Superintegrability with second- and third-order integrals of motion

Superintegrable systems in two-dimensional Euclidean space E ₂ with scalar potentials and integrals of motion that are second- or third-order polynomials in the momenta are discussed. We review old results, obtained in collaboration with Ya.A. Smorodinsky more than 40 years ago and present new ones that are their continuation. The text was submitted by the author in English.     

A prototype for a tomography system using third-order acoustic nonlinear effects

A prototype of a tomography system for reconstructing the distributions of acoustic nonlinear parameters is developed and manufactured on the basis of the effect of nonlinear noncollinear interaction of three primary waves. Application of coded primary signals with further correlation processing of a detected combination signal makes it possible to reconstruct the complete image of an object as a result of a single experiment using a small number of transducers, i.e., three radiators and one receiver. A mirror system is proposed, consisting of two coaxial conical acoustic mirrors that make it possible to transform the front of a wave from a cylindrical transducer into a homogeneous quasi-plane beam with a large width close to the real medical diagnostics requirements. Results of physical experiments are given.     

Factorization of quantities arising in second- and third-order perturbation theory for atoms

A closed expression is found for the real representation of the Coulomb Green's function for negative energy. Equations are derived to take into account the atomic characteristics in second- and third-order perturbation theory for atoms of the first and second groups in the periodic table.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 54–58, December, 1969.     

Microscopic theory of the second- and third-order elastic constants of cobalt in a thermodynamic module

The total potential energy of a crystal is presented as an expansion by irreducible interactions in clusters containing pairs of atomic triplets and quadruplets. The potential energy of the clusters in the adiabatic approximation is a function of vectors \(\vec r_{ik}\) connecting the centers of atoms in the clusters. Arguments (basic functions) affecting the potential energy of clusters are found using the model with allowance for the exchange symmetry of atoms and the irreducibility of the considered energies of doublet, triplet, and quadruplet atoms and interactions. This allows us to present arguments of the potential energy in the form of summed integral numbers (latticed sums) multiplied by a fixed value of the crystal unit cell parameter, and to set the numerical values of the potential energy arguments as a function of vectors \(\vec r_{ik}\). A potential corresponding to the thermodynamic additivity concept of crystal energy is selected as the model potential of pairwise interaction. Secondand third-order moduli of the rigidity of Co crystals with A1, A2, and HCP structures are calculated using this model of multiatom interactions.     

Effect of concentration gradient on the first-, second- and third-order polarizabilities

The effect of concentration gradients on the polarizabilities of chemical systems of mixed composition is examined theoretically. It is found that the first-, second- and third-order polarizabilities all exhibit linear dependences on concentration gradient but nonlinear dependences on dielectric gradient. Furthermore, sample calculations imply that under selected conditions, concentration and/or dielectric gradient effects should be readily observable experimentally and indeed, under some conditions could dominate the experimental response. Examples of systems that might display the predicted effects include compositionally graded polymer, semiconductor, composite films, as well as thermally graded liquid mixtures and colloidal suspensions.     

Dynamics of wave packets and soliton interaction in terms of the third-order nonlinear Schrödinger equation

We present the results of numerical study of the evolution of wave packets and envelope soliton interaction in terms of the third-order nonlinear Schrödinger equation. It is shown that an arbitrary initial pulse evolves to a few solitons and a linear quasiperiodic wave. The interaction of solitons is accompanied by the radiation of part of the wave field in the form of a linear quasiperiodic wave from the interaction region, amplification of the soliton with larger amplitude and attenuation of the soliton with smaller amplitude.     

