Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation

Rogue waves are a class of nonlinear waves with extreme amplitudes, which usually appear suddenly and disappear without any trace. Recently, the parity-time ($\mathcal {PT}$)-symmetric vector rogue waves (RWs) of multi-component nonlinear Schrödinger equation ($n$-NLSE) are usually derived by the methods of integrable systems. In this paper, we utilize the multi-stage physics-informed neural networks (MS-PINNs) algorithm to derive the data-driven $\mathcal {PT}$ symmetric vector RWs solution of coupled NLS system in elliptic and X-shapes domains with nonzero boundary condition. The results of the experiment show that the multi-stage physics-informed neural networks are quite feasible and effective for multi-component nonlinear physical systems in the above domains and boundary conditions.     

The influence of collision energy on magnetically tuned 6Li-6Li Feshbach resonance

The effect of collision energy on the magnetically tuned $^{6}$Li-$^{6}$Li Feshbach resonance (FR) is investigated theoretically by using the coupled-channel (CC) method for the collision energy ranging from 1 μ$ {\rm K} \cdot {k}_{\rm B}$ to 100 μ$ {\rm K} \cdot {k}_{\rm B}$. At the collision energy of 1 μ$ {\rm K} \cdot {k}_{\rm B}$, the resonance positions calculated are 543.152 Gs (s wave, the unit $1 {\rm Gs}=10^{-4} {\rm T}$), 185.109 Gs (p wave $|m_{l}| = 0$), and 185.113 Gs (p wave $|m_{l}| = 1$), respectively. The p-wave FR near 185 Gs exibits a doublet structure of 4 mGs, associated with dipole-dipole interaction. With the increase of the collision energy, it is found that the splitting width remains the same (4 mGs), and that the resonance positions of s and p waves are shifted to higher magnetic fields with the increase of collision energy. The variations of the other quantities including the resonance width and the amplitude of the total scattering section are also discussed in detail. The thermally averaged elastic rate coefficients at $T=10$, 15, 20, 25 K are calculated and compared.     

Investigation of third-order nonlinear optical properties of two azo-nickel chelate compounds

This paper investigates the third-order nonlinear optical properties of two azo-nickel chelate compounds by the optical Kerr gate method at 830 nm wavelength with pulse duration of 120 fs. Both of the two compounds exhibited large third-order optical nonlinearity. The second-order hyperpolarizability,γ, of Compound 1 is of 1.0 × 10^-31 esu. Due to the charge transfer, the γ of Compound 2 with electron donor and acceptor group is 4.9 × 10^-31 esu, which is a four-time enhancement in comparison with Compound i. The absorption spectra show that the electron push-pull effect, which induces intramolecular charge transfer, leads to the increased optical nonlinearity.     

Elastic scattering of two ground-state N atoms

An interaction potential for an N₂(X¹σ_g⁺) molecule is constructed by using the highly accurate valence internally contracted multireference configuration interaction method and the largest basis set, aug-cc-pV6Z, in the valence range. The potential is used to investigate the elastic scattering of two N atoms at energies from 1.0× 10^-11 to 1.0× 10^-4 a.u. The derived total elastic cross sections are very large and almost constant at ultralow temperatures, and the shape of total elastic cross section curve is mainly dominated by the s-partial wave at very low collision energies. Three shape resonances are found in the total elastic cross sections. Concretely, the first one is very sharp and strong. It results from the g-partial-wave contribution and the resonant energy is 3.645× 10^-6 a.u. The second one is contributed by the h-partial wave and the resonant energy is 1.752× 10^-5 a.u. This resonance is broadened by those from the d- and f-partial waves. The third one comes from the l = 6 partial wave contribution and the resonant energy is 3.522× 10^-5 a.u. This resonance is broadened by those from the g- and h-partial waves. The N₂(X¹σ_g⁺) molecular parameters, which are determined at the current theoretical level, achieve very high accuracy due to the employment of the largest correlation-consistent basis set in the valence range.     

Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential in the deep learning.     

