Two model equations with a second degree logarithmic nonlinearity and their Gaussian solutions

In the paper, we try to study the mechanism of the existence of Gaussian waves in high degree logarithmic nonlinear wave motions. We first construct two model equations which include the high order dispersion and a second degree logarithmic nonlinearity. And then we prove that the Gaussian waves can exist for high degree logarithmic nonlinear wave equations if the balance between the dispersion and logarithmic nonlinearity is kept. Our mathematical tool is the logarithmic trial equation method.     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

架设阶段悬索桥静力问题的随机有限元分析

架设阶段的悬索桥不仅具有显著的几何非线性行为，而且存在材料特性的变异。本文建立了以增量形式表达的非线性摄动法随机有限元列式，并编制了悬索桥静力分析程序ＳＮＡＰ。针对一座国内在建的悬索桥，计算了该桥在某一施工阶段考虑缆索弹性模量随机性时主梁跨中点位移的变化。结果表明，材料随机性的影响十分明显。     

Global existence of solutions of the critical semilinear wave equations with variable coefficients outside obstacles

In this paper, we consider the exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove the global existence of smooth solutions. As in the constant coefficients case, we show that the energy cannot concentrate at any point (t, x) ∈ (0, ∞) ×Ω. For that purpose, following Ibrahim and Majdoub's paper in 2003, we use a geometric multiplier similar to the well-known Morawetz multiplier used in the constant coefficients case. We then use the compari...     

Interface kink solitons in defocusing saturable nonlinear media

We report on the dynamics of semi-localized nonlinear optical modes supported by an interface separating a uniform defocusing saturable medium and an imprinted semi-infinite photonic lattice. Out-of-phase and in-phase kink solitons composed by dark-soliton-like pedestals and oscillatory tails are found. Two branches of out-of-phase kink solitons exist in shallow lattices. Saturable nonlinearity enhances the pedestal height and renormalized energy flow of kink solitons evidently. While in-phase kink solitons are always unstable, out-of-phase kink solitons will be completely stable provided that lattice depth exceeds a critical value. Furthermore, stable kink solitons in the higher band gaps are also possible. Our results may give a helpful hint for understanding the dynamics of kink solitons with high pedestals in other fields.     

The fluctuating phase error analysis in the digital grating phase-shifting profilometry

The nonlinear response of the experimental system and the saturation of fringe patterns can induce the fluctuating phase error in the projection grating phase-shifting profilometry. Two major factors of the fluctuating phase error are discussed by simulation. The fluctuating phase error caused by the nonlinear response of the system is four times the frequency of the fringe pattern when the conventional four-frame phase extracting algorithm is used. However, such error can be decreased by five-frame algorithm. On the other hand, the fluctuating phase error caused by the fringe saturation is five times the frequency of the fringe pattern by using conventional five-frame phase extracting algorithm. A novel phase recovering algorithm is used to decrease the phase error caused by the saturation. Furthermore, the applicability range of the proposed phase recovering algorithm is analyzed by simulation and experiments with different saturation degree of the fringe pattern and nonlinearity of the measurement system.     

Efficient preparation of a four-photon cluster state with homodyne measurement via cross-Kerr nonlinearity

A scheme is proposed for generating a multiphoton entangled cluster state among four modes. The scheme only uses Kerr medium, beam splitter and homodyne measurements on coherent light fields, which can be efficiently made in quantum optical laboratories. The photon in the signal mode is prepared in a superposition state of the vacuum state and one-photon state while the probe beam is initially set in a coherent state superposition. The strong probe mode interacts successively with multiple signal-mode photo...     

Field history dependence of nonlinear dielectric properties of Ba0.6Sr0.4TiO3 ceramics under bias electric field: Polarization behavior of polar nano-regions

Nonlinear dielectric properties of Ba_0.6Sr_0.4TiO₃ ceramics prepared by citrate method were investigated under bias electric field with respect to field history. X-ray diffraction analysis and temperature dependence of the dielectric constant (ε_r) confirmed a macroscopically paraelectric state for the specimen at room temperature. A slim polarization versus electric field (P-E) hysteresis loop of the specimen at room temperature indicated the existence of polar nano-regions (PNRs) superimposed on the paraelectric background. The nonlinear dielectric properties in continuous cycles of bias field sweep displayed a strong sensitivity to the field history. This phenomenon was qualitatively explained in terms of an irreversible polarization evolution of the PNRs under the bias fields. A considerable decline of the tunability with the cycle number suggests an appreciable contribution of the PNRs to the dielectric nonlinearity. The polarization and size of the PNRs were determined by fitting the dielectric constants to a multipolarization mechanism model.     

Second-harmonic pressure generation of a non-diffracting acoustical high-order Bessel vortex beam of fractional type α

Background and motivation

Generalized Bessel vortex beams are regaining interest from the standpoint of acoustic scattering and radiation force theories for applications in particle rotation, mixing and manipulation. Other possible applications may include medical and nondestructive imaging. To manipulate heavy particles in a host medium, a minimum threshold of the incident sound field intensity is required at relatively high wave amplitudes such that nonlinear wave propagation occurs and the generation of harmonics may be detected. Thus, predictions of the harmonics generation become crucial from the standpoint of experimental design, and the present analysis should assist in the development of more complete models related to the (nonlinear) scattering and radiation forces under such circumstances. The purpose of this research is to construct a theoretical model for the second-harmonic pressure generation associated with a category of non-diffracting Bessel vortex beams known as high-order Bessel vortex beams of fractional typeα (HOBVBs-Fα).

Method

The weakly nonlinear wave propagation of a HOBVB-Fα is investigated based on Lighthill’s formalism. Analytical solutions up to the second-order level of approximation are derived and discussed. Closed-form solutions are obtained, which are expressed as a function of first-order quantities available from the classical linear theory. Lateral profiles of the HOBVB-Fα are computed and compared.

Results and conclusion

The results show that the beam’s width reduces and becomes narrower, the side-lobes decrease in magnitude, and the hollow region diameter (or null in magnitude) increases as the order of nonlinearity increases. Furthermore, the nonlinearity of the medium preserves the non-diffracting feature of the beam’s second-harmonic generation within the pre-shock range.     

Estimation of anharmonic parameters, molecular radius, Mr and Beyer's non-linearity parameter, B/A in TBnA series

The thermodynamic parameters like Moelwyn-Hughes parameter (C₁), reduced molar volume (V^~), isochoric temperature coefficient of internal pressure (X), Huggin's parameter (F), Gruneisen parameter (Γ_p), isothermal microscopic Gruneisen parameter, Sharma parameter (S_o), fraction free volume (f) and (A^?) a dimensionless thermal parameter, etc and the Beyer's nonlinearity parameter (B/A) were deduced employing the thermal expansion coefficient derived from density data for the homologues series of compounds terephthalyidene-bis-p-n-alkyl anilines, TBnA with n = 5 to 10,12,14,16 and 18. Further, it is found that the variation of molecular radius, M_r increases with the increase of alkyl chain number, with a core radius of 4.78 Å and the increment for methylene unit is 0.086 Å . The results were discussed with the body of the data available.