耦合相对转动非线性动力系统的稳定性与近似解

研究了一类含三次非线性耦合项的相对转动非线性动力系统的动力学行为. 建立了具有非线性弹性力、广义摩阻力耦合项的系统动力学方程. 运用多尺度法求解谐波激励下耦合非自治系统的近似解,通过讨论系统的主共振和内共振特性,分析了耦合项对系统响应的影响. 应用奇异性理论研究了主振稳态响应分岔方程的稳定性,得到了系统的转迁集和分岔曲线的拓扑结构. 关键词： 相对转动 非线性耦合动力系统 奇异性理论 稳定性     

A cobalt (II) complex with 6-methylpicolinate: Synthesis,characterization, second- and third-order nonlinear optical properties,and DFT calculations

A cobalt(II) complex of 6-methylpicolinic acid, [Co(6-Mepic)₂(H₂O)₂]·2H₂O, was prepared and fully determined by single crystal X-ray crystal structure analysis as well as FT-IR, FT-Raman. UV–vis spectra were recorded within different solvents, to illustrate electronic transitions and molecular charge transfer within complex 1. The coordination sphere of complex 1 is a distorted octahedron according to single crystal X-ray results. Moreover, DFT (density functional theory) calculations with HSEH1PBE/6-311 G(d,p) level were carried out to back up the experimental results, and form base for future work in advanced level. Hyperconjugative interactions, intramolecular charge transfer (ICT), molecular stability and bond strength were researched by the using natural bond orbital (NBO) analysis. X-ray and NBO analysis results demonsrate that O−H···O hydrogen bonds between the water molecules and carboxylate oxygen atoms form a 2D supramolecular network, and also adjacent 2D networks connected by C−H···π and π···π interactions to form a 3D supramolecular network. Additionally, the second− and third−order nonlinear optical parameters of complex 1 were computed at DFT/HSEH1PBE/6-311 G(d,p) level. The refractive index (n) was calculated by using the Lorentz−Lorenz equation in order to investigate polarization behavior of complex 1 in different solvent polarities. The first−order static hyperpolarizability (β) value is found to be lower than pNA value because of the inversion symmetry around Co (II). But the second−order static hyperpolarizability (γ) value is 2.45 times greater than pNA value (15×10⁻³⁰ esu). According to these results, Co(II) complex can be considered as a candidate to NLO material. Lastly molecular electrostatic potential (MEP), frontier molecular orbital energies and related molecular parameters for complex 1 were evaluated.     

Comparison of the third-order dispersion terms of interaction between chiral molecules

Using a quantum electrodynamic approach, the coupling between chiral molecules derived from the exchange of three virtual photons of the electric dipole type is investigated. The resulting interaction potential, which involves the first nonlinear polarizability of the molecules, is compared with a previously derived interaction potential involving the optical activity of the molecules; the latter is accounted for by the exchange of two virtual photons, one being electric dipolar and the other being electric quadrupolar or magnetic dipolar. The symmetry properties of the new term is considered and its quasi-static limit is derived. It appears that the two interaction terms, which both are third-order dispersion processes, might have the same order of magnitude; thus none can a priori be neglected and both might equally contribute to the possibility of a helical ordering.     

The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals

The third-order nonlinear susceptibilities and nonlinear refractive indices of KDP, BBO, and LiNbO₃ crystals at wavelengths of 1064 and 532 nm are measured using the Z-scan technique. The measurements are made for different energies of incident radiation and different focusing conditions and crystal lengths. It is found that, as the angle between the light beam and the principal optic axis of a KDP crystal increases, the nonlinearity drops, with its magnitude at a wavelength of 1064 nm being higher than at 532 nm. For a BBO crystal, the nonlinearity dispersion is normal. The mechanisms of nonlinear losses in KDP, BBO, and LiNbO₃ crystals are identified. The values of the nonlinear susceptibilities thus obtained are analyzed and compared with the results of calculations based on an empirical model and with the data of other authors.     

Novel correction method for X-ray beam energy fluctuation of high energy DR system with a linear detector

A high energy digital radiography (DR) testing system has generated diverse scientific and technological interest in the field of industrial non-destructive testing. However, due to the limitations of manufacturing technology for accelerators, an energy fluctuation of the X-ray beam exists and leads to bright and dark streak artifacts in the DR image. Here we report the utilization of a new software-based method to correct the fluctuation artifacts. The correction method is performed using a high pass filtering operation to extract the high frequency information that reflects the X-ray beam energy fluctuation, and then subtracting it from the original image. Our experimental results show that this method is able to rule out the artifacts effectively and is readily implemented on a practical scanning system.     

