Envelope Solitons Perturbed by Stochastic Noise

We perform langevin dynamics simulation for envelope solitons in anFPU-β lattice, with the nearest-neighbor interaction and quartic anharmonicity. We get the motion equations of our discrete system by adding noise and damping to the set of deterministic motion equations. We define ``half-time' as the time when the amplitude of the envelope soliton decreases by half due to damping. And then the mass, center and half-time of the perturbed envelope soliton are numerically simulated, beginning with the discrete envelope soliton at rest. Results show successfully how noise affects behavior of the envelope soliton.     

Solitary waves in the Madelung's fluid: Connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation

An investigation to deepen the connection between the family of nonlinear Schr?dinger equations and the one of Korteweg-de Vries equations is carried out within the context of the Madelung's fluid picture. In particular, under suitable hypothesis for the current velocity, it is proven that the cubic nonlinear Schr?dinger equation, whose solution is a complex wave function, can be put in correspondence with the standard Korteweg-de Vries equation, is such a way that the soliton solutions of the latter are the squared modulus of the envelope soliton solution of the former. Under suitable physical hypothesis for the current velocity, this correspondence allows us to find envelope soliton solutions of the cubic nonlinear Schr?dinger equation, starting from the soliton solutions of the associated Korteweg-de Vries equation. In particular, in the case of constant current velocities, the solitary waves have the amplitude independent of the envelope velocity (which coincides with the constant current velocity). They are bright or dark envelope solitons and have a phase linearly depending both on space and on time coordinates. In the case of an arbitrarily large stationary-profile perturbation of the current velocity, envelope solitons are grey or dark and they relate the velocity u₀ with the amplitude; in fact, they exist for a limited range of velocities and have a phase nonlinearly depending on the combined variable x-u_{0 s} (s being a time-like variable). This novel method in solving the nonlinear Schr?dinger equation starting from the Korteweg-de Vries equation give new insights and represents an alternative key of reading of the dark/grey envelope solitons based on the fluid language. Moreover, a comparison between the solutions found in the present paper and the ones already known in literature is also presented. Received 20 February 2002 and Received in final form 22 April 2002 Published online 6 June 2002     

Long-range effects on superdiffusive algebraic solitons in anharmonic chains

Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam (FPU)-like lattices were recently generalized to the case of dispersive long-range interactions (LRI) of the Kac-Baker form. The variance of the soliton position shows a stronger than linear time-dependence (superdiffusion) as found earlier for lattice solitons on FPU chains with nearest-neighbour interactions (NNI). Since the superdiffusion seems to be generic for nontopological solitons, we want to illuminate the role of the soliton shape on the superdiffusive mechanism. Therefore, we concentrate on an FPU-like lattice with a certain class of power-law long-range interactions where the solitons have algebraic tails instead of the exponential tails in the case of FPU-type interactions (with or without Kac-Baker LRI).Despite of structurally similar Langevin equations which hold for the soliton position and width of the two soliton types, the algebraic solitons reach the superdiffusive long-time limit with a characteristic t^3/2 time-dependence much faster than exponential solitons. The soliton shape determines the diffusion constant in the long-time limit that is approximately a factor of π smaller for algebraic solitons. Our results appear to be generic for nonlinear excitaitons in FPU-chains, because the same superdiffusive time-dependence was also observed in simulations with discrete breathers.     

Dynamics of generalized sine-Gordon soliton in inhomogeneous media

In this paper we introduce a few novel generalized sine-Gordon equations and study the dynamics of its solitons in inhomogeneous media. We consider length, mass, gravitational acceleration and spring stiffness of a coupled pendulums chain as a function of position x. Then in the continuum limit we derive semi-analytical and numerical soliton solutions of the modified sine-Gordon equation in the inhomogeneous media. The obtained results confirm that the behavior of solitons in these media is similar to that of a classical point particle moved in an external potential.     

有耗孤子的最小作用量原理及其在二维光子带隙结构中的应用

采用变分方法对于二维矩形光子带隙结构中X点孤子的耗散以及相干作用进行了动力学分析.通过对由作用量原理得到的孤子参量联立方程组的分析可知，除了耗散作用导致孤子的幅度衰减，并引起孤子横向的展宽以外，同向孤子的相互作用负势函数为耗散作用所减弱，从而使得孤子在x方向的束缚态被削弱.孤子中心间距随x的增加呈现出增幅变化且波动周期增大，当纵向距离增大到一定值后，孤子中心间距将一直增大下去.     

Dispersionless envelope lattice solitons on a D-dimensional nonlinear lattice

We show by using the real exponential approach that the d-dimensional discrete nonlinear Schrödinger equation has more general dispersionless envelope lattice soliton solutions than the known bright soliton and kink solutions. Depending on the values of the parameters, the new solutions can describe both bright and dark lattice solitons. Especially, we find novel “W”-like envelope lattice solitons.     

