Forced nonlinear Schrödinger equation with arbitrary nonlinearity (original) (raw)
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We investigate the generalized nonlinear Schrödinger equation (NLSE) of third order analytically, that accept the single-parameter family of one hump embedded soliton. This equation has been utilized to model ultra-short pulses in optical fibers. The physical phenomenon of this dynamical equation is revealed through the exact solutions for example solitons, elliptic function solutions, etc. These types of novel explicit solutions are constructed of generalized third-order NLSE via the modified extended direct algebraic method (MEDAM), that have key applications in engineering and applied sciences. Movements of some achieved solutions are described graphically, that assists to know the physical interpretation of this equation. Furthermore, the configuration conditions of dark-bright solitons are also presented. The influence of this scheme is shown in computational work and achieved results.
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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.
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The nonlinear Schrbdinger equation (NLS), with its modified forms, is the central equation for the description of nonlinear pulse propagation in optical fibers. There are a number of different physical situations in which coupling between waves leads to energy transfer. In such systems, ultrashort pulses have been observed to form during propagation. In this paper we show that much of this behavior can be understood by considering the effects of gain in the NLS. We also show that perturbations of the NLS do not destroy these results, provided that the modified equation possesses solitary-wave solutions.
In this paper, the higher-order generalized nonlinear Schrödinger equation, which describes the propagation of ultrashort optical pulse in optical fibers, is analytically investigated. By virtue of the Darboux transformation constructed in this paper, some exact soliton solutions on the continuous wave (cw) background are generated. The following propagation characteristics of those solitons are mainly discussed: (1) Propagation of two types of breathers which delineate modulation instability and bright pulse propagation on a cw background respectively; (2) Two types propagation characteristics of twosolitons: elastic interactions and mutual attractions and repulsions bound solitons. Those results might be useful in the study of ultrashort optical solitons in optical fibers.
Nonlinear Waves: Classical and Quantum Aspects
NATO Science Series II: Mathematics, Physics and Chemistry, 2005
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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.
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.
Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation
Physical Review Letters, 1997
We study solitary wave solutions of the higher order nonlinear Schrödinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions always exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of N -soliton solutions (N ≥ 2) are determined; when these conditions are met the equation becomes the modified KdV equation. A proper subset of these conditions meet the Painlevé plausibility conditions for integrability. 42.81.Dp, 02.30.Jr, 42.65.Tg, 42.79.Sz