Enhanced optical nonlinear absorption of graded Au--TiO2 composite films

This paper reports that single-layer and graded Au-TiO2 granular composite films with Au atom content 15%- 66% were prepared by using reactive co-sputtering technique. The third-order optical nonlinearity of single-layer and graded composite films was investigated by using s- and p-polarized Z-scans in femtosecond time scale. The nonlinear absorption coefficient βeff of single-layer Au-TiO2 films is measured to be -2.3×10^3-0.76×10^3 cm/GW with Au atom content 15%-66%. The βeff value of the 10-layer Au-TiO2 graded film is enhanced to be -2.1×10^4cm/GW calculated from p-polarized Z-scans, which is about ten times the maximum βeff of single-layer films. Broadened response in the wavelength region 730-860 nm of the enhanced optical nonlinearity of graded Au-TiO2 composite films was also investigated.     

Quantum Statistical Properties of k-Quantum Nonlinear Coherent States

In our preceding work, a class of k-quantum nonlinear coherent states, i.e., the k eigenstates of the powers B^^k (k≥3) of the annihilation operator B^ =a^ 1/f(N^) of f-oscillators, are introduced. In this paper, we introduce a new kind of higher-order squeezing and an antibunching effect. The quantum statistical properties of the k states are studied. The result shows that the M-th order [M=(n 1/2)k; n=0, 1,...] squeezing effects exist in all of the k states when k is even. There is the antibunching effect in all of the k states.     

A New Kind of k-Quantum Nonlinear Coherent States: Their Generation and Physical Meaning

A new kind of k-quantum nonlinear coherent states, i.e., the k eigenstates of the k-th power $\hat{B}^{k}$ (k≥3) of the generalized annihilation operator $\hat{B}=\hat{a}\frac1{f(\hat{N})}$ of f-oscillators, are obtained and their properties are discussed. The completeness of the k states is investigated. An alternative method to construct them is proposed. It is shown that these states may form a complete Hilbert space, and all of them can be generated by a linear superposition of k Roy-type nonlinear coherent states. Physically, they can be generated by a linear superposition of the time-dependent Roy-type nonlinear coherent states at different instants.     

Mutual recombination in slow Si^＋ ＋ H^- collisions

This paper studies the process of mutual neutralization of Si^＋ and H^- ions in slow collisions within the multichannel Landau-Zener model. All important ionic-covalent couplings in this collision system are included in the collision dynamics. The cross sections for population of specific final states of product Si atom are calculated in the CM energy range 0.05 e∨/u-5 ke∨/u. Both singlet and triplet states are considered. At collision energies below -10 e∨/u, the most populated singlet state is Si（3p4p, ^1S0）, while for energies above -150e∨/u it is the Si（3p, 4p, ^1P1） state. In the case of triplet states, the mixed 3p4p（^3S1 ＋^3P0） states are the most populated in the entire collision energy range investigated. The total cross section exhibits a broad maximum around 200 300e∨/u and for ECM ≤ 10e∨/u it monotonically increases with decreasing the collision energy, reaching a value of 8 × 10^-13 cm^2 at ECM = 0.05 e∨/u. The ion-pair formation process in Si（3p^2 ^3PJ）＋H（1s） collisions has also been considered and its cross section in the considered energy range is very small （smaller than 10^-20 cm^2 in the energy region below 1 ke∨/u）.     

Exact solution and exotic coherent solition structures of the （2＋1）—dimensional generalized nonlinear Schroedinger equation

In this paper,a variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized nonlinear Schroedinger equation:Iφt-(α-β）φxx (α β）φyy-2λφ[（α β）（∫-∞^x|φ|y^2ydx u1(y,t))-(α-β）(∫-∞^y|φ|x^2dy u2(x,t))]=0,By applying a special Baecklund transformation and introducing arbitrary functions of the seed solutions,the abundance of the localized structures of this model are derived.By selecting the arbitrary function appropriately,some special types of localized excitations such as dromions,dromion lