depending on the competition among nonlinearity, diffraction, temporal dispersion and waveguide geometry.