Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse

The sine-Gordon (SG) equation and perturbed nonlinear Schrödinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called light bullets) in nonlinear dispersive optical media. We begin with the (2 + 1) SG equation obtained as an asymptotic reduction in the two level dissipationless Maxwell-Bloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing-defocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet solutions of SG and perturbed NLS as well as critical NLS are reported, which validate that the solution of the perturbed NLS as well as its finite-term truncations are in qualitative and quantitative agreement with the solution of SG for the light bullets propagation even after the critical collapse of cubic focusing NLS. In contrast, standard critical NLS is in qualitative agreement with SG only before its collapse. As a benefit of such observations, pulse propagations are studied via solving the perturbed NLS truncated by reasonably many nonlinear terms, which is a much cheaper task than solving SG equation directly.     

Invariant tori in the discrete NLS with small amplitude diffraction management

We consider the question of persistence of breather solutions of the discrete NLS equation under time-periodic perturbations corresponding to small amplitude diffraction management. The question is formulated as a problem of continuation of tori in an infinite-dimensional Hamiltonian system with symmetries and we show that one-peak breathers of the discrete NLS with zero residual diffraction can be continued to periodic or quasiperiodic solutions of the discrete NLS with small residual diffraction and small amplitude diffraction management, provided that a nonresonance condition is satisfied. We also present numerical evidence that a similar continuation should be possible for certain single-, and multi-peak breathers of the discrete NLS with small diffraction.     

On the Incompressible Fluid Limit and the Vortex Motion Law of the Nonlinear Schrödinger Equation

The nonlinear Schr?dinger equation (NLS) has been a fundamental model for understanding vortex motion in superfluids. The vortex motion law has been formally derived on various physical grounds and has been around for almost half a century. We study the nonlinear Schr?dinger equation in the incompressible fluid limit on a bounded domain with Dirichlet or Neumann boundary condition. The initial condition contains any finite number of degree ± 1 vortices. We prove that the NLS linear momentum weakly converges to a solution of the incompressible Euler equation away from the vortices. If the initial NLS energy is almost minimizing, we show that the vortex motion obeys the classical Kirchhoff law for fluid point vortices. Similar results hold for the entire plane and periodic cases, and a related complex Ginzburg–Landau equation. We treat as well the semi-classical (WKB) limit of NLS in the presence of vortices. In this limit, sound waves propagate through steady vortices. Received: 1 December 1997 / Accepted: 27 June 1998     

Spectral stability analysis for periodic traveling wave solutions of NLS and CGL perturbations

We consider cnoidal traveling wave solutions to the focusing nonlinear Schrödinger equation (NLS) that have been shown to persist when the NLS is perturbed to the complex Ginzburg-Landau equation (CGL). We show that while these periodic traveling waves are spectrally stable solutions of NLS with respect to periodic perturbations, they are unstable with respect to bounded perturbations. Furthermore, we use an argument based on the Fredholm alternative to find an instability criterion for the persisting solutions to CGL.     

傅里叶变换红外光谱法研究核辐照对三七总皂苷粉成分的影响

采用傅里叶变换红外光谱(卷积谱)法对经不同剂量y-射线核辐照的三七总皂苷粉进行了对比研究.辐照剂量不高于9kGy时,三七总皂苷粉样品的化学成分几乎没有发生变化;三七总皂苷粉样品经15kGy及以上的辐照剂量辐照后,可能产生新的化学成分;经21kGy的剂量辐照,三七总皂苷粉产生了人参、三七粉所含的普通成分(非三七总皂苷成分...     

ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.     

Dark Solitons for the Defocusing Cubic Nonlinear Schrödinger Equation with the Spatially Periodic Potential and Nonlinearity

We study the existence of dark solitons of the defocusing cubic nonlinear Schrödinger (NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincaré map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.     

Nonlinear photonic crystals: IV. Nonlinear Schrödinger equation regime

We study here the nonlinear Schrödinger (NLS) equation as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell (NLMs) equations. The dielectric medium is assumed to be periodic, with a cubic nonlinearity, and with its linear background possessing inversion symmetric dispersion relations. The medium is excited by a current J producing an EM wave. The wave nonlinear evolution is analysed based on the modal decomposition and an expansion of the exact solution to the NLM into an asymptotic series with respect to three small parameters α, β and ?. These parameters are introduced through the excitation current J to scale, respectively (i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the range of wavevectors involved in its modal composition, with β^?1 scaling its spatial extension; (iii) its frequency bandwidth, with ?^?1 scaling its time extension. We develop a consistent theory of approximations of increasing accuracy for the NLM with its first term governed by the NLS. We show that such NLS regime is the medium response to an almost monochromatic excitation current J. The developed approach not only provides rigorous estimates of the approximation accuracy of the NLM with the NLS in terms of powers of α, β and ?, but it also produces a new extended NLS (ENLS) providing better approximations. Remarkably, quantitative estimates show that properly tailored ENLS can significantly improve the approximation accuracy of the NLM compared with the classical NLS equation.     

