Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

The appearance of gap solitons in a nonlinear Schrödinger lattice

We study the appearance of discrete gap solitons in a nonlinear Schrödinger model with a periodic on-site potential that possesses a gap evacuated of plane-wave solutions in the linear limit. For finite lattices supporting an anti-phase (q=π/2) gap edge phonon as an anharmonic standing wave in the nonlinear regime, gap solitons are numerically found to emerge via pitchfork bifurcations from the gap edge. Analytically, modulational instabilities between pairs of bifurcation points on this “nonlinear gap boundary” are found in terms of critical gap widths, turning to zero in the infinite-size limit, which are associated with the birth of the localized soliton as well as discrete multisolitons in the gap. Such tunable instabilities can be of relevance in exciting soliton states in modulated arrays of nonlinear optical waveguides or Bose-Einstein condensates in periodic potentials. For lattices whose gap edge phonon only asymptotically approaches the anti-phase solution, the nonlinear gap boundary splits in a bifurcation scenario leading to the birth of the discrete gap soliton as a continuable orbit to the gap edge in the linear limit. The instability-induced dynamics of the localized soliton in the gap regime is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsically localized modes (discrete breathers) from the extended out-gap soliton reveals a phase transition of the solution.     

Exact Solutions of Nonlinear Dynamics Equation in a New Double-Chain Model of DNA

The exact solutions of the general nonlinear dynamic system in a new double-chain model of DNA are studiedkink shape excitations can be found in both the Conte's truncation expansion and the Pickering's truncation expansion.Three types of new localized excitations, the asymmetric kink-kink excitations, the soliton-kink excitation, and thekink-soliton excitations, are found by using the Pickering's nonstandard truncation expansion.     

计及次近邻非谐相互作用下原子链中的非线性元激发

利用多重尺度法，研究了谐振、非谐的近邻和次近邻相互作用下单原子链的晶格振动行为， 发现非传播孤子在单原子链Brillouin区的任何位置均有可能存在.与仅考虑最近邻相互作用 下的单原子链的孤子群速相比，此时链中孤子的群速在Brillouin中央增大且衰减得更快； 在Brillouin区会出现另一个群速为零的位置.同时，孤子的幅度也相应增大且最大幅度孤子 的位置趋向于Brillouin中央. 关键词： 单原子链 非谐次近邻相互作用 非传播的孤子     

Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity

We study the discrete nonlinear Schrödinger lattice model with the onsite nonlinearity of the general form, |u|^2σu. We systematically verify the conditions for the existence and stability of discrete solitons in the one-dimensional version of the model predicted by means of the variational approximation (VA), and demonstrate the following: monostability of fundamental solitons (FSs) in the case of the weak nonlinearity, 2σ+1<3.68; bistability, in a finite range of values of the soliton’s power, for 3.68<2σ+1<5; and the presence of a threshold (minimum norm of the FS), for 2σ+1≥5. We also perform systematic numerical simulations to study higher-order solitons in the same general model, i.e., bound states of the FSs. While all in-phase bound states are unstable, stability regions are identified for antisymmetric double solitons and their triple counterparts. These numerical findings are supplemented by an analytical treatment of the stability problem, which allows quantitively accurate predictions for the stability features of such multipulses. When these waveforms are found to be unstable, we show, by means of direct simulations, that they self-trap into a persistent lattice breather, or relax into a stable FS, or sometimes decay completely.     

Double Wronskian Solutions for a Generalized Nonautonomous Nonlinear Equation in a Nonlinear Inhomogeneous Fiber

A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed and verified with the Wronskian technique. Collisions among the three solitons are discussed and illustrated, and effects of the coefficients σ₁(x, t), σ₂(x, t), σ₃(x, t) and v(x, t) on the collisions are graphically analyzed, where σ₁(x, t), σ₂(x, t), σ₃(x, t) and v(x, t) are the first-, second-, third-order dispersion parameters and an inhomogeneous parameter related to the phase modulation and gain(loss), respectively. The head-on collisions among the three solitons are observed, where the collisions are elastc. When σ₁(x, t) is chosen as the function of x, amplitudes of the solitons do not alter, but the speed of one of the solitons changes. σ₂(x, t) is found to affect the amplitudes and speeds of the two of the solitons. It reveals that the collision features of the solitons alter with σ₃(x, t)=-1.8x. Additionally, traveling directions of the three solitons are observed to be parallel when we change the value of v(x, t).     

