Stability of solitons in nonlinear composite media

An analysis is made of the dynamic stability of soliton solutions of the Hamilton equations describing plane waves in nonlinear elastic composite media in the presence and absence of anisotropy. In the anisotropiccase two two-parameter soliton families, fast and slow, are obtained in analytic form; in the absence of anisotropy there is a single three-parameter soliton family. It is shown that solitons from the slow family in an anisotropic composite and solitons in an isotropic composite are dynamically stable if their velocities lie in a certain range known as the range of stability. The analysis of stability is based on the spectral properties of the “linearized Hamiltonian” ?. It is shown that the operator ? is positively semidefinite on some linear subspace of the main solution space from which stability follows. Problems of instability of the fast soliton family in the anisotropic case and representatives of soliton families whose velocities lie outside the range of stability in the presence and absence of anisotropy are discussed.     

Asymptotic Stability in Geometric σ–models

Geometric –models have been defined as purely geometric theories of scalar fields in interaction with gravity. By construction, these theories possess soliton solutions with topologically nontrivial scalar sectors. We perform a detailed analysis of the stability of the effective scalar field theory far from the soliton core. It is shown that the requirement for the asymptotic stability is consistent with the existence of massive, static, spherically symmetric soliton solutions.     

Modulational instability and gap solitons in periodic ferromagnetic films

The modulational instability and gap solitons are theoretically studied in the ferromagnetic films under a periodic magnetic field. By multiple scale expansion, the envelope soliton solutions are obtained naturally. Due to the periodic modulation of dispersion, the solitons may be pushed into the gap region. For the easy-axis magnetic film, the red-shift of frequency leads to a modulational instability in the bottom of band and generates a bright gap soliton. For the easy-plane case, the blue-shift leads to an instability in the top of band and a dark gap soliton emerges. The weak damping produces an attenuation factor and a small oscillation.     

KdV方程的左行孤子解及其相互作用

本文指出,KdV方程允许存在左行孤子、静态孤子和左、右行孤子的相互作用解。左行孤子的振幅越大,运动得越慢。 关键词：     

Two-Dimensional Soliton Interaction for Multi-component Bose-Einstein Condensates

For two-component disk-shaped Bose-Einstein condensates with repulsive atom-atom interaction, the small amplitude, finite and long wavelength nonlinear waves can be described by a Kadomtsev-Petviashvili-I equation at the lowest order from the original coupled Gross-Pitaevskii equations. One- and two-soliton solutions of the Kadomtsev-Petviashvili-I equation are given, therefore, the wave functions of both atomic gases are obtained as well. The instability of a soliton under higher-order long wavelength disturbance has been investigated. It is found that the instability depends on the angle between two directions of both soliton and disturbance.     

Nonlinear propagation of vector extremely short pulses in a medium of symmetric and asymmetric molecules

The nonlinear propagation of extremely short electromagnetic pulses in a medium of symmetric and asymmetric molecules placed in static magnetic and electric fields is theoretically studied. Asymmetric molecules differ in that they have nonzero permanent dipole moments in stationary quantum states. A system of wave equations is derived for the ordinary and extraordinary components of pulses. It is shown that this system can be reduced in some cases to a system of coupled Ostrovsky equations and to the equation intagrable by the method for an inverse scattering transformation, including the vector version of the Ostrovsky–Vakhnenko equation. Different types of solutions of this system are considered. Only solutions representing the superposition of periodic solutions are single-valued, whereas soliton and breather solutions are multivalued.     

Solitonic mechanism of structural transformations in a nondegenerate chain of particles with anharmonic and competing interactions

A new type of dynamics of an infinite atomic chain of particles with anharmonic and competing interactions is investigated in the general case when its homogeneous equilibrium states have different energies. Cooperative transformations realized by topological and nontopological solitons are revealed. The soliton velocity spectrum is calculated in the framework of an approximate continuous second-order theory. Solitons with vanishing velocity are shown to be in good asymptotical correspondence with the exact static solutions of the Reichert-Schilling model.     

