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

Derivation of a modified Korteweg–de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian

Propagation of few-cycles optical pulses in a centrosymmetric nonlinear optical Kerr (cubic) type material described by a general Hamiltonian of multilevel atoms is considered. Assuming that all transition frequencies of the nonlinear medium are well above the typical wave frequency, we use a long-wave approximation to derive an approximate evolution model of modified Korteweg–de Vries type. The model derived by rigorous application of the reductive perturbation formalism allows one the adequate description of propagation of ultrashort (few-cycles long) solitons

Extending the solid step fixed-charge transportation problem to consider two-stage networks and multi-item shipments

This paper develops a new mathematical model for a capacitated solid step fixed-charge transportation problem. The problem is formulated as a two-stage transportation network and considers the option of shipping multiple items from the plants to the distribution centers (DC) and afterwards from DCs to customers. In order to tackle such an NP-hard problem, we propose two meta-heuristic algorithms; namely, Simulated Annealing (SA) and Imperialist Competitive Algorithm (ICA). Contrary to the previous studies, new neighborhood strategies maintaining the feasibility of the problem are developed. Additionally, the Taguchi method is used to tune the parameters of the algorithms. In order to validate and evaluate the performances of the model and algorithms, the results of the proposed SA and ICA are compared. The computational results show that the proposed algorithms provide relatively good solutions in a reasonable amount of time. Furthermore, the related comparison reveals that the ICA generates superior solutions compared to the ones obtained by the SA algorithm

Derivation of a generalized double-sine-Gordon equation describing ultrashort-soliton propagation in optical media composed of multilevel atoms

We consider the propagation of ultrashort optical solitons in media described by a general Hamiltonian of multilevel atoms. Assuming that all transition frequencies of the medium are well below the typical wave frequency, i.e., only the contribution of infrared transitions is taken into account, we use a short-wave approximation and a rigorous application of the reductive perturbation formalism to derive a cumbersome coupled system of nonlinear partial differential equations describing ultrashort soliton evolution in such systems. The rather complicated set of coupled equations can be simplified to a generic double-sine-Gordon equation for a special case of identical three-level atoms, whereas for a special case of identical four-level atoms the system of coupled equations can be reduced to a generalized double-sine-Gordon equation. Numerical simulations showing the formation of robust breather-type solutions of both the standard double-sine-Gordon and of the generalized double-sine-Gordon equations from sinusoidal inputs with Gaussian envelopes are also presented