Nonlinear edge modes in a honeycomb electrical lattice near the Dirac points

We examine - both experimentally and numerically - a two-dimensional nonlinear driven electrical lattice with honeycomb structure. Drives are considered over a range of frequencies both outside (below and above) and inside the band of linear modes. We identify a number of discrete breathers both existing in the bulk and also (predominantly) ones arising at the domain boundaries, localized either along the arm-chair or along the zig-zag edges. The types of edge-localized breathers observed and computed emerge in distinct frequency bands near the Dirac-point frequency of the dispersion surface while driving the lattice subharmonically (in a spatially homogeneous manner). These observations/computations can represent a starting point towards the exploration of the interplay of nonlinearity and topology in an experimentally tractable system such as the honeycomb electrical lattice.     

Exact solution of the one-dimensional Klein-Gordon equation with scalar and vector linear potentials in the presence of a minimal length

Using the momentum space representation, we solve the Klein-Gordon equation in one spatial dimension for the case of mixed scalar and vector linear potentials in the context of deformed quantum mechanics characterized by a finite minimal uncertainty in position. The expressions of bound state energies and the associated wave functions are exactly obtained.     

Nonlinear Longitudinal Strain Waves in a Solid Exposed to Pulsed Laser Radiation

The propagation of longitudinal strain waves in a solid with quadratic nonlinearity of elastic continuum was studied in the context of a model that takes into account the joint dynamics of elastic displacements in the medium and the concentration of the laser-induced point defects. The input equations of the problem are reformulated in terms of only the total displacements of the medium points. In this case, the presence of structural defects manifests itself in the emergence of a delayed response of the system to the propagation of the strain-related perturbations, which is characteristic of media with relaxation or memory. The model equations describing the nonlinear displacement wave were derived with allowance made for the values of the relaxation parameter. The influence of the generation, relaxation, and the strain-induced drift of defects and the flexoelectricity on the propagation of this wave was analyzed. It is shown that, for short relaxation times of defects, the strain can propagate in the form of both shock fronts and solitary waves (solitons). Exact solutions depending on the type of relation between the coefficients in the equation and describing both the shock-wave structures and the evolution of solitary waves are presented. In the case of longer relaxation times, shock waves do not form and the strain wave propagates only in the form of solitary waves or a train of solitons. The contributions of the finiteness of the defect-recombination rate and the flexoelectricity to linear elastic moduli and spatial dispersion are determined.     

On a Nonlinear Model in Adiabatic Evolutions

In this paper, we study a kind of nonlinear model of adiabatic evolution in quantum search problem. As will be seen here, for this problem, there always exists a possibility that this nonlinear model can successfully solve the problem, while the linear model can not. Also in the same setting, when the overlap between the initial state and the final stare is sufficiently large, a simple linear adiabatic evolution can achieve O(1) time efficiency, but infinite time complexity for the nonlinear model of adiabatic evolution is needed. This tells us, it is not always a wise choice to use nonlinear interpolations in adiabatic algorithms. Sometimes, simple linear adiabatic evolutions may be sufficient for using.     

Chaotic Solutions of a Typical Nonlinear Oscillator in a Double Potential Trap

We have obtained a general unstable chaotic solution of a typical nonlinear oscillator in a double potential trap with weak periodic perturbations by using the direct perturbation method. Theoretical analysis reveals that the stable periodic orbits are embedded in the Melnikov chaotic attractors. The corresponding chaotic region and orbits in parameter space are described by numerical simulations.     

New Exact Jacobi Elliptic Function Solutions of Three—Dimensional Nonlinear Helmholtz Equation in a Nonlinear Kerr—Type Medium

In this letter the three-dimensional nonlinear Helmholtz equation is investigated.which describes electromagnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic function solutions are obtained,by using our extended Jacobian elliptic function expansion method.When the modulus m-→1 or 0,the corresponding solitary waves including bright solitons,dark solitons and new line solitons and singly periodic solutions can be also found.     

New Exact Jacobi Elliptic Function Solutions of Three-Dimensional Nonlinear Helmholtz Equation in a Nonlinear Kerr-Type Medium

In this letter the three-dimensional nonlinear Helmholtz equation is investigated, which describes electro-magnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic functionsolutions are obtained, by using our extended Jacobian elliptic function expansion method. When the modulus m → 1or0, the corresponding solitary waves including bright solitons, dark solitons and new line solitons and singly periodicsolutions can be also found.     

Symmetry Reductions,Group‐Invariant Solutions,and Conservation Laws of a (2+1)‐Dimensional Nonlinear Schrödinger Equation in a Heisenberg Ferromagnetic Spin Chain

Nonlinear spin excitations in ferromagnetic spin chains are studied for spintronic and magnetic devices including magnetic‐field sensors and for high‐density data storage. Here, (2+1)‐dimensional nonlinear Schrödinger equation is investigated, which describes the nonlinear spin dynamics for a Heisenberg ferromagnetic spin chain. Lie point symmetry generators and Lie symmetry groups of that equation are derived. Lie symmetry groups are related to the time, space, scale, rotation transformations, and Galilean boosts of that equation. Certain solutions, which are associated with the known solutions, are constructed. Based on the Lie symmetry generators, the reduced systems of such an equation are obtained. Based on the polynomial expansion and through one of the reduced systems, group‐invariant solutions are constructed. Soliton‐type group‐invariant solutions are graphically investigated and effects of the magnetic coupling coefficients, that is, α₁, α₂, α₃, and α₄, on the soliton's amplitude, width, and velocity are discussed. It is seen that α₁, α₂, α₃, and α₄ have no influence on the soliton's amplitude, but can affect the soliton's velocity and width. Lax pair and conservation laws of such an equation are derived.     

Exact Solutions of a Generalized Multi-Fractional Nonlinear Diffusion Equation in Radical Symmetry

This paper is devoted to investigating exact solutions of a generalized fractional nonlinear anomalous diffusion equation in radical symmetry. The presence of external force and absorption is also considered. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solutions, and the solutions found here can have a compact behavior or a long tailed behavior.     

Electric Black Holes in a Model of Nonlinear Electrodynamics

The electric counterpart of the magnetic black hole solution found in nonlinear electrodynamics (NED) is presented. The electric field emerges regular and confined whereas the spacetime which satisfies all the energy conditions is singular. Our result is in conformation with a theorem proved before about the existence of regular electric black holes. The thermal properties of the black hole including the first law, Smarr's formula, and the thermal stability are investigated. This provides a chance to compare the electric and magnetic types of black holes in a particular model of NED.     

双模环型激光增益噪声模型的非线性效应

讨论了一个双模环型激光增益噪声模型，其中考虑了完全饱和效应且乘法噪声由增益系数涨落引起。在共振及两模具有相同泵参数时，获得了光强联合定态分布的精确解析表达式。通过与现有的双模激光摸型（其中乘法噪声由损失系数涨落引起）的比较，发现乘法噪声系数的非线性效应减弱了乘法噪声给激光光强统计性质带来的反常特性，并且这种减弱随着乘法噪声增强或损失系数减小而愈加明显。     

Liley模型的模拟EEG信号的非线性预测和分析

分析Liley模型的模拟脑电(Electroencephalogram,EEG)信号的非线性预测和径向基函数(Radial Basis Functions,RBF)神经网络预测,利用相图分析和非线性正交预测(Nonlinear Cross-Prediction,NLCP)方法研究模拟EEG信号.结果发现:①RBF神经网络预测的效果要好于非线性预测;②NLCP方法对含有强周期分量的高维系统具有较好的适用性;③支持了EEG中存在混沌运动的观点.