The mutual interaction of molecular rotation and translation

The dynamics of molecular rototranslation are treated with an equation of motion with a non-Markovian, stochastic force/torque. It is shown that this Mori/Kubo/Zwanzig representation is equivalent to a multidimensional Markov equation which may be identified with analytical models of the molecular motion. Langevin and Fokker-Planck equations for two such models are derived from the general equations of motion. The analytical results are compared with a computer simulation of the velocity/angular velocity mixed autocorrelation function, C _vω(t) = <v(0) . ω(t)> for a triatomic of C _2v symmetry.     

Brownian motion in inhomogeneous media and with interacting particles

Brownian motion in media with a space-dependent temperature and density is described by Langevin equations in phase space. Elimination of the velocity shows that diffusion inx-space cannot in general be characterized by a single diffusion parameter, nor can space-dependence always be accounted for by mere assignment of some sense of stochastic integration to the Langevin equation which has been reduced as in the homogeneous case. Steady solutions of the resulting equations agree with thermodynamics. Interactions between Brownian particles (giving rise to nonlinear Fokker-Planck equations) lead to a generalization of Einstein's relation.Work supported by the Swiss National Science Foundation     

Modulated spin waves and robust quasi-solitons in classical Heisenberg rings

We investigate the dynamical behavior of finite rings of classical spin vectors interacting via nearest-neighbor isotropic exchange in an external magnetic field. Our approach is to utilize the solutions of a continuum version of the discrete spin equations of motion (EOM) which we derive by assuming continuous modulations of spin wave solutions of the EOM for discrete spins. This continuum EOM reduces to the Landau-Lifshitz equation in a particular limiting regime. The usefulness of the continuum EOM is demonstrated by the fact that the time-evolved numerical solutions of the discrete spin EOM closely track the corresponding time-evolved solutions of the continuum equation. It is of special interest that our continuum EOM possesses soliton solutions, and we find that these characteristics are also exhibited by the corresponding solutions of the discrete EOM. The robustness of solitons is demonstrated by considering cases where initial states are truncated versions of soliton states and by numerical simulations of the discrete EOM equations when the spins are coupled to a heat bath at finite temperatures.     

The quantum Fokker-Planck equation for certain boson systems

Quantum relations between a class of boson Langevin equations and the associated Fokker-Planck equations are derived. The Fokker-Planck equations for the Wigner distribution Φ_sym related with symmetric ordering of the boson operators, the distribution Φ_A related with antinormal ordering, and the distribution Φ_N related with normal ordering (P-representation) are given.     

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.     

Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation

We investigated the soliton solution for N$N$ coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management.     

Soliton relaxation in antiferromagnets

We study the relaxation two-parameter one-dimensional solitons in antiferromagnets using the phenomenological theory. Allowing for relaxation terms of a relativistic and exchange nature, we set up a system of evolution equations for the constants of the motion of a soliton and calculate the corresponding integral curves, which describe the variation of the soliton parameters in the relaxation process. Zh. éksp. Teor. Fiz. 111, 1633–1650 (May 1997)     

Transverse envelope solutions in an atomic chain

We derive a set of discrete nonlinear equations for transverse waves in an atomic chain under longitudinal stress. Circularly and linearly polarized envelope solitons are obtained, numerical simulations are used to check the stability and the collisions.     

Kinetic and hydrodynamic models of chemotactic aggregation

We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean-field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean-field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed.     

Fokker-Planck approximation for N-dimensional nonmarkovian langevin equations

The Fokker-Planck approximation for n-dimensional nonmarkovian Langevin equations is discussed through an expansion in powers of the correlation time of the noise. Exact cases are considered and an application to brownian motion is presented.     

LiNbO3晶体中屏蔽光伏孤子自偏转的 时空演化与可控因素

基于带输运模型理论建立了 LiNbO₃ 晶体屏蔽光伏孤子的时空演化动力学方程, 用有限差分方法求解发现, LiNbO₃ 晶体中明、暗屏蔽光伏孤子存在大的自偏转, 并且光孤子形状变得具有不对称性, 偏转方向的曲线斜率绝对值变大, 偏转反方向的曲线斜率绝对值变小. 分析研究表明影响其自偏转度和形变的因素包括受主浓度 N_A, 暗辐射强度 I_d 和外加电场 E₀ . 其他条件不变的情况下N_A 越大, 明孤子的自偏转度与形变越小, 暗孤子的自偏转度与形变反而越大; 对于 I_d , 它对明暗孤子的影响是相同的, I_d 越小, 晶体里诱导出的空间电荷场越容易达到饱和, 当信号光中心光强与暗辐射强度之比为 10^-1时无饱和现象产生; 随着 E₀ 数值的增大, 明孤子的自偏转度和形变减小, 而暗孤子的自偏转度和形变反而增大.     

Exact stationary probability distribution for a simple model of a nonequilibrium phase transition

We derive the exact stationary probability distribution for the coupled system of Langevin equationsd _t u=u–u s,d _t s=–s+d ²+F(t).