Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

A universal separatrix map for weak interactions of solitary waves in generalized nonlinear Schrödinger equations

It is known that weak interactions of two solitary waves in generalized nonlinear Schrödinger (NLS) equations exhibit fractal dependence on initial conditions, and the dynamics of these interactions is governed by a universal two-degree-of-freedom ODE system [Y. Zhu J. Yang, Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations, Phys. Rev. E 75 (2007) 036605]. In this paper, this ODE system is analyzed comprehensively. Using asymptotic methods along separatrix orbits, a simple second-order map is derived. This map does not have any free parameters after variable rescalings, and thus is universal for all weak interactions of solitary waves in generalized NLS equations. Comparison between this map’s predictions and direct simulations of the ODE system shows that the map can capture the fractal-scattering phenomenon of the ODE system very well both qualitatively and quantitatively.     

一类广义N维非线性Schr?dinger方程的孤子解及其性质研究

研究一类N维广义非线性Schr?dinger方程的孤子解及其性质，研究非线性参数α变化（α→0及α→∞）时孤子性态的变化规律，同时研究该问题的数值解法，得到了该方程的P-R差分格式的收敛性和稳定性条件． 关键词：     

On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation

In this Letter, the generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. A generalized Darboux transformation (DT) of the GNLS equation is constructed with the help of the gauge transformation for an Ablowitz–Kaup–Newell–Segur (AKNS) type GNLS spectral problem, from which a unified formula of Nth-order rogue wave solution to the GNLS equation is given. In particular, the first and second-order rogue wave solutions to the GNLS equation are explicitly illustrated through some figures.     

The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations

With the aid of the ordinary differential equation (ODE) involving an arbitrary positive power of dependent variable proposed by Li and Wang and an indirect F-function method very close to the F-expansion method, we solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p) and the generalized nonlinear Schrödinger equation with nonlinear dispersion GNLS(l,n,p,q). Taking advantage of the new subsidiary ODE, this F-function method is used to map the solutions of C(l,n,p) and GNLS(l,n,p,q) equations to those of that nonlinear ODE. As result, we can successfully obtain in a unified way, many exact solutions.     

An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

An integrable discrete system obtained by the algebraization of the difference operator is studied. The system is named discrete generalized nonlinear Schrödinger (GNLS) equation, which can be reduced to classical discrete nonlinear Schrödinger (NLS) equation. Furthermore, all of the linear reductions for the discrete GNLS equation are given through the theory of circulant matrices and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones.     

混沌背景下信号的盲分离

混沌信号与确定性小信号叠加生成的混合信号是一更高维的混沌信号,因而不能用一般的混沌信号噪声抑制的方法进行分离.提出了一种这类信号盲分离的方法.在重构未知的混沌信号的动力方程时,充分利用混沌吸引子的几何特性,并且限定动力映射为原混沌吸引子所在流形的内部映射,从而保证了重构的动力系统方程对应于原混沌信号,而不是同样具有混沌特性的混合信号.然后利用重构的动力方程,借用混沌信号中的噪声抑制思想,估计出原混沌信号对应的轨道,实现信号分离.通过对Lorenz系统中谐波信号、Henon映象中自回归过程,以及脑电信号中谐波信号进行提取的数值实验,验证了信号盲分离方法的有效性和可行性. 关键词： 混沌 非线性 信号处理 盲分离     

From the Torsion Tensor for Spinors to the Weak Forces for Leptons

We consider a geometric approach to field theory in which torsion is present beside gravity and also electrodynamics for the matter field equations, and we develop the consequences of the torsion-spin coupling for the spinor fields; we show that these interactions have the structure of the weak interactions acting among leptons: we discuss the implications for the standard model of fundamental interactions of elementary fields in the perspective of the foundations of unification in theoretical physics.     

Energy-Momentum Tensor and Equations of Motion of Glashow-Salam-Weinberg-Theory in Curved Space-Time

The equations of motion of the unified gauge theory of weak and electromagnetic interactions, when minimally coupled to the gravitational field, are given.     

DYNAMICAL SYMMETRY BREAKINGS IN QUANTUM CHROMODYNAMICS

Dynamical symmetry breakings in quantum chromodynamics （QCD） are investigated by means of the renormalization-group equations and the Dyson equations. It is shown that: （1） The electromagnetic （and weak） interactions can not be neglected in studying those nonperturbative problems as dynamical symmetry breakings for type-A models in QCD. （2） When the electromagnetic interactions between stratons are taken into account in the weak coupling approximation, dynamical symmetry breakings in QCD occur for many realistic models （type-A models） with QCD asymptotically free. This conclusion may hold beyond the weak coupling approximation of the electromagnetic interactions if there can still be a self-consistent solution of Johnson-Baker-Willey in quantum electrodynamics or β_e has an UV-stable fixed point. e_∞≠0 （a simple zero of β_e） and e_∞is not too large.     

