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- 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.
- 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.
- 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.
- ItemSasa-Satsuma equation: Soliton on a background and its limiting cases(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Bandelow, Uwe; Akhmediev, NailWe present a multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the NLSE limit and the limit of rogue wave solutions
- 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.
- ItemSasa-Satsuma hierarchy of integrable evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bandelow, Uwe; Ankiewicz, Adrian; Amiranashvili, Shalva; Pickartz, Sabrina; Akhmediev, NailWe present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixthorder terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly.
- ItemGeneralized Sasa-Satsuma equation: Densities approach to new infinite hierarchy of integrable evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Ankiewicz, Adrian; Bandelow, Uwe; Akhmediev, NailWe derive the new infinite Sasa-Satsuma hierarchy of evolution equations using an invariant densities approach. Being significantly simpler than the Lax-pair technique, this approach does not involve ponderous 3 x3 matrices. Moreover, it allows us to explicitly obtain operators of many orders involved in the time evolution of the Sasa-Satsuma hierarchy functionals. All these operators are parts of a generalized Sasa-Satsuma equation of infinitely high order. They enter this equation with independent arbitrary real coefficients that govern the evolution pattern of this multi-parameter dynamical system.
- ItemSolitons on a background, rogue waves, and classical soliton solutions of Sasa-Satsuma equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bandelow, Uwe; Akhmediev, NailWe present the most general multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution contains a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits nontrivial limiting cases, such as rogue waves and classical solitons, that are considered in detail.