Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schrödinger equation

We consider in detail the self-trapping of a soliton from a wave pulse that passes from a defocusing region into a focusing one in a spatially inhomogeneous nonlinear waveguide, described by a nonlinear Schrödinger equation in which the dispersion coefficient changes its sign from normal to anomalous. The model has direct applications to dispersion-decreasing nonlinear optical fibers, and to natural waveguides for internal waves in the ocean. It is found that, depending on the (conserved) energy and (nonconserved) “mass” of the initial pulse, four qualitatively different outcomes of the pulse transformation are possible: decay into radiation; self-trapping into a single soliton; formation of a breather; and formation of a pair of counterpropagating solitons. A corresponding chart is drawn on a parametric plane, which demonstrates some unexpected features. In particular, it is found that any kind of soliton(s) (including the breather and counterpropagating pair) eventually decays into pure radiation with an increase of energy, the initial “mass” being kept constant. It is also noteworthy that a virtually direct transition from a single soliton into a pair of symmetric counterpropagating ones seems possible. An explanation for these features is proposed. In two cases when analytical approximations apply, viz., a simple perturbation theory for broad initial pulses and the variational approximation for narrow ones, comparison with direct simulations shows reasonable agreement.