Dynamics near a minimal-mass soliton for a Korteweg–de Vries equation

We study soliton solutions to a generalized Korteweg–de Vries (KdV) equation with a saturated nonlinearity, following the line of inquiry of Marzuola, Raynor and Simpson for the nonlinear Schrödinger equation (NLS). KdV with such a nonlinearity is known to possess a minimal-mass soliton. We consider a small perturbation of a minimal-mass soliton and numerically shadow a system of ordinary differential equation (ODEs), which models the behaviour of the perturbation for short times. This connects nicely to analytic works of Comech, Cuccagna and Pelinovsky as well as of Grimshaw and Pelinovsky. These ODEs form a simple dynamical system with a single unstable hyperbolic fixed point with two possible dynamical outcomes. A particular feature of the dynamics is that they are non-oscillatory. This distinguishes the KdV problem from the analogous NLS one.     

Bright and dark solitons in the normal dispersion regime of inhomogeneous optical fibers

In this letter, under investigation is a nonlinear Schrödinger equation with varying dispersion, nonlinearity and loss for the propagation of ultra-short optical pulses in the normal dispersion regime of optical fibers. By virtue of the modified Hirota's method and symbolic computation, the analytic two-soliton solution is explicitly obtained. Both the bright and dark solitons are observed in the normal dispersion regime of optical fibers with dispersion management. An asymptotic analysis to verify the elastic collision between solitons is performed and the stability of the soliton solutions is investigated. Besides, a new bright solitonic generator for generating high-power and narrow bandwidth pulses is advised. Furthermore, possible applicable soliton control techniques which might be used for the design of optical switch and dispersion-managed systems are proposed.     

Nonlinear wave modulation in fluid filled distensible tubes

Summary The present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic or elastic tube filled with an incompressible, inviscid fluid. Using the reductive perturbation technique, and assuming the weakness of dissipative effects, the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a dissipative nonlinear Schrödinger equation (NLS). In the absence of dissipative effects, this equation reduces to the classical NLS equation. The examination of the coefficients of the dissipative and classical NLS equations reveals the significance of the tube wall inertia to obtain a balance between nonlinearity and dispersion. Some special solutions of the NLS equation are given and the modulational instability of the plane wave solution is discussed for various incompressible hyperelastic materials.     

Wave propagation in fluid filled nonlinear viscoelastic tubes

Summary The present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic tube filled with an incompressible, inviscid fluid. In order to include the geometric dispersion in the analysis, the tube wall inertia effects are added to the pressure-area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the long-wave approximation is examined. In the long-wave approximation, a general equation is obtained, and it is shown that by a proper scaling this equation reduces to the well-known nonlinear evolution equations. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained. In the absence of nonlinear viscoelastic effects all the equations reduce to those of the linear viscoelastic tube.     

A note on the amplitude modulation of symmetric regularized long-wave equation with quartic nonlinearity

We study the amplitude modulation of a symmetric regularized long-wave equation with quartic nonlinearity through the use of the reductive perturbation method by introducing a new set of slow variables. The nonlinear Schr?dinger (NLS) equation with seventh order nonlinearity is obtained as the evolution equation for the lowest order term in the perturbation expansion. It is also shown that the NLS equation with seventh order nonlinearity assumes an envelope type of solitary wave solution.     

Analytical investigation of solitary waves in nonlinear Kerr medium

We study analytically the solution of nonlinear equation which result from the propagation of electromagnetic waves within a nonlinear Kerr medium. The medium is characterized by a dielectric constant which varies periodically and depends on the local field intensity. As a first step, we detail the resolution of the nonlinear equations with a quadratic nonlinearity. After that, we apply the slowly varying envelope approximation to obtain a Sine–Gordon equation. In this kind of nonlinearity, a gap solitons occurs. Moreover we verify that the solutions of the nonlinear equation for all frequencies within the gap are solitons solutions. After that we study the conditions of apparition of these particular solutions (i.e. soliton or anti-soliton) which are very suitable in experiments.     

