Ultrashort optical solitons in nonlinear media with arbitrary dispersion

We consider the propagation of ultrashort optical pulses in nonlinear fibers and suggest a new theoretical framework for the description of pulse dynamics and exact characterization of solitary solutions. Our approach deals with a proper complex generalization of the nonlinear Maxwell equations and completely avoids the use of the slowly varying envelope approximation. The only essential restriction is that fiber dispersion does not favor both the so-called Cherenkov radiation, as well as the resonant generation of the third harmonics, as these effects destroy ultrashort solitons. Assuming that it is not the case, we derive a continuous family of solitary solutions connecting fundamental solitons to nearly single-cycle ultrashort ones for arbitrary anomalous dispersion and cubic nonlinearity.

We showed recently [1] that a single soliton solution of the highly-nonlinear Schrödinger equation becomes multi-stable for a certain class of nonlinearities. This implies that more than one amplitude profile and propagation constant of a single soliton may exist for the same total power carried by the soliton. The multistable solitons exist only if the nonlinear component of the susceptibility as a function of intensity is either changing its sign or its derivative has a sufficiently sharp peak. This can result in such effects as bistable (or multistable, in general) self-trapping of light in nonlinear media as well as bistable propagation of soliton pulses in nonlinear optical fibers. Both of these processes may be described by the same nonlinear Schrödinger equation with a nonlinear term in the form Ef(|E|2) where f(|E|2) is an arbitrary function of the field intensity |E|2 with f(0) = 0. The so called cubic nonlinear Schrödinger equation with f(|E|2) proportional |E|2, corresponds to Kerr-nonlinearity in optical propagation and does not give rise to bistable solitons). By evaluating the propagation constant of a solitary solution δ as a function of its total power P for each given f(|E|)2), we found a class of nonlinear functions f(I) for which the dependence δ(P) becomes multivalued. We conjectured earlier [1] that stable (unstable) solitary solutions are those with dδ/dP > 0 (dδ/dP < 0). The first computer results by Enns and Rangnekar [2] supported this conjecture for a certain non-Kerr nonlinearity. In our further collaborative research [3], we discovered that depending on the nonlinear function f(I), the stable solitary solutions (dδ/dP > 0) fall into two classes: “weak” solitons (stable only against sufficiently small perturbation), and “robust” solitons (stable against any perturbation, in particular, against collisions with other solitons). It was shown [3] that while the condition dδ/dP > 0 is sufficient for the “weak” stability, some extra conditions imposed on the function f(I) warrant existence of “robust” solitons (one of them e.g. is f(I)/I2 = o(1) as I → ∞; stability against collapsing behavior). It was shown most recently by Enns [4] that at least for some of the nonlinear functions of (|E|2) giving rise to bistable solitons, the nonlinear Schrödinger equation passes the so called Painleve test and is therefore completely integrable. This confirms that the “robust” solitary solutions are indeed solitons. The propagation of single pulses in nonlinear fibers with an appropriate nonlinearity may provide the first known opportunity to obtain a temporal (dynamic) bistability as contrasted to the known kinds of optical bistability which are so far based on the equilibrium regimes.

Propagation of ultra-short optical pulses in cubic nonlinear media

We propose a simple scheme to generate high-energy ultrashort pulses by combination and compression of multiple input pulses which share the same chirp profile. First, the multiple raised-cosine pulses in the input pulse train are modulated by a phase modulator, in which each modulation cycle covers two, three, four, or five pulses. Then, the modulated pulses are launched into a nonlinear fiber with the exponentially decreasing dispersion. We find that these pulses initially coalesce into a single pulse whose pulse profile is nearly hyperbolic secant, which then undergoes self-similar compression. Thus in the proposed method, first the combination of the multiple optical pulses occurs and then self-similar compression takes over. Besides, we also report the generation of ultrashort pulses by combination and compression of multiple hyperbolic secant pulses with the same chirp. The numerical results reveal that the resulting ultrashort pulse possesses a large portion of the input pulses for both raised-cosine and hyperbolic secant pulses. However, the compression factor and energy ratio are relatively higher for the hyperbolic secant pulses when compared with the raised-cosine pulses.

We study the propagation of few-cycle optical solitons in nonlinear media with an anomalous, but otherwise arbitrary, dispersion and a cubic nonlinearity. Our approach does not derive from the slowly varying envelope approximation. The optical field is derived directly from Maxwell's equations under the assumption that generation of the third harmonic is a nonresonant process or at least cannot destroy the pulse prior to inevitable linear damping. The solitary wave solutions are obtained numerically up to nearly single-cycle duration using the spectral renormalization method originally developed for the envelope solitons. The theory explicitly distinguishes contributions between the essential physical effects such as higher-order dispersion, self-steepening, and backscattering, as well as quantifies their influence on ultrashort optical solitons.