Conservation laws,modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber

Under investigation in this paper is a sextic nonlinear Schrödinger equation, which describes the pulses propagating along an optical fiber. Based on the symbolic computation, Lax pair and infinitely-many conservation laws are derived. Via the modiied Hirota method, bilinear forms and multi-soliton solutions are obtained. Propagation and interactions of the solitons are illustrated graphically: Initial position and velocity of the soliton are related to the coefficient of the sixth-order dispersion, while the amplitude of the soliton is not affected by it. Head-on, overtaking and oscillating interactions between the two solitons are displayed. Through the asymptotic analysis, interaction between the two solitons is proved to be elastic. Based on the linear stability analysis, the modulation instability condition for the soliton solutions is obtained.     

Stability of vortex solitons in the cubic-quintic model

We investigate one-parameter families of two-dimensional bright spinning solitons (ring vortices) in dispersive media combining cubic self-focusing and quintic self-defocusing nonlinearities. In direct simulations, the spinning solitons display a symmetry-breaking azimuthal instability, which leads to breakup of a soliton into a set of fragments, each being a stable nonspinning soliton. The fragments fly out tangentially to the circular crest of the original vortex ring. If the soliton’s energy is large enough, the instability develops so slowly that the spinning solitons may be regarded as virtually stable ones, in accord with earlier published results. Growth rates of perturbation eigenmodes with different azimuthal “quantum numbers” are calculated as a function of the soliton’s propagation constant κ from a numerical solution of the linearized equations. As a result, a narrow (in terms of κ) stability window is found for extremely broad solitons with values of the “spin” s=1 and 2. However, analytical consideration of a special perturbation mode in the form of a spontaneous shift of the soliton’s central “bubble” (core of the vortex embedded in a broad soliton) demonstrates that even extremely broad solitons are subject to an exponentially weak instability against this mode. In actual simulations, a manifestation of this instability is found in a three-dimensional soliton with s=1. In the case when the two-dimensional spinning solitons are subject to tangible azimuthal instability, the number of the nonspinning fragments into which the soliton splits is usually, but not always, equal to the azimuthal number of the instability eigenmode with the largest growth rate.     

Soliton of Bose–Einstein condensate in a trap with rapidly oscillating walls: II. Analysis of the soliton behavior upon a decrease in the wall oscillation frequency

This work is a continuation of our study [1], in which a two-scale analytical approach to the investigation of a soliton oscillon in a trap with rapidly oscillating walls has been developed. In terms of this approach, the solution to the equation of motion of the soliton center is sought as a series expansion in powers of a small parameter, which is a ratio of the intrinsic frequency of slow soliton oscillations to the frequency of fast trap wall oscillations. In [1], we have examined the case ε ? 1, in which, to describe the motion of the soliton, it is sufficient to restrict the consideration to the zero approximation of the sought solution. However, when the frequency of wall oscillations begins to decrease, while the parameter begins to increase, it is necessary to take into account corrections to the zero approximation. In this work, we have calculated corrections of the first and second orders in to this approximation. We have shown that, with an increase in, the role played by the corrections related to fast oscillations of the trap walls increases, which results in a complex shape of the envelope of oscillations of the soliton center. It follows from our calculations that, if the difference between the amplitudes of wall oscillations is not too large, the analytical solution of the equation of motion of the soliton center will coincide very well with the numerical solution. However, with an increase in this difference, as well as with a decrease in the wall oscillation frequency, the discrepancy between the numerical and analytical solutions generally begins to increase. Regimes of irregular oscillations of the soliton center arise. With a decrease in the frequency of wall oscillations, the instability boundary shows a tendency toward a smaller difference between the wall oscillation amplitudes. In general, this leads to enlargement of the range of irregular regimes. However, at the same time, stability windows can arise in this range in which the analytical and numerical solutions correlate rather well with each other. Our comparative analysis of the analytical and numerical solutions has allowed us not only to study their properties in detail, but also to draw conclusions on the limits of applicability of the analytical approach.     

线性和二次电光效应共同主导的非相干耦合空间孤子族

基于加偏压的单光子光折变晶体,理论推导了线性和二次电光效应共同主导下的亮孤子族和暗孤子族的解,数值研究了亮孤子族和暗孤子族的强度包络和稳定特性,讨论了线性和二次电光效应在孤子族形成中的不同作用.结果表明:线性和二次电光效应的相互作用能够增强亮孤子族的光折变非线性,而减弱暗孤子族的光折变非线性.此外,在传输过程中,亮孤子族的各个分量能够稳定传输;暗孤子族各个分量在较长传输距离时表现出不稳定性.     

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.     

