Soliton solutions of nonlinear diffusion–reaction-type equations with time-dependent coefficients accounting for long-range diffusion

We investigate three variants of nonlinear diffusion–reaction equations with derivative-type and algebraic-type nonlinearities, short-range and long-range diffusion terms. In particular, the models with time-dependent coefficients required for the case of inhomogeneous media are studied. Such equations are relevant in a broad range of physical settings and biological problems. We employ the auxiliary equation method to derive a variety of new soliton-like solutions for these models. Parametric conditions for the existence of exact soliton solutions are given. The results demonstrate that the equations having time-varying coefficients reveal richness of explicit soliton solutions using the auxiliary equation method. These solutions may be of significant importance for the explanation of physical phenomena arising in dynamical systems described by diffusion–reaction class of equations with variable coefficients

Theoretical studies of ultrashort-soliton propagation in nonlinear optical media from a general quantum model

We overview some recent theoretical studies of dynamical models beyond the framework of slowly varying envelope approximation, which adequately describe ultrashort-soliton propagation in nonlinear optical media. A general quantum model involving an arbitrary number of energy levels is considered. Model equations derived by rigorous application of the reductive perturbation formalism are presented, assuming that all transition frequencies of the nonlinear medium are either well above or well below the typical wave frequency. We briefly overview (a) the derivation of a modified Korteweg-de Vries equation describing the dynamics of few-cycle solitons in a centrosymmetric nonlinear optical Kerr (cubic) type material, (b) the analysis of a coupled system of Korteweg-de Vries equations describing ultrashort-soliton propagation in quadratic media, and (c) the derivation of a generalized double-sine-Gordon equation describing the dynamics of few-cycle solitons in a generic optical medium. The significance of the obtained results is discussed in detail

Derivation of a coupled system of Korteweg–de Vries equations describing ultrashort soliton propagation in quadratic media by using a general Hamiltonian for multilevel atoms

We consider the propagation of ultrashort solitons in noncentrosymmetric quadratically nonlinear optical media described by a general Hamiltonian of multilevel atoms. We use a long-wave approximation to derive a coupled system of Korteweg–de Vries equations describing ultrashort soliton evolution in such materials. This model was derived by using a rigorous application of the reductive perturbation formalism (multiscale analysis). The study of linear eigenpolarizations in the degenerate case and the corresponding formation of half-cycle solitons from few-cycle-pulse inputs are also presented

Circularly polarized few-optical-cycle solitons in the short-wave-approximation regime

We consider the propagation of few-cycle pulses (FCPs) beyond the slowly varying envelope approximation in media in which the dynamics of constituent atoms is described by a two-level Hamiltonian by taking into account the wave polarization. We consider the short-wave approximation, assuming that the resonance frequency of the two-level atoms is well below the inverse of the characteristic duration of the optical pulse. By using the reductive perturbation method (multiscale analysis), we derive from the Maxwell-Bloch-Heisenberg equations the governing evolution equations for the two polarization components of the electric field in the first order of the perturbation approach. We show that propagation of circularly polarized (CP) few-optical-cycle solitons is described by a system of coupled nonlinear equations, which reduces in the scalar case to the standard sine Gordon equation describing the dynamics of linearly polarized FCPs in the short-wave-approximation regime. By direct numerical simulations, we calculate the lifetime of CP FCPs, and we study the transition to two orthogonally polarized single-humped pulses as a generic route of their instability

Studies of existence and stability of circularly polarized few-cycle solitons beyond the slowly-varying envelope approximation

In this chapter, we provide an overview of recent studies of theoretical models which adequately describe the temporal dynamics of circularly polarized few-cycle optical solitons in both long-wave- and short-wave approximation regimes, beyond the framework of slowly varying envelope approximation. In the long-wave-approximation regime, i.e., when the frequency of the transition is far above the characteristic wave frequency, by using the multiscale analysis (reductive perturbation method), we show that propagation of circularly polarized (vectorial) few-cycle pulses, is described by the nonintegrable complex modified Korteweg–de Vries equation. In the short-wave-approximation regime, i.e., when the frequency of the transition is far below the characteristic wave frequency, by using the multiscale analysis, we derive from the Maxwell-Bloch equations the governing nonlinear evolution equations for the two polarization components of the electric field, in the first order of the perturbation approach. In this latter case we show that propagation of circularly-polarized few-optical-cycle solitons is described by a system of coupled nonlinear evolution equations, which reduces, for the particular case of scalar solitons, to the completely integrable sine-Gordon equation describing the dynamics of linearly polarized few-cycle pulses in the short-wave-approximation regime. It is seen that, from the slowly varying envelope approximation down to a few cycles, circularly polarized solitons are very robust, according to rotation symmetry and conservation of the angular momentum. However, in the sub-cycle regime, they become unstable, showing a spontaneous breaking of the rotation symmetry

Robust circularly polarized few-optical-cycle solitons in Kerr media

We consider the propagation of circularly polarized few-cycle pulses (FCPs) in Kerr media beyond the slowly varying envelope approximation. Assuming that the frequency of the transition is far above the characteristic wave frequency (long-wave-approximation regime), we show that propagation of FCPs, taking into account the wave polarization, is described by the nonintegrable complex modified Korteweg–de Vries (cmKdV) equation. By direct numerical simulations, we get robust localized solutions to the cmKdV equation, which describe circularly polarized few-cycle-optical solitons and strongly differ from the breather soliton of the modified Korteweg–de Vries equation, which represents linearly polarized FCP solitons. The circularly polarized FCP soliton becomes unstable when the angular frequency is less than 1.5 times the inverse of the pulse length. The unstable subcycle pulses decay into linearly polarized half-cycle pulses, the polarization direction of which slowly rotates around the propagation axis