The solitary wave ansatz method for finding the exact bright and dark soliton solutions of two nonlinear Schrödinger equations

The dynamics of an ultrashort optical pulse in order—disorder ferroelectrics in the presence of defects without limitations on the pulse power is considered in the framework of the microscopic pseudospin formalism within the unidirectional propagation approximation. The effective equations for the electric field amplitude and the ferroelectric polarization are derived and numerically solved. Quasi-soliton modes of propagation of ultrashort optical pulses are revealed, and the dependences on the microscopic Hamiltonian parameters are analyzed. The influence of defects of different types on the propagation of ultrashort pulses is established.

In this paper, the rogue waves of the higher-order dispersive nonlinear Schrödinger (HDNLS) equation are investigated, which describes the propagation of ultrashort optical pulse in optical fibers. The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method. The explicit first and second-order rogue wave solutions are presented under the plane wave seeding solution background. The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions. The influence of little perturbation ε on the rogue waves is discussed with the help of graphical simulation.

Spectral stability of elliptic solutions to the short-pulse equation

In this study, we investigate the generalized stochastic nonlinear Schrödinger equation, which models the propagation of ultrashort optical pulses in nonlinear and dispersive media, incorporating both Kerr effect and higher-order nonlinearities. To construct exact analytical solutions, we employ the tan ( ϖ ( ξ ) 2 ) \tan \left( {{{\varpi (\xi )} \over 2}} \right) -expansion method and the ( G ′/ G , 1/ G )-expansion method. These methods yield a variety of exact solutions, including dark, singular, and singular periodic soliton solutions, each representing different physical wave behaviors. We further perform a stability analysis to determine the robustness of these solutions under perturbations and examine their temporal evolution to better understand their propagation dynamics. Graphical illustrations of selected solutions are provided to visualize their dynamics and to demonstrate how the passage of time influences the structure and stability of the resulting wave forms.

The subsidiary elliptic-like equation and the exact solutions of the higher-order nonlinear Schrödinger equation

In this paper, we propose a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrödinger (NLS) equation in the ultrashort-pulse regime. Furthermore, we construct the multi-dark-soliton solution for the defocusing complex short-pulse equation through the Darboux transformation and reciprocal (hodograph) transformation. One- and two-dark-soliton solutions are given explicitly, whose properties and dynamics are analyzed and illustrated.

In this paper, we utilize the first integral method with symbolic computation to constructing periodic and solitary wave solutions of complex nonlinear evolution equations. With the aid of Maple, new exact solutions of two nonlinear evolution equations in physics, namely, generalized Zakharov equations (GZ) and higher-order nonlinear Schrodinger equation (HONLS) describing propagation of ultrashort pulses in nonlinear optical fibers are obtained.

This research undertakes a comprehensive investigation of the optical soliton solutions of the Focusing Non-linear Schr¨odinger Equation (NLSE), a fundamental model describing the propagation of optical solitons in nonlinear media. To achieve this, we employ two versatile and efficient methods: the Ricatti-Bernoulli Sub Ordinary Differential Equation (RBSODE) method and the Bernoulli Sub Ordinary Differential Equation (BSODE) method. These methods enable us to derive a wide range of optical soliton solutions.We examine two distinct nonlinearities: the Kerr law nonlinearity and the quadratic-cubic nonlinearity. These nonlinearities are crucial in determining the behavior of optical solitons in various nonlinear optical media. Our analysis reveals that the derived soliton solutions exhibit distinct characteristics, such as amplitude, width, and velocity, which are influenced by the type of nonlinearity and the parameters of the NLSE.Furthermore, we utilize the multiplier approach to obtain the conservation laws of the NLSE. These conservation laws provide valuable insights into the underlying dynamics of the optical solitons and have significant implications for the design and optimization of nonlinear optical system.

In this paper, we study the generalized coupled nonlinear Schrödinger equations with the cubic-quintic nonlinearity and arbitrary coupling parameters which describe the effects of the quintic nonlinearity on the ultrashort optical soliton pulse propagation in the non-Kerr media. Dark–dark N- and bright–dark two-soliton solutions are derived with symbolic computation. The bilinear Bäcklund transformation and the corresponding one-soliton solution are also given. Interactions of the dark–dark solitons including the repulsive and coherent interactions are investigated analytically and graphically, which are different from those in previous studies. The relationship between the amplitude and the velocity of the dark soliton is studied, especially the soliton with the small amplitude catches up with the one with the large amplitude. Head-on and overtaking interactions of dark–dark solitons are presented. Interactions between a stationary dark soliton and another dark one are shown. The asymptotic analysis shows that the interaction of dark–dark two solitons is elastic. The periodic attraction and repulsion of the bright–dark two solitons in the bound state are also analyzed, which are in accordance with those in the bright–dark vector soliton studies but different from the asymmetric and symmetric stationary bound states of Manakov bright–dark solitons.