Wave turbulent statistics in non-weak wave turbulence

In wave turbulence, which is made by nonlinear interactions among waves, it has been believed that statistical properties are well described by the weak turbulence theory, where separation of linear and nonlinear time scales derived from weak nonlinearity is assumed. However, the separation of the time scales is often violated. To get rid of this inconsistency, closed equations are derived in wave turbulence without assuming the weak nonlinearity according to Direct-Interaction Approximation (DIA), which has been successful in Navier-Stokes turbulence. The DIA equations is a natural extension of the conventional kinetic equation to not-necessarily-weak wave turbulence.     

Perturbation theory for double-time temperature-dependent Green functions

A perturbation theory is developed for normal many-particle systems by expanding the inverse of the advanced and retarded Green functions. Both systems with direct interactions and systems with interactions through a quantum field are considered. For the former, the infinite set of coupled equations is written down and the various methods of decoupling discussed. Systems with weak, hard sphere type, or coulomb interactions are considered and the connections with other methods discussed. For systems with quantum field interactions, the chain of equations involves both fermion and boson Green functions and the first approximation can be much better than in earlier cases. This latter approach seems to be more convenient when collective oscillations are important.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Chaos in discrete fractional difference equations

Recently, the discrete fractional calculus (DFC) is receiving attention due to its potential applications in the mathematical modelling of real-world phenomena with memory effects. In the present paper, the chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied numerically. We analyse the chaotic behaviour of these fractional difference equations and compare them with their integer counterparts. It is observed that fractional difference equations for the Gauss and tent maps are more stable compared to their integer-order version.     

Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales

We study the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with frequency nominally within the Propagation Zone (i.e., propagation occurs in the absence of the damping). The period of the structure consists of a linear stiffness and a weak linear/nonlinear damping. Adapting the transfer matrix method and using harmonic balance for the nonlinear terms, a four-dimensional linear/nonlinear map governing the dynamics is obtained. We analyze this map by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The approximations are validated by comparing with other analytical solutions for the linear case and full numerics for the nonlinear case. Good agreement is obtained. The method of analysis presented here can be extended to more complex structures.     

Bewegungsgesetze nicht abgeschlossener Quantensysteme

Starting from so-called dynamical mappings which map the set of density operators onto itself, a possibility is discussed for deriving stochastic equations of motion for quantum systems. Two simple applications are given, the relaxation of a nuclear spin, and a quantummechanical equation of motion for a damped harmonic oscillator. The method can also be used in Quantum Electrodynamics. This theory is usually considered as closed in the following sense: Any Feynman graph the external lines of which describe photons and electrons only contains also in the virtual states these two kinds of particles only. It seems to be more realistic to replace this postulate of nonexistence of other interactions by the weaker assumption that such interactions exist but are not completely known. In this way a modification of Quantum Electrodynamics is achieved for high energy processes.     

Hadronic Parity Violation in Few-Nucleon Systems

The weak interaction between quarks induces a parity-violating component in the interactions between nucleons, which is typically suppressed by a factor of \({\approx 10^{-7}}\) compared to the dominant parity-conserving part. Because of the short range of the weak interactions, it provides a unique probe of the strong dynamics that confine quarks into nucleons. An experimental program to map out this weak component of the nuclear force is underway at a number of facilities, including the Spallation Neutron Source at Oak Ridge National Laboratory. The corresponding observables are related to few-nucleon processes at very low energies, at which pionless effective field theory provides a reliable and model-independent theoretical approach to hadronic parity violation. Results in two- and three-nucleon systems, the role of parity-violating three-nucleon forces, and possible extensions to other few-nucleon systems are discussed.     

Freely decaying weak turbulence for sea surface gravity waves

We study the long-time evolution of deep-water ocean surface waves in order to better understand the behavior of the nonlinear interaction processes that need to be accurately predicted in numerical models of wind-generated ocean surface waves. Of particular interest are those nonlinear interactions which are predicted by weak turbulence theory to result in a wave energy spectrum of the form of [k](-2.5). We numerically implement the primitive Euler equations for surface waves and demonstrate agreement between weak turbulence theory and the numerical results.     

Generalized Darboux maps and massless spin-s fields

We consider conformally invariant massless spin-s field equations on a spherically symmetrical space-time. Precisely when these equations are consistent appropriately defined field components are shown to satisfy wave equations related by a generalization of the classical Darboux map.