Bright,dark and dark-singular soliton solutions of nonlinear Schrödinger's equation with spatio-temporal dispersion

This article presents an analytical study of the propagation of solitons through optical fibres. We consider a nonlinear Schrödinger equation with spatio-temporal dispersion and quadratic–cubic nonlinearity. Jacobi elliptic functions are used as an ansatz to extract optical dark and bright solitons as well as Jacobi elliptic solutions. The extended direct algebraic method gives dark and dark-singular soliton solutions. The constraint conditions which guarantee the existence of soliton solutions are listed.     

Kuznetsov–Ma waves train generation in a left-handed material

We analyze the behavior of an electromagnetic wave which propagates in a left-handed material. Second-order dispersion and cubic–quintic nonlinearities are considered. This behavior of an electromagnetic wave is modeled by a nonlinear Schrödinger equation which is solved by collective coordinates theory in order to characterize the light pulse intensity profile. More so, a specific frequency range has been outlined where electromagnetic wave behavior will be investigated. The perfect combination of second-order dispersion and cubic nonlinearity leads to a robust soliton. When the quintic nonlinearity comes into play, it provokes strong and long internal perturbations which lead to Benjamin–Feir instability. This phenomenon, also called modulational instability, induces appearance of a Kuznetsov–Ma waves train. We numerically verify the validity of Kuznetsov–Ma theory by presenting physical conditions which lead to Kuznetsov–Ma waves train generation. Thereafter, some properties of such waves train are also verified.     

Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod

Summary In this paper, we study nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. The aim is to derive simplified model equations in the far field which include both nonlinearity and dispersion. We consider disturbances in an initially pre-stressed rod. For long finite-amplitude waves, the Korteweg-de Vries (KdV) equation arises as the model equation. However, in a critical case, the coefficient of the dispersive term in the KdV equation vanishes. As a result, the dispersion cannot balance the nonlinearity. On the other hand, the latter has the tendency to make the wave profile steeper and steeper. The attention is then focused on finite-length and finite-amplitude waves. A new nonlinear dispersive equation which includes extra nonlinear terms involving second-order and third-order derivatives is derived as the model equation. In the case that the rod is composed of a compressible neo-Hookean material, that equation is further reduced to the Benjamin-Bona-Mahony (BBM) equation, which is known as an alternative to the KdV equation for modelling long finite-amplitude waves. To the author's knowledge, it is the first time that the BBM equation is found to arise as a model equation for finite-length and finite-amplitude waves.     

Rogue waves generation in a left-handed nonlinear transmission line with series varactor diodes

We investigate the electromagnetic wave behavior and its characterization using collective variables technique. Second-order dispersion, first- and second-order nonlinearities, which strongly act in a left-handed nonlinear transmission line with series varactor diodes, are taken into account. Four frequency ranges have been found. The first one gives the so-called energetic soliton due to a perfect combination of second-order dispersion and first-order nonlinearity. The second frequency range presents a dispersive soliton leading to the collapse of the electromagnetic wave at the third frequency range. But the fourth one shows physical conditions which are able to provoke the appearance of wave trains generation with some particular waves, the rogue waves. Moreover, we demonstrate that the number of rogue waves increases with frequency. The soliton, thereafter, gains a relative stability when second-order nonlinearity comes into play with some specific values in the fourth frequency range. Furthermore, the stability conditions of the electromagnetic wave at high frequencies have been also discussed.     

Stochastic dark solitons for a higher-order nonlinear Schrödinger equation in the optical fiber

In this paper, a stochastic higher-order nonlinear Schrödinger equation, which describes the wave propagation in the optical fiber with stochastic dispersion and nonlinearity, is investigated analytically. Via the symbolic computation and white noise functional approach, the stochastic dark one- and two-soliton solutions are obtained, and effects of the Gaussian white noise on the stochastic dark one and two solitons are discussed. For the stochastic dark one soliton, velocity and phase change randomly because of the Gaussian white noise, but the energy, shape and amplitude keep unchanged during the soliton propagation. For the stochastic dark two solitons, effect of the Gaussian white noise leads to the inversion of the velocity directions, while the velocities have the same varying trend so that the interaction appears that the stochastic dark two solitons keep parallel.     

