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- ItemDispersion of nonlinear group velocity determines shortest envelope solitons(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Amiranashvili, Shalva; Bandelow, Uwe; Akhmediev, Nail N.We demonstrate that a generalized nonlinear Schrödinger equation (NSE), that includes dispersion of the intensity-dependent group velocity, allows for exact solitary solutions. In the limit of a long pulse duration, these solutions naturally converge to a fundamental soliton of the standard NSE. In particular, the peak pulse intensity times squared pulse duration is constant. For short durations this scaling gets violated and a cusp of the envelope may be formed. The limiting singular solution determines then the shortest possible pulse duration and the largest possible peak power. We obtain these parameters explicitly in terms of the parameters of the generalized NSE.
- ItemAdiabatic theory of champion solitons(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Pickartz, Sabrina; Bandelow, Uwe; Amiranashvili, ShalvaWe consider scattering of small-amplitude dispersive waves at an intense optical soliton which constitutes a nonlinear perturbation of the refractive index. Specifically, we consider a single-mode optical fiber and a group velocity matched pair: an optical soliton and a nearly perfectly reflected dispersive wave, a fiber-optical analogue of the event horizon. By combining (i) an adiabatic approach that is used in soliton perturbation theory and (ii) scattering theory from Quantum Mechanics, we give a quantitative account for the evolution of all soliton parameters. In particular, we quantify the increase in the soliton peak power that may result in spontaneous appearance of an extremely large, so-called champion soliton. The presented adiabatic theory agrees well with the numerical solutions of the pulse propagation equation. Moreover, for the first time we predict the full frequency band of the scattered dispersive waves and explain an emerging caustic structure in the space-time domain.
- ItemEfficient all-optical control of solitons(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Pickartz, Sabrina; Bandelow, Uwe; Amiranashvili, ShalvaWe consider the phenomenon of an optical soliton controlled (e.g. amplified) by a much weaker second pulse which is efficiently scattered at the soliton. An important problem in this context is to quantify the small range of parameters at which the interaction takes place. This has been achieved by using adiabatic ODEs for the soliton characteristics, which is much faster than an empirical scan of the full propagation equations for all parameters in question.
- ItemNumerical methods for accurate description of ultrashort pulses in optical fibers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Amiranashvili, Shalva; Radziunas, Mindaugas; Bandelow, Uwe; C̆iegis, RaimondasWe consider a one-dimensional first-order nonlinear wave equation (the so-called forward Maxwell equation, FME) that applies to a few-cycle optical pulse propagating along a preferred direction in a nonlinear medium, e.g., ultrashort pulses in nonlinear fibers. The model is a good approximation to the standard second-order wave equation under assumption of weak nonlinearity. We compare FME to the commonly accepted generalized nonlinear Schrödinger equation, which quantifies the envelope of a quickly oscillating wave field based on the slowly varying envelope approximation. In our numerical example, we demonstrate that FME, in contrast to the envelope model, reveals new spectral lines when applied to few-cycle pulses. We analyze and compare pseudo-spectral numerical schemes employing symmetric splitting for both models. Finally, we adopt these schemes to a parallel computation and discuss scalability of the parallelization.
- ItemUnusual ways of four-wave mixing instability(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Amiranashvili, Shalva; Bandelow, UweA pump carrier wave in a dispersive system may decay by giving birth to blue- and red-shifted satellite waves due to modulation or four-wave mixing instability. We analyse situations where the satellites are so different from the carrier wave, that the red-shifted satellite either changes its propagation direction (k < 0, ω > 0) or even gets a negative frequency (k, ω < 0). Both situations are beyond the envelope approach and require application of Maxwell equations.
- ItemSpectral properties of limiting solitons in optical fibers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Amiranashvili, Shalva; Bandelow, Uwe; Akhmediev, NailIt seems to be self-evident that stable optical pulses cannot be considerably shorter than a single oscillation of the carrier field. From the mathematical point of view the solitary solutions of pulse propagation equations should loose stability or demonstrate some kind of singular behavior. Typically, an unphysical cusp develops at the soliton top, preventing the soliton from being too short. Consequently, the power spectrum of the limiting solution has a special behavior: the standard exponential decay is replaced by an algebraic one. We derive the shortest soliton and explicitly calculate its spectrum for the so-called short pulse equation. The latter applies to ultra-short solitons in transparent materials like fused silica that are relevant for optical fibers.
- ItemStabilization of optical pulse transmission by exploiting fiber nonlinearities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Bandelow, Uwe; Amiranashvili, Shalva; Pickartz, SabrinaWe prove theoretically, that the evolution of optical solitons can be dramatically influenced in the course of nonlinear interaction with much smaller group velocity matched pulses. Even weak pump pulses can be used to control the solitons, e.g., to compensate their degradation caused by Raman-scattering.
- ItemGeneralized integrable evolution equations with an infinite number of free parameters(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Akhmediev, Nail; Ankiewicz, Adrian; Amiranashvili, Shalva; Bandelow, UweEvolution equations such as the nonlinear Schrödinger equation (NLSE) can be extended to include an infinite number of free parameters. The extensions are not unique. We give two examples that contain the NLSE as the lowest-order PDE of each set. Such representations provide the advantage of modelling a larger variety of physical problems due to the presence of an infinite number of higher-order terms in this equation with an infinite number of arbitrary parameters. An example of a rogue wave solution for one of these cases is presented, demonstrating the power of the technique.
- ItemFew-cycle optical solitons in dispersive media beyond the envelope approximation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Amiranashvili, Shalva; Bandelow, Uwe; Akhmediev, NailWe study the propagation of few-cycle optical solitons in nonlinear media with an anomalous, but otherwise arbitrary, dispersion and a cubic nonlinearity. Our theory extends beyond the slowly varying envelope approximation. The optical field is derived directly from the Maxwell equations under the assumption that generation of the third harmonic is a non-resonant 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 a modification of the spectral renormalisation method originally developed for the envelope solitons.
- ItemUltrashort optical solitons in transparent nonlinear media with arbitrary dispersion(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Amiranashvili, Shalva; Bandelow, Uwe; Akhmediev, NailWe 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.