One-parameter localized traveling waves in nonlinear Schrödinger lattices

We address the existence of traveling single-humped localized solutions in the spatially discrete nonlinear Schrödinger (NLS) equation. A mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero. The technique is based on using the Implicit Function Theorem for solution of an appropriate differential advance-delay equation in exponentially weighted spaces. The resulting Melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse, which are checked numerically. We apply the technique to the so-called Salerno model and the translationally invariant discrete NLS equation with a cubic nonlinearity. We show that the traveling solutions terminate in the Salerno model whereas they generally persist in the translationally invariant NLS lattice as a one-parameter family of solutions. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions under time evolution of the discrete NLS equation.     

Breathers and solitons for the coupled nonlinear Schr?dinger system in three-spine α-helical protein

We mainly investigate the variable-coefficient 3-coupled nonlinear Schr?dinger(NLS) system, which describes soliton dynamics in the three-spine α-helical protein with inhomogeneous effect. The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly. The Hirota method is used to solve the constant coefficient NLS equation, and then we get the one-and two-breather solutions of the variable-coefficient NLS equation. The results show that, in the background of plane waves and periodic waves, the breather can be transformed into some forms of combined soliton solutions. The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail. Our results are helpful to study the soliton dynamics inα-helical protein.     

Electron-acoustic supernonlinear waves and their multistability in the framework of the nonlinear Schrödinger equation

Finite-amplitude supernonlinear electron-acoustic waves (EAWs) are investigated under the nonlinear Schrödinger (NLS) equation in a plasma system that is composed of cold electron fluid, immobile ions and q-nonextensive hot electrons. Using the wave transfiguration, the NLS equation is deduced in a dynamical system. The presence of finite-amplitude nonlinear and supernonlinear EAWs is shown by phase plane analysis. The effects of the nonextensive parameter (q) and the speed of waves (v) on different traveling wave solutions of EAWs are presented. Furthermore, by introducing a small external periodic force in the dynamical system, multistability behaviors of EAWs under the NLS equation are shown for the first time in classical plasmas.     

The theorem of permutability for the nonlinear schrödinger equation

The theorem of permutability for the nonlinear Schrödinger (NLS) equation is proved, in the general case, by introducing a real potential function which satisfies a nonlinear evolution equation equivalent to the NLS equation.     

Transformation from the nonautonomous to standard NLS equations

In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schrödinger (NLS) equation. An integrable condition is first obtained by the Painlevé analysis, which is shown to be consistent with that obtained by the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.     

Explicit N-Solit on Solution of the Unstable Nonlinear Schrödinger Equation

The unstable nonlinear Schrodinger (NLS) equation is solved by the inverse scattering transform. Based on the constructed Zakharov-Shabat equation, it is shown that the soliton solution of the unstable NLS equation can be known from the soliton solution of the usual NLS equation by simply exchanging the tariables. The explicit N-soliton solution and the position shifts due to the collision are thus calculated.     

色散缓减光纤的模面积对孤子传输的影响

本文求解了色散缓减光纤的NLS方程,由其孤子解讨论了色散缓减光纤模面积对孤子传输的影响。     

On the Asymptotic Stability of {N}-Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation for finite density type initial data. Using the \({\overline{\partial}}\) generalization of the nonlinear steepest descent method of Deift and Zhou, we derive the leading order approximation to the solution of NLS for large times in the solitonic region of space–time, \({|x| < 2 t}\), and we provide bounds for the error which decay as \({ t \rightarrow \infty}\) for a general class of initial data whose difference from the non vanishing background possesses a fixed number of finite moments and derivatives. Using properties of the scattering map of NLS we derive, as a corollary, an asymptotic stability result for initial data that are sufficiently close to the N-dark soliton solutions of NLS.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

New geometries associated with the nonlinear Schrödinger equation

We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr?dinger equation (NLS). We show that any given solution of the NLS gets associated with three distinct space curve evolutions. The tangent vector of the first of these curves, the binormal vector of the second and the normal vector of the third, are shown to satisfy the integrable Landau-Lifshitz (LL) equation = ×, ( = 1). These connections enable us to find the three surfaces swept out by the moving curves associated with the NLS. As an example, surfaces corresponding to a stationary envelope soliton solution of the NLS are obtained. Received 5 December 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: radha@imsc.ernet.in     

On instability of excited states of the nonlinear Schrödinger equation

We introduce a new notion of linear stability for standing waves of the nonlinear Schrödinger equation (NLS) which requires not only that the spectrum of the linearization be real, but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive. We prove that excited states of the NLS are not linearly stable in this more restrictive sense. We then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability.     

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrdinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.