Individual and joint influence of second- and third-order dispersion on the transmission quality in the presence of coherent interference

The shape of a resulting pulse (coherent interference superimposed to Gaussian pulse) along the linear optical fiber is determined by using attained analytical expressions. These analytical expressions enable the investigation of the influence of arbitrary dispersion order on pulse propagation. Individual and joint influence of second- and third-order dispersion on the transmission quality is examined in this paper. Pulse shape, eye diagram and bit error rate (BER) are used to represent in the best manner the above-mentioned influences.     

Tomography of the spatial distribution of a scatterer in third-order nonlinear processes

The possibility of using the third-order nonlinear effect of generation of combination waves in medical tomography has been considered. Simultaneous evolution of two competing processes, giving similar effects (which nevertheless can be separated) is analyzed. A tomography scheme based on only three sources and one receiver is proposed.     

Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system

The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses. The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.     

Analytic treatment of a system with a vibro-impact nonlinear energy sink

This Communication reports on an approximate analytic procedure for analysis of transient damped response in a system comprising a primary linear oscillator and vibro-impact nonlinear energy sink (NES). Multiple-scale expansion reduces the problem in the first approximation to low-dimensional vibro-impact system with forcing and damping; the latter problem yields to direct analytic approach. The results computed by this way fairly correlate with direct numeric simulations. The analysis takes advantage of small ratio of the NES and the primary masses, but does not require the restitution coefficient to be close to unity. The proposed approach paves the way for efficient design of systems with vibro-impact NESs.     

Relationship between third-order nonlinear optical properties and magnetic interactions in open-shell systems: a new paradigm for nonlinear optics

Although most third-order nonlinear optical (NLO) materials are closed-shell singlet systems, this Letter theoretically shows that the third-order NLO properties are drastically enhanced in symmetric open-shell diradical systems with intermediate diradical character and further reveals that this enhancement is associated with the ferromagnetic and antiferromagnetic interactions. This new paradigm opens up a promising area for designing new third-order NLO materials.     

一种侧链共轭聚噻吩衍生物薄膜的三阶非线性光学响应

合成了一种侧链共轭的聚噻吩衍生物聚3-（5′辛基-噻吩乙烯基）噻吩（POTVTh）,并通过溶液旋涂制备了聚合物薄膜.吸收光谱显示该聚合物薄膜具有较小的禁带宽度和宽的光谱响应.采用Z扫描技术在800 nm下用飞秒激光器研究了该聚合物薄膜的三阶非线性光学特性,非线性吸收系数为5.63×10^-7cm/W,非线性折射率为-6.38×10^-11 cm²/W,三阶非线性极化率为4.21×10^-9esu,比侧链未共轭的 关键词： 聚噻吩衍生物 侧链共轭 三阶非线性极化率 非线性折射率     

Dispersive effects and correction term in two-mode phonon conduction model for Ge

A complete analysis of the phonon conductivity κ, still lacking in the literature, is presented in the two-mode conduction model for Germanium. First a method is derived from which the correction term κ_c of the Callaway model is separated into its longitudinal and transverse parts and then the effects of strong phonon dispersion and the role of longitudinal and transverse phonons on κ_c are studied. For this purpose we have also proposed some new empirical expressions for the three phonon relaxation rates τ_3ph⁻¹'s which are valid in the entire temperature range. This improvised model, when applied simultaneously to the phonon conductivity data of both normal and enriched Ge, yields some new results. These are (i) κ_c neglected by the earlier workers in the two-mode phonon conduction model, gives a substantial contribution beyond the conductivity maximum and (ii) the longitudinal phonons are the major carriers of heat at high temperature.