Nonlinear waves: a review Vector $0\pi$ pulse and the generalized perturbative reduction method

Abstract

In this review, a more general theory of self-induced transparency (SIT) in comparison with the theory of McCall and Hahn is considered. Using the recently developed generalized perturbative reduction method (GPRM) the SIT equations are reduced to vector (coupled) nonlinear Schrodinger equations for auxiliary functions. This approach demonstrates that, unlike McCall and Hahn SIT theory in which single-component scalar breather can propagate independently, in the more general theory of SIT the second derivatives with respect to the spatial coordinate and time of the wave equation play a significant role and describe the interaction of two scalar SIT breathers forming a coupled pair. This is a vector 0\pi pulse of SIT - a two-component vector breather oscillating with the sum and difference of frequencies and wave numbers. The profile, parameters and properties of this pulse differ significantly from the characteristics of the McCall and Hahn pulses. Using GPRM it is shown that besides the scalar soliton and the scalar breather, the vector $0\pi$ pulse is also a universal nonlinear wave, arising in virtually all areas of physics where nonlinear phenomena are described by nonlinear equations containing second-order and higher-order derivatives. A number of such nonlinear equations are presented. Among them almost all well known nonlinear equations, as well as the general fourth-order nonlinear partial differential equation and the sixth-order generalized Boussinesq-type equations.