Eigenvalues of the Zakharov-Shabat scattering problem for two separated sech-shaped pulses

It is shown that the initial condition of two separated sech-shaped in-phase pulses in the nonlinear Schrödinger equation, may give rise to not only stationary solitons, but also symmetrically separating solitons, provided the initial distance of separation is large enough. The critical distance between the pulses for which a separating soliton pair can be found for certain amplitudes is derived using a variational approach.     

Cascade compression induced by nonlinear barriers in propagation of optical solitons

We investigate the nonlinear tunneling of optical solitons through both dispersion and nonlinear barriers by employing the exact solution of the generalized nonlinear Schrödinger equation with variable coefficients. The extensive numerical simulations show that the optical solitons can be efficiently compressed when they pass through adequate engineered nonlinear barriers. A cascade compression system in a dispersion decreasing fiber with nonlinear barriers on an exponential background is proposed and the cascade compression of optical pulses is further investigated in detail. Finally, the stability to various initial perturbations of the cascade compressed optical soliton and the interaction between two neighboring compressed solitons were investigated too.     

Soliton coupling in birefringent optical fiber with third-order dispersion

Nonlinear coupling of polarized solitons in birefringent optical fiber in the presence of third-order dispersion is considered in the framework of the coupled nonlinear Schrödinger equations. The influence of third-order dispersion on the interaction between solitons is investigated. For sufficiently strong third-order dispersion the interaction may even become repulsive. The stable conditions for solitons of partial pulses are analyzed and amplitude threshold, which decreases with third-order dispersion coefficient decreasing, for the capture of solitons of partial pulses into a coupled two-component pulse is obtained.     

The modified propagation equation for TM polarized subwavelength spatial solitons in a nonlinear planar waveguide

The wave equation of TM polarized subwavelength beam propagations in a nonlinear planar waveguide is derived beyond the paraxial approximation. This modified equation contains more higher-order linear and nonlinear terms than the nonlinear Schrödinger equation. The propagation of fundamental subwavelength spatial solitons is numerically studied. It is shown that the effect of the higher nonlinear terms is significant. That is, for the propagation of narrower beam the modified nonlinear Schrödinger equation is more suitable than the nonlinear Schrödinger equation.     

具有色散系数的(2+1)维非线性Schrödinger方程的有理解和空间孤子

非线性Schrödinger方程是物理学中具有广泛应用的非线性模型之一. 本文采用相似变换, 将具有色散系数的(2+1)维非线性Schrödinger方程简化成熟知的Schrödinger方程, 进而得到原方程的有理解和一些空间孤子.     

Exact chirped gray soliton solutions of the nonlinear Schrödinger equation with variable coefficients

In this paper, we consider the nonlinear Schrödinger equation with variable coefficients, and by using direct transformation of variables and functions, the explicit chirped gray one- and two-soliton solutions are presented. Based on the exact solutions, we in detail analyze the propagation characteristics of the chirped gray soliton, including the stability against either the deviation from integrable condition or the initial perturbation, and interaction between the chirped gray solitons. The results show that the gray soliton can be compressed by choosing the appropriate initial chirp, and the chirped gray pulses can stably propagate along optical fibers remaining the character of solitons.     

Angular pseudomomentum theory for the generalized nonlinear Schrödinger equation in discrete rotational symmetry media

We develop a complete mathematical theory for the symmetrical solutions of the generalized nonlinear Schrödinger equation based on the concept of angular pseudomomentum. We consider the symmetric solitons of a generalized nonlinear Schrödinger equation with a nonlinearity depending on the modulus of the field. We provide a rigorous proof of a set of mathematical results justifying that these solitons can be classified according to the irreducible representations of a discrete group. Then we extend this theory to non-stationary solutions and study the relationship between angular momentum and pseudomomentum. We illustrate these theoretical results with numerical examples.     

Short Envelope Solitons. Isotropic and Anisotropic Media

Within the framework of the third-order approximation of the nonlinear wave dispersion theory, we find new classes of short scalar and vector solitons of lengths about several wavelengths. Short scalar solitons are found within the framework of a third-order nonlinear Schrödinger equation (NSE-3) including both the nonlinear dispersion terms and the third-order linear dispersion term. The interaction of such solitons is studied, and the soliton stability is proved. Short vector solitons are found within the framework of a coupled third-order nonlinear Schröodinger equation (CNSE-3). Interaction and stability of such solitons are studied.     

Propagation of optical pulses at an absorption line in the presence of a weak nonlinearity

We examine the propagation of short pulses of light in a resonantly absorbing, weakly nonlinear medium within the limits of a model described by the nonlinear Schrödinger equation. The possibility of transforming pulses of various forms into a soliton signal due to the effects of self-interaction is studied. On the basis of the study of spectra for the associated linear problem, we investigate the break-up of an initial pulse into solitons. We have obtained solutions for two particular cases of the initial pulse.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 36–39, February, 1989.     

Interaction of nonlinearity and polarization mode dispersion

The derivation of the coupled nonlinear Schrödinger equation and the Manakov-PMD equation is reviewed. It is shown that the usual scalar nonlinear Schrödinger equation can be derived from the Manakov-PMD equation when polarization mode dispersion is negligible and the signal is initially in a single polarization state as a function of time. Applications of the Manakov-PMD equation to studies of the interaction of the Kerr nonlinearity with polarization mode dispersion are then discussed.     

