Trapping of two-component light bullets in a gradient waveguide with normal group dispersion

In this paper we consider the process of the second harmonic generation in a gradient waveguide, taking into account diffraction and relatively weak temporal dispersion. Using the slowly varying envelope approximation and neglecting the dispersion of the nonlinear part of the response of the medium we obtain the system of parabolic equations for the envelopes of both harmonics. We also derive integrals of motion of this system. To solve it numerically we construct a nonlinear finite-difference scheme based on the Crank-Nicolson method preserving the integrals. Primarily, we focus our investigations on the processes of a two-component light bullets generation. We demonstrate that the generation of a coupled pair is possible in a planar waveguide even at normal group velocity dispersion.