Theory of Nonlinear Dispersive Waves and Selection of the Ground State (original) (raw)
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Physica D: Nonlinear Phenomena, 2003
We introduce a model of a two-core system, based on an equation of the Ginzburg-Landau (GL) type, coupled to another GL equation, which may be linear or nonlinear. One core is active, featuring intrinsic linear gain, while the other one is lossy. The difference from previously studied models involving a pair of linearly coupled active and passive cores is that the stabilization of the system is provided not by a linear diffusion-like term, but rather by a cubic or quintic dissipative term in the active core. Physical realizations of the models include systems from nonlinear optics (semiconductor waveguides or optical cavities), and a double-cigar-shaped Bose-Einstein condensate with a negative scattering length, in which the active "cigar" is an atom laser. The replacement of the diffusion term by the nonlinear loss is principally important, as diffusion does not occur in these physical media, while nonlinear loss is possible. A stability region for solitary pulses is found in the system's parameter space by means of direct simulations. One border of the region is also found in an analytical form by means of a perturbation theory. Moving pulses are studied too. It is concluded that collisions between them are completely elastic, provided that the relative velocity is not too small. The pulses withstand multiple tunneling through potential barriers. Robust quantum-rachet regimes of motion of the pulse in a time-periodic asymmetric potential are found as well.
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.
Physical Review Letters, 2005
It is proven that periodically varying and sign definite nonlinearity in a general case does not prevent collapse in two-and three-dimensional nonlinear Schrödinger equations: at any oscillation frequency of the nonlinearity blowing up solutions exist. Contrary to the results known for a sign alternating nonlinearity, increase of the frequency of oscillations accelerates collapse. The effect is discussed from the viewpoint of scaling arguments. For the three-dimensional case a sufficient condition for existence of collapse is rigorously established. The results are discussed in the context of the meanfield theory of Bose-Einstein condensates with time dependent scattering length.
Physical Review E, 1996
A nonlinear theory describing the long-term dynamics of unstable solitons in the generalized nonlinear Schrodinger (NLS) equation is proposed. An analytical model for the instability-induced evolution of the soliton parameters is derived in the framework of the perturbation theory, which is valid near the threshold of the soliton instability. As a particular example, we analyze solitons in the NLS-type equation with two power-law nonlinearities. For weakly subcritical perturbations, the analytical model reduces to a second-order equation with quadratic nonlinearity that can describe, depending on the initial conditions and the model parameters, three possible scenarios of the longterm soliton evolution: (i) periodic oscillations of the soliton amplitude near a stable state, (ii) soliton decay into dispersive waves, and (iii) soliton collapse. We also present the results of numerical simulations that con6rm excellently the predictions of our analytical theory.
Physical Review A, 2020
We show theoretically that highly dispersive optical media characterized by a Kerr nonlinear response may support the existence of quartic and dipole solitons in the presence of the self-steepening effect. The existence and stability properties of these localized pulses are examined in the presence of all the material parameters. Regimes for the modulation instability of a continuous-wave signal propagating inside the nonlinear medium are investigated and an analytic expression for the gain spectrum is obtained and shown to be dependent on the self-steepening parameter in addition to second-and fourth-order group velocity dispersion parameters. Self-similar soliton solutions are constructed for a generalized nonlinear Schrödinger equation with distributed second-, third-, and fourth-order dispersions, self-steepening nonlinearity, and gain or loss describing ultrashort pulse propagation in the inhomogeneous nonlinear media via the similarity transformation method. The evolutional dynamics of the self-similar structures are investigated in a periodic distributed waveguide system and an exponential dispersion decreasing waveguide.
Annals of Physics, 2015
The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose-Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin-Feir index, i.e. a nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.
Nonlinear Waves: Classical and Quantum Aspects
NATO Science Series II: Mathematics, Physics and Chemistry, 2005
Towards algebro-geometric integration of the Gross-Pitaevskii equation V.Z. Enolskii 1 Introduction 2 Utilization of the Schrödinger equation 3 Solutions in terms of hyperelliptic functions 4 Two component Gross-Pitaevskii equation and the Manakov system 9 On modeling adiabatic N-soliton interactions V.S. Gerdjikov 1 Introduction 2 N-soliton trains of the NLS and HNLS equations 3 N-soliton trains of the MNLS equation 4 The importance of the CTC model 18 5 Dynamical regimes of the HNLS soliton trains 6 The perturbed NLS and perturbed CTC 6.1 Second order dispersion and nonlinear gain 6.2 Quadratic and periodic potentials 7 Analysis of the Perturbed CTC 8 Discussion Dynamical stabilization of nonlinear waves F. Abdullaev 1 Introduction 2 Dynamics of solitons in BEC with rapidly oscillating trap 3 Stable two dimensional bright soliton under Feschbach resonance management 4 Stable two dimensional dispersion-managed soliton 5 Conclusions v vi
The dispersion-managed soliton as a ground state of a macroscopic nonlinear quantum oscillator
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2001
High‐frequency nonlinear solitary wave propagation in media with strongly varying dispersion is considered. A dynamical model is proposed which allows us to determine the propagation characteristics of a strongly dispersion‐managed soliton without applying any averaging procedure. This approach is compared with previously developed models, focusing on the dependence of the energy on the quasimomentum. An analogy with a macroscopic nonlinear quantum oscillator model is briefly discussed. The analytical ...
Resonant nonlinearity management for nonlinear Schrödinger solitons
Physical Review E, 2004
We consider effects of a periodic modulation of the nonlinearity coefficient on fundamental and higher-order solitons in the one-dimensional NLS equation, which is an issue of direct interest to Bose-Einstein condensates in the context of the Feshbach-resonance control, and fiber-optic telecommunications as concerns periodic compensation of the nonlinearity. We find from simulations, and explain by means of a straightforward analysis, that the response of a fundamental soliton to the weak perturbation is resonant, if the modulation frequency ω is close to the intrinsic frequency of the soliton. For higher-order n-solitons with n = 2 and 3, the response to an extremely weak perturbation is also resonant, if ω is close to the corresponding intrinsic frequency. More importantly, a slightly stronger drive splits the 2-or 3-soliton, respectively, into a set of two or three moving fundamental solitons. The dependence of the threshold perturbation amplitude, necessary for the splitting, on ω has a resonant character too. Amplitudes and velocities of the emerging fundamental solitons are accurately predicted, using exact and approximate conservation laws of the perturbed NLS equation. PACS numbers: 03.75.Lm, 05.45.Yv, 42.65.Tg In particular, solitons in fiber-optic telecommunications [3] and quasi-1D Bose-Einstein condensates (BECs) with attractive interactions between atoms [4], have drawn a great deal of interest.
Modulation analysis and optical solitons of perturbed nonlinear Schrodinger equation
Revista Mexicana De Fisica, 2021
We investigate modulation analysis and optical solitons of perturbed nonlinear Schrodinger equation (PNLSE). The PNLSE has terms of cubic nonlinearity and self-steepening and spatio-temporal dispersion (STD). Proposed model has been studied by [14, 15] without self-steepening term. The presence of the STD and self-steepening can help to compensate the low GVD to the model. Bright and dark solitary waves, trigonometric, periodic andsingular optical solitons are obtained by some expansion methods including exponential and sinh-Gordon. Obtained results will hold a significant place in the field of nonlinear optical fibers, where solitons are used to codify data.