Acoustic Solit on in One-Dimensional Ferromagnetic Systems

The local nonlinear excitation caused by the magnon-phonon interactions as an anisotropic source of nonlinearity is studied. The nonlinear equation for the Schrodinger probability amplitude of spin motion is given, and its soliton solutions are obtained in a weak coupling approximation. The existence conditions are discussed. It is shown that the soliton excitation energy can be less than a one-magnon state and a gap appears in the energy spectrum. The effect of the magnon-rnagnon interactions on the local acouitic soliton excitation is also discussed.     

Soliton stability in equations of the KdV type

A criterion for soliton stability is presented for equations of the KdV type with arbitrary nonlinearity. It is shown that for the power nonlinearity, soliton instability is closely connected with wave collapse.     

Ginzburg-Landau theory and solitary waves in shape-memory alloys

A one-dimensional dynamic Ginzburg-Landau theory of the martensitic phase transition in shape-memory alloys is established. The nonlinear equations of motion yield solitary wave solutions of kink and of soliton type. The kink solutions which cannot move without external force represent single domain walls either between austenite and martensite or between two martensite variants. The soliton solutions correspond to a matrix of austenite or of martensite containing a moving sheet of the other phase. The velocity of the solitons depends on their amplitude. In the static case they reduce to the critical nucleus. The energy of each type of solitary waves is calculated.     

Time-dependent solutions with localized energy of the Einstein system of equations and a nonlinear scalar field

A soliton-like time-dependent solution in the form of a running wave is derived of a self-consistent system of the gravitational field equations of Einstein and Born-Infeld type of equations of a nonlinear scalar field in a conformally flat metric. This solution is localized in space and possesses a localized energy. It is shown that both the gravitational field and the nonlinearity of the scalar field are essential to the presence of such a localized solution. In recent years various classical particle models have been widely discussed which are static or time-independent solutions of nonlinear equations with localization in space and which possess a finite field energy. In particular, soliton solutions [1], solutions in the form of eddies [2], and so on have been derived and investigated. All these solutions were treated in a flat space-time. It is of interest to derive the analogous particle-like solutions with the gravitational field taken into account; in particular it is of interest to investigate the roles of the gravitational field in connection with the formation of localized objects. These problems have been discussed in [3] in the static case. We will present below a soliton-like time-dependent solution in the form of a solitary running wave as an example of the inter-action of a Born-Infeld type of nonlinear scalar field and an Einstein gravitational field in a conformally flat metric.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 12–17, May, 1979.     

Optical soliton solutions of the generalized higher-order nonlinear Schrödinger equations and their applications

The propagation of the optical solitons is usually governed by the higher order nonlinear Schrödinger equations (NLSE). In optics, the NLSE modelizes light-wave propagation in an optical fiber. In this article, modified extended direct algebraic method with add of symbolic computation is employed to construct bright soliton, dark soliton, periodic solitary wave and elliptic function solutions of two higher order NLSEs such as the resonant NLSE and NLSE with the dual-power law nonlinearity. Realizing the properties of static and dynamic for these kinds of solutions are very important in various many aspects and have important applications. The obtaining results confirm that the current method is powerful and effectiveness which can be employed to other complex problems that arising in mathematical physics.     

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.     

Soliton Asymptotics of Solutions of the Sine-Gordon Equation

An asymptotic analysis of the Marchenko integral equation for the sine-Gordon equation is presented. The results are used for a construction of soliton asymptotics of decreasing and some non-decreasing solutions of the sine-Gordon equation. The soliton phases are shown to have an additional shift with respect to the reflectionless case caused by the non-zero reflection coefficient of the corresponding Dirac operator. Explicit formulas for the phases are also obtained. The results demonstrate an interesting phenomenon of splitting of non-decreasing solutions into an infinite series of asymptotic solitons.     

Painlevé analysis and exact solutions of the resonant Davey-Stewartson system

It is shown that the resonant Davey-Stewartson (RDS) system can pass the Painlev test. By truncating the Laurent series to a constant level term, a dependent variable transformation is naturally derived, which leads to the bilinear forms of the RDS system. From the bilinear equations, through making suitable assumptions, some new soliton solutions are obtained. Some representative profiles of the solitary waves are graphically displayed including the two-line soliton solution, “Y” soliton solution, “V” soliton solution, solitoff, etc. The solutions might be useful to describe the nonlinear phenomena in Madelung fluids, capillarity fluids, and so on.