Solitary Waves of Maxwell's Equations in Nonlinear Anisotropic Media

Abstract

The (1 + 1)-D solitary wave solutions of Maxwell's equations in nonlinearity induced anisotropic media (in liquids such as carbon disulphide, and in crystals, etc.) are investigated. We find that there is no arbitrarily linearly polarized (in the x-y plane perpendicular to the propagation direction z) soliton solution from Maxwell's equations except that with linear polarization either in alignment with or orthogonal to the geometric axis of the light induced refractive index change. This contradicts the prediction of the vector nonlinear Schroedinger equation (an approximation of Maxwell's equations) which yields soliton solutions with an arbitrary linear polarization. However, Maxwell's equations are found to admit stable elliptically polarized solitary wave solutions which reduce to the stable circularly polarized solitary wave solutions of the vector nonlinear Schroedinger equation when the induced refractive index change approaches zero.     

Tailored dispersion profile in controlling optical solitons in a tapered parabolic index fiber

We investigate the soliton dynamics in tapered parabolic index fibers via symbolic computation for a variety of dispersion profiles to inspect how a specific dispersion profile controls the optical soliton. By means of AKNS procedure, Lax pair is constructed for nonlinear Schrödinger equation with variable coefficients. Using obtained Lax pair, multi-soliton solutions are generated via Darboux transformation technique. Using multi-soliton solutions, soliton dynamics in tapered parabolic index fiber with the hyperbolic, Gaussian, exponential, and linear profiles are discussed. Results obtained in this study will be of certain potential application on construction of the nonlinear optical devices by soliton control. Results obtained in this study will be of certain value to the studies on the propagation and application of the soliton in the tapered parabolic index fiber and dispersion-managed fiber system.     

Analytic study on soliton-effect pulse compression in dispersion-shifted fibers with symbolic computation

The soliton-effect pulse compression of ultrashort solitons in a dispersion-shifted fiber (DSF) is investigated based on solving the higher-order nonlinear Schrödinger equation with the effects of third-order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). By using Hirota's bilinear method with a set of parametric conditions, the analytic one-, two- and three-soliton solutions of this model are obtained. According to those solutions, the higher-order soliton is shown to be compressed in the DSF for the pulse with width in the range of a few picoseconds or less. An appealing feature of the soliton-effect pulse compression is that, in contrast to the second-order soliton compression due to the combined effects of negative TOD and SRS, the third-order soliton can significantly enhance the soliton compression in the DSF with small values of the group-velocity dispersion (GVD) at the operating wavelength.     

What makes the Peregrine soliton so special as a prototype of freak waves?

The formation of breathers as prototypes of freak waves is studied within the framework of the classic ‘focussing’ nonlinear Schrödinger (NLS) equation. The analysis is confined to evolution of localised initial perturbations upon an otherwise uniform wave train. For a breather to emerge out of an initial hump, a certain integral over the hump, which we refer to as the “area”, should exceed a certain critical value. It is shown that the breathers produced by the critical and slightly supercritical initial perturbations are described by the Peregrine soliton which represents a spatially localised breather with only one oscillation in time and thus captures the main feature of freak waves: a propensity to appear out of nowhere and disappear without trace. The maximal amplitude of the Peregrine soliton equals exactly three times the amplitude of the unperturbed uniform wave train. It is found that, independently of the proximity to criticality, all small-amplitude supercritical humps generate the Peregrine solitons to leading order. Since the criticality condition requires the spatial scale of the initially small perturbation to be very large (inversely proportional to the square root of the smallness of the hump magnitude), this allows one to predict a priori whether a freak wave could develop judging just by the presence/absence of the corresponding scales in the initial conditions. If a freak wave does develop, it will be most likely the Peregrine soliton with the peak amplitude close to three times the background level. Hence, within the framework of the one-dimensional NLS equation the Peregrine soliton describes the most likely freak-wave patterns. The prospects of applying the findings to real-world freak waves are also discussed.     

Analytic study on bound solitons and soliton collisions for the coupled nonlinear Schrödinger equations

A type of mode-locked fiber laser, which can generate the bound solitons in the anomalous group-velocity dispersion regime, is suggested in this paper via symbolic computation. A transformation is given to convert the coupled nonlinear Schrödinger equations to a simpler form. Analytic two-, three- and N-soliton solutions for the coupled nonlinear Schrödinger equations are obtained. Based on those solutions, formation of the bound solitons is analyzed, and methods for the soliton control are presented. The soliton intensity and collision period can be controlled with the changes in the nonlinearity and group-velocity dispersion of optical fibers. The slight phase shift has a great influence on the bound states of solitons after the soliton collision. To support the results, numerical simulations of three-soliton propagation are performed. Finally, the modulational instability of bound solitons is analyzed.     

光纤中光孤子传输演化及相互作用研究

基于非线性薛定谔方程,采用分步傅立叶方法模拟了基阶、二阶和三阶皮秒光孤子以及光孤子对在光纤中的传输演化.结果表明,在长距离传输过程中基阶孤子的幅度和脉宽基本不变,是进行光孤子通信的理想载体.高阶孤子的幅度和脉宽变化较大,并呈现一种周期性的变化;孤子对的传输与两个脉冲的初始相位和输入强度相关.考虑三阶色散的飞秒量级孤子在光纤中的传输将不再出现对称性和周期性,脉冲的中心位置将发生偏移,同时脉冲的波形也会发生扭曲.     

Inelastic vector soliton collisions: a lattice-based quantum representation

Lattice-based quantum algorithms are developed for vector soliton collisions in the completely integrable Manakov equations, a system of coupled nonlinear Schr?dinger (coupled-NLS) equations that describe the propagation of pulses in a birefringent fibre of unity cross-phase modulation factor. Under appropriate conditions the exact 2-soliton vector solutions yield inelastic soliton collisions, in agreement with the theoretical predictions of Radhakrishnan et al. (1997 Phys. Rev. E56, 2213). For linearly birefringent fibres, quasi-elastic solitary-wave collisions are obtained with emission of radiation. In a coupled integrable turbulent NLS system, soliton turbulence is found with mode intensity spectrum scaling as kappa(-6).     

On the range of validity of a simple wave approximation of a non-linear set of diffusive wave equations

Summary The set of wave equations considered is an intermediate approximation of the Navier-Stokes equations. A further approximation leads to Burgers' equation. The range of validity of this simple wave approximation has been studied. The method used is especially useful for small nonlinearity.     

The multiple-scale averaging and dynamics of dispersion-managed optical solitons

Multiple-scale averaging is applied to the nonlinear Schrödinger equation with rapidly varying coefficients, and use the results to analyze pulse propagation in an optical fiber when a periodic dispersion map is employed. The effects of fiber loss and repeated amplification are taken into account by use of a coordinate transformation to relate the pulse dynamics in lossy fibers to that in equivalent lossless fibers. Second-order averaging leads to a general evolution equation that is applicable to both return-to-zero (soliton) and non-return-to-zero encoding schemes. The resulting equation is then applied to the specific case of solitons, and an asymptotic theory for the pulse dynamics is developed. Based upon the theory, a simple and effective design of two-step dispersion maps that are advantageous for wavelength-division-multiplexed soliton transmission is proposed. Theuse of these specifically designed dispersion maps allows simultaneous minimization of dispersive radiation in several different channels.