Narrow Solitons Research Articles

Propagation of ion-acoustic solitary waves in a relativistic electron-positron-ion plasma

The propagation of large amplitude ion-acoustic solitary waves (IASWs) in a fully relativistic plasma consisting of cold ions and ultra-relativistic hot electrons and positrons is investigated using the Sagdeev pseudopotential method in a relativistic hydrodynamics model. The effects of streaming speed of the plasma fluid, thermal energy, positron density, and positron temperature on large amplitude IASWs are studied by analysis of the pseudopotential structure. It is found that in regions in which the streaming speed of the plasma fluid is larger than that of the solitary wave, by increasing the streaming speed of the plasma fluid, the depth and width of the potential well increase, resulting in narrower solitons with larger amplitude. This behavior is opposite to the case where the streaming speed of the plasma fluid is less than that of the solitary wave. On the other hand, an increase in the thermal energy results in wider solitons with smaller amplitude, because the depth and width of the potential well decrease in that case. Additionally, the maximum soliton amplitude increases and the width becomes narrower as a result of an increase in positron density. It is shown that varying the positron temperature does not have a considerable effect on the width and amplitude of IASWs. The existence of stationary soliton-like arbitary amplitude waves is also predicted in fully relativistic electron-positron-ion (EPI) plasmas. The effects of streaming speed of the plasma fluid, thermal energy, positron density, and positron temperature on these kinds of solitons are the same for large amplitude IASWs.

Solitons in combined linear and nonlinear lattice potentials

We study ordinary solitons and gap solitons (GSs) in the effectively one-dimensional Gross-Pitaevskii equation, with a combination of linear and nonlinear lattice potentials. The main points of the analysis are effects of the (in)commensurability between the lattices, the development of analytical methods, viz., the variational approximation (VA) for narrow ordinary solitons, and various forms of the averaging method for broad solitons of both types, and also the study of mobility of the solitons. Under the direct commensurability (equal periods of the lattices, the family of ordinary solitons is similar to its counterpart in the free space. The situation is different in the case of the subharmonic commensurability, with L_{lin}=(1/2)L_{nonlin}, or incommensurability. In those cases, there is an existence threshold for the solitons, and the scaling relation between their amplitude and width is different from that in the free space. GS families demonstrate a bistability, unless the direct commensurability takes place. Specific scaling relations are found for them too. Ordinary solitons can be readily set in motion by kicking. GSs are mobile too, featuring inelastic collisions. The analytical approximations are shown to be quite accurate, predicting correct scaling relations for the soliton families in different cases. The stability of the ordinary solitons is fully determined by the VK (Vakhitov-Kolokolov) criterion, while the stability of GS families follows an inverted ("anti-VK") criterion, which is explained by means of the averaging approximation.

Soliton dynamics at an interface between a uniform medium and a nonlinear optical lattice

We study trapping and propagation of a matter-wave soliton through the interface between uniform medium and a nonlinear optical lattice. Different regimes for transmission of a broad and a narrow solitons are investigated. Reflections and transmissions of solitons are predicted as a function of the lattice phase. The existence of a threshold in the amplitude of the nonlinear optical lattice, separating the transmission and reflection regimes, is verified. The localized nonlinear surface state, corresponding to the soliton trapped by the interface, is found. Variational approach predictions are confirmed by numerical simulations for the original Gross-Pitaevskii equation with nonlinear periodic potentials.

Solitons in the discrete nonpolynomial Schrödinger equation

We introduce a species of the discrete nonlinear Schr\"odinger (DNLS) equation, which is a model for a self-attractive Bose-Einstein condensate confined in a combination of a cigar-shaped trap and deep optical lattice acting in the axial direction. The equation is derived as a discretization of the respective nonlinear nonpolynomial Schr\"odinger equation. Unlike previously considered varieties of one-dimensional DNLS equations, the present discrete model admits on-site collapse. We find two families of unstaggered on-site-centered discrete solitons, stable and unstable ones, which include, respectively, broad and narrow solitons, their stability exactly complying with the Vakhitov-Kolokolov criterion. Unstable on-site solitons either decay or transform themselves into robust breathers. Intersite-centered unstaggered solitons are unstable to collapse; however, they may be stabilized by the application of a sufficiently strong kick, which turns them into moving localized modes. Persistently moving solitons can be readily created too by the application of the kick to stable on-site unstaggered solitons. In the same model, staggered solitons, which are counterparts of gap solitons in the continuum medium, are possible if the intrinsic nonlinearity is self-repulsive. All on-site staggered solitons are stable, while intersite ones have a small instability region. The staggered solitons are immobile.

Analytic theory of narrow lattice solitons

The profiles of narrow lattice solitons are calculated analytically using perturbation analysis. A stability analysis shows that solitons centred at a lattice (potential) maximum or saddle point are unstable, as they drift towards the nearest lattice minimum. This instability can, however, be so weak that the soliton is ‘mathematically unstable’ but ‘physically stable’. Stability of solitons centred at a lattice minimum depends on the dimension of the problem and on the nonlinearity. In the subcritical and supercritical cases, the lattice does not affect the stability, leaving the solitons stable and unstable, respectively. In contrast, in the critical case (e.g. a cubic nonlinearity in two transverse dimensions), the lattice stabilizes the (previously unstable) solitons. The stability in this case can be so weak, however, that the soliton is ‘mathematically stable’ but ‘physically unstable’.

Stable circulation modes in a dual-core matter-wave soliton laser

We consider a model of a matter-wave laser generating a periodic array of solitary-wave pulses. The system, a general version of which was recently proposed by us (2005 J. Phys. B: At. Mol. Opt. Phys. 38 4221), is composed of two parallel tunnel-coupled cigar-shaped traps (a reservoir and a lasing cavity), solitons being released through a valve at one edge of the cavity. We report a stable lasing mode accounted for by circulations of a narrow soliton in the cavity, which generates an array of strong pulses (with 103–104 atoms in each, the array's duty cycle being ≃30%) when the soliton periodically hits the valve.

Two-dimensional solitons in saturable media with a quasi-one-dimensional lattice potential

We study families of solitons in a two-dimensional model of the light transmission through a photorefractive medium equipped with a (quasi-)one-dimensional photonic lattice. The soliton families are bounded from below by finite minimum values of the peak and total power. Narrow solitons have a single maximum, while broader ones feature side lobes. Stability of the solitons is checked by direct simulations. The solitons can be set in motion across the lattice (actually, made tilted in the spatial domain), provided that the respective boost parameter does not exceed a critical value. Collisions between moving solitons are studied too. Collisions destroy the solitons, unless their velocities are sufficiently small. In the latter case, the colliding solitons merge into a single stable pulse.

Matter-wave solitons in nonlinear optical lattices

We introduce a dynamical model of a Bose-Einstein condensate based on the one-dimensional (1D) Gross-Pitaevskii equation (GPE) with a nonlinear optical lattice (NOL), which is represented by the cubic term whose coefficient is periodically modulated in the coordinate. The model describes a situation when the atomic scattering length is spatially modulated, via the optically controlled Feshbach resonance, in an optical lattice created by interference of two laser beams. Relatively narrow solitons supported by the NOL are predicted by means of the variational approximation (VA), and an averaging method is applied to broad solitons. A different feature is a minimum norm (number of atoms), N = N(min), necessary for the existence of solitons. The VA predicts N(min) very accurately. Numerical results are chiefly presented for the NOL with the zero spatial average value of the nonlinearity coefficient. Solitons with values of the amplitude A larger than at N = N(min) are stable. Unstable solitons with smaller, but not too small, A rearrange themselves into persistent breathers. For still smaller A, the soliton slowly decays into radiation without forming a breather. Broad solitons with very small A are practically stable, as their decay is extremely slow. These broad solitons may freely move across the lattice, featuring quasielastic collisions. Narrow solitons, which are strongly pinned to the NOL, can easily form stable complexes. Finally, the weakly unstable low-amplitude solitons are stabilized if a cubic term with a constant coefficient, corresponding to weak attraction, is included in the GPE.

New solutions for the one-dimensional nonconservative inviscid Burgers equation

The propagation of travelling waves is a relevant physical phenomenon. As usual the understanding of a real propagating wave depends upon a correct formulation of a idealized model. Discontinuous functions, Dirac- δ measures and their distributional derivatives are, respectively, idealizations of sharp jumps, localized high peaks and single sharp localised oscillations. In the present paper we study the propagation of distributional travelling waves for Burgers inviscid equation. This will be afforded by our theory of distributional products, and is based on a rigorous and consistent concept of solution we have introduced in [C.O.R. Sarrico, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003) 641–656]. Our approach exhibit Dirac- δ travelling solitons (they are just the “infinitesimal narrow solitons” of Maslov, Omel'yanov and Tsupin [V.P. Maslov, O.A. Omel'yanov, Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys 36 (1981) 73–149; V.P. Maslov, V.A. Tsupin, Necessary conditions for the existence of infinitely narrow solitons in gas dynamics, Soviet Phys. Dokl. 24 (1979) 354–356]) and also solutions which are not measures such as for instance u ( x , t ) = b + δ ′ ( x − b t ) , a wave of constant speed b. Moreover, for signals with two jump discontinuities we have, in our setting, the propagation of more solitons and more values for the signal speed are allowed than those afforded within classical framework.

Ultra-narrow bright spatial solitons interacting with left-handed surfaces

A vectorial FDTD method is used to present a numerical study of very narrow spatial solitons interacting with the surface of what has become known as a left-handed medium. After a comprehensive discussion of the background and the family of surface modes to be expected on a left-handed material, bounded by dispersion-free right-handed material, it is demonstrated that robust outcomes of the FDTD approach yield dramatic confirmation of these waves. The FDTD results show how the linear and nonlinear surface modes are created and can be tracked in time as they develop. It is shown how they can move backwards, or forwards, depending either upon a critical value of the local nonlinear conditions at the interface, or the ambient linear conditions. Several examples are given to demonstrate the power and versatility of the method, and the sensitivity to the launching conditions.

Dynamics of Narrow Bright Solitons in an Array of Attractive Atoms in a Bose-Einstein Condensate

We study the dynamics of a narrow bright soliton in a one-dimensional lattice of condensed attractive atoms when the soliton width is comparable to the lattice spacing. If a momentum is imprinted to a stationary state, the soliton can have oscillations around a site or it can undergo a random motion along the array. The motion is very sensitive to the atomic background distribution, and a thermal cloud or quantum field fluctuations can induce a random motion of the soliton.

Solitons in inharmonic lattices: Application to achiral carbon nanotubes

In this paper we demonstrate that soliton solutions, whose existence is well known in one-dimensional (1D) model anharmonic systems, can also exist in more complex systems such as carbon nanotubes (CNT). We investigate armchair and zigzag nanotubes and show that both can be simplified to anharmonic 1D lattice models with reduced internal degrees of freedom. Because of the differences in chirality, the armchair and zigzag nanotubes are represented, respectively, by homogeneous (one site per unit cell) and dimerized chains. Starting with the Brenner potential we expand the tube energy into Taylor series and construct effective interaction potentials for model lattices of armchair and zigzag CNTs. We then construct a continuous approximation for the model lattices and derive the Korteweg--de Vries (KdV) equation by applying the reductive perturbation method. The stability of KdV solitons are checked by molecular dynamics simulations of model lattices with parameters specific for CNTs. Numerical simulations attest to the stability of even narrow solitons in most CNTs. We show that solitons can be generated by shock compression at one end of the model chains. The formalism developed can also be applied to many quasi-1D objects such as polymers, nanowires, and others.

Calculation of Integrals of the Hugoniot–Maslov Chain for Singular Vortical Solutions of the Shallow-Water Equation

We discuss the problems of the Hugoniot-Maslov chain integrability for singular vortical solutions of the shallow-water equations on the β plane. We show that the complex variables used to derive the chain automatically give most of the integrals of the complete and the truncated chains. We also study how some of these integrals are related to the Lagrangian invariant (potential vorticity) .W e discuss how to choose solutions of the chain that can be used to describe the actual trajectories of tropical cyclones. Here, the scalar "phase" S(x, t), the vector (or scalar) "background" f (x, t), and the "amplitude" g(x, t) are smooth (infinitely differentiable) functions. The scalar function F (τ ) smoothly depends on τ for τ � and has a singularity at τ = 0. The singularities of the function w(x, t) are thus determined by the zeros of the phase S. Hereafter, the left superscript t denotes transposition. These solutions also include shock waves; in this case, F =Θ (τ ), where Θ(τ ) is the Heaviside function (Θ =0f or τ< 0a nd Θ=1f orτ ≥ 0). These solutions also include infinitely narrow solitons F =S ol(τ ), where Sol(τ )=0f orτ � 0 and Sol(τ )=1f orτ = 0. In the one-dimensional problems, the phase S has the form S = x− X(t) in both cases. Another example is given by weak singular solutions whose singularity has the type of the square root of a quadratic form. In this case, F = √ τ , and in the first approximation, S is a nondegenerate positive quadratic form with respect to x centered at the points x = X(t )t hat determine the trajectory Γ = � x = X(t) � of the singularity motion. Some properties of singular solutions of such systems of shallow-water equations on the β plane are studied in this paper. In this case, x = t (x1 ,x 2) ∈ R 2 , the vectors w, f ,a ndg have three components, and we have the formulas   η u1 u2

Functionals with values in the non‐archimedean field of laurent series and their applications to the equations of elasticity theory. I 1

Abstract Functionals with values in Non‐Archimedean field of Laurent series applied to the definition of generalized solution (in the form of soliton) of the Hopf equation. Calculation method for the profile of infinitely narrow soliton is proposed. Applying this method, calculations of profiles are reduced to the nonlinear system of algebraic equations in R n+1, n > 1. It is shown that there is a possibility to find out some of the solutions of this system using the Newton iteration method. Example and numerical test are considered.