Conservation laws,bilinear forms and solitons for a fifth-order nonlinear Schrödinger equation for the attosecond pulses in an optical fiber

Under investigation in this paper is a fifth-order nonlinear Schrödinger equation, which describes the propagation of attosecond pulses in an optical fiber. Based on the Lax pair, infinitely-many conservation laws are derived. With the aid of auxiliary functions, bilinear forms, one-, two- and three-soliton solutions in analytic forms are generated via the Hirota method and symbolic computation. Soliton velocity varies linearly with the coefficients of the high-order terms. Head-on interaction between the bidirectional two solitons and overtaking interaction between the unidirectional two solitons as well as the bound state are depicted. For the interactions among the three solitons, two head-on and one overtaking interactions, three overtaking interactions, an interaction between a bound state and a single soliton and the bound state are displayed. Graphical analysis shows that the interactions between the two solitons are elastic, and interactions among the three solitons are pairwise elastic. Stability analysis yields the modulation instability condition for the soliton solutions.     

Integrability of the Gross-Pitaevskii equation with Feshbach resonance management

In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schrödinger equation. By this transformation, each exact solution of the standard nonlinear Schrödinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Explicit Exact Solutions for Some Nonlinear Partial Differential Equations with Nonlinear Terms of Any Order

In this paper, by introducing some appropriate transformation and with the help of symbolic computation, we study exact travelling wave solutions for the high-order modified Boussinesq equation, a single nonlinear reaction-diffusion equation and a generalized nonlinear Schrödinger equation with nonlinear terms of any order by use of the extended-tanh method. Thus, some new exact travelling-wave solutions, which contain kink-shaped solitons, bell-shaped solitons, periodic solutions, combined formal solitons, rational solutions and singular solitons for these equations, are obtained.     

Effect of initial pulse shape modulation on spontaneous soliton formation in the NSE model

An initial pulse with fairly steep fronts whose evolution is described by the nonlinear Schrödinger equation, splits into soliton-like pulses (spontaneous soliton formation). The number of solitons formed in this process can be estimated by the number of spectral points of the associated linear Zakharov-Shabat problem for the initial pulse. Exact solutions of the Zakharov-Shabat problem are constructed for some classes of initial piecewise-continuous pulses by using the Darboux method. This allows us to estimate the effect of the shape of the initial pulse on the number of formed solitions and their parameters.V. V. Kuibyshev State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 19–24, June, 1992.     

Self-compression and controllable guidance of multi-millijoule femtosecond laser pulses

Self-compression of multi-millijoule femtosecond laser pulses and dramatic increase of the peak intensity are found in pressurized helium and neon within a range of intensity in which the ionization modification of the material parameters by the pulse is negligible. The pulse propagation is studied by the (3 + 1)-dimensional nonlinear Schrödinger equation including basic lowest order optical processes - diffraction, second order of dispersion, and third order of nonlinearity. Smooth and well controllable pulse propagation dynamics is found. Construction of compressed pulses of controllable parameters at given space target point by a proper chose of the pulse energy and/or gas pressure is predicted.     

Modulational instability in the cubic-quintic nonlinear Schrödinger equation through the variational approach

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.     

Generation and propagation of pulse trains with ultrashort pulse separation

We investigate the nonlinear Schrödinger equation with variable coefficients by employing perturbation method. The analysis solution of the harmonic form is presented. The solution is one of forms to describe pulse trains with ultrashort pulse separation, which is about two orders of magnitude shorter than one of sech-type solitons considered before. And we could systematically adjust the perturbation parameter to obtain different pulse separation. As an example, we consider a nonlinear dispersive system with spatial parameter variations, and the results show that, the pulse train with ultrashort pulse separation presented by analysis solution may keep its shape even if the velocity is changed. The stability of the solution is discussed numerically, and the results reveal that the finite initial perturbations, such as white noise could not influence the main character of the solution. In addition, the stability of the solution is also discussed under more general conditions.     

Nonlinear Dynamics of Wave Fields in Nonresonant Media: From Envelope Solitons toward Video Solitons

We analyze a new class of soliton solutions for a wave field, which describes propagation of soliton-like structures of a circularly polarized electromagnetic field comprising a finite number of field-oscillation periods in a transparent nonresonant medium. The considered solutions feature a smooth transition from the soliton solutions of Schröodinger type, which correspond to long pulses with a large number of field oscillations, to extremely short, virtually single-cycle video pulses. We show that such solutions can also be important for linearly polarized laser fields. The structural stability of few-optical-cycle solitons is demonstrated numerically, including the case of their collision. Based on stability analysis and with allowance for the genealogic relation between the obtained wave solitons and the solitons of the nonlinear Schröodinger equation, we argue that the former solitons can play the same fundamental role in the nonlinear dynamics of the considered wave fields. In particular, it is shown by numerical simulations that the few-optical-cycle solutions turn out to be the basic elementary components of such a dynamical process as the temporal compression of an initially long pulse to a pulse of very short duration. In this case, the minimum duration of a compressed pulse is determined by soliton structures of about minimal duration.     

Laser pulse sequence effects on selective ionization in three-level systems driven by coherent radiation fields

Using the time-dependent Schrödinger equation we investigate the effect of different pulse sequences of two lasers on selective ionization in three-level systems driven by coherent radiation fields. It is assumed that inhomogeneous broadening as well as the radiative decay can be neglected during the laser pulses. We demonstrate to what an extent the selectivity given by the absorption or relaxation process can be increased by coherent interaction during the first step and different timing of the incoherent interaction during the second step of a two-step photoionization. A comparison is made with the rate-equation results. We could show that utilization of coherent excitation for high ionization selectivity is most efficient if the two laser pulses do not overlap.