FUNCTIONALS WITH VALUES IN THE NON‐ARCHIMEDEAN FIELD OF LAURENT SERIES AND THEIR APPLICATIONS TO THE EQUATIONS OF ELASTICITY THEORY. I

Functionals with values in Non‐Archimedean field of Laurent series applied to the definition of generalized solution (in the form of soliton) of the Hopf equation. Calculation method for the profile of infinitely narrow soliton is proposed. Applying this method, calculations of profiles are reduced to the nonlinear system of algebraic equations in R n+1, n &gt; 1. It is shown that there is a possibility to find out some of the solutions of this system using the Newton iteration method. Example and numerical test are considered.

Spatial solitons in a medium composed of self-focusing and self-defocusing layers

We introduce a model combining Kerr nonlinearity with a periodically changing sign (“nonlinearity management”) and a Bragg grating (BG). The main result, obtained by means of systematic simulations, is presented in the form of a stability diagram on the parameter plane of the model; the diagram turns out to be a universal one, as it practically does not depend on the soliton's power. Moreover, simulations of the nonlinear Schrödinger (NLS) model subjected to the same “nonlinearity management” demonstrate that the same diagram determines the stability of the NLS solitons, unless they are very narrow. The stability region of very narrow NLS solitons is much smaller, and soliton splitting is readily observed in that case. The universal diagram shows that a minimum nonzero average value of Kerr coefficient is necessary for the existence of stable solitons. Interactions between identical solitons with an initial phase difference between them are simulated too in the BG model, resulting in generation of stable moving solitons. A strong spontaneous symmetry breaking is observed in the case when in-phase solitons pass through each other due to attraction between them.

Nonlinear atom optics and bright-gap-soliton generation in finite optical lattices

We theoretically investigate the transmission dynamics of coherent matter wave pulses across finite optical lattices in both the linear and the nonlinear regimes. The shape and the intensity of the transmitted pulse are found to strongly depend on the parameters of the incident pulse, in particular its velocity and density: a clear physical picture of the main features observed in the numerical simulations is given in terms of the atomic band dispersion in the periodic potential of the optical lattice. Signatures of nonlinear effects due to the atom-atom interaction are discussed in detail, such as atom-optical limiting and atom-optical bistability. For positive scattering lengths, matter waves propagating close to the top of the valence band are shown to be subject to modulational instability. A scheme for the experimental generation of narrow bright gap solitons from a wide Bose-Einstein condensate is proposed: the modulational instability is seeded starting from the strongly modulated density profile of a standing matter wave and the solitonic nature of the generated pulses is checked from their shape and their collisional properties.

One- and two-dimensional subwavelength solitons in saturable media

Very narrow spatial bright solitons in (1+1)D and (2+1)D versions of cubic-quintic and full saturable models are studied, starting from the full system of the Maxwell's equations, rather than from the paraxial (NLS) approximation. For the solitons with both TE and TM polarizations, it is shown that there always exists a finite minimum width, and they cease to exist at a critical value of the propagation constant, at which their width diverges. Full similarity of the results obtained for both nonlinearities suggests that the same general conclusions apply to narrow solitons in any non-Kerr model.

Subwavelength spatial solitons in optical media with non-Kerr nonlinearities

We consider very narrow time-independent spatially localized solutions (spatial solitons), whose width may be comparable to or smaller than the carrier wavelength, in two-and three-dimensional waveguides with the cubic-quintic or saturable nonlinearity. Equations describing the shape of the solitons are derived directly from the Maxwell equations, taking into account terms which are neglected in the usual paraxial (nonlinear Schrodinger) approximation, which is valid for solitons whose width is much larger than the wavelength. We consider both transverse electric and transverse magnetic solitons, and show that, in all cases, there is a finite minimum value of the soliton's width, which is attained at a certain value of the propagation constant (an internal parameter of the family of soliton solutions), and the width diverges at some larger value of the propagation constant, which is an existence border for the soliton family. The full similarity of the results obtained for both the cubic-quintic and saturable nonlinearities suggests that the same general conclusions apply to narrow bright solitons in any model featuring nonlinearity saturation in some form.

Quantization of weakly nonlinear lattices: envelope solitons.

A method of quantizing weakly nonlinear lattices is proposed. It is based on introducing "pseudofield" operators. In this formalism quantum envelope solitons together with phonons are regarded as elementary quasiparticles making up a boson gas. In the classical limit the excitations corresponding to frequencies above a linear cutoff frequency are reduced to conventional envelope solitons. The approach allows one to identify a quantum soliton that is localized in space and to understand the existence of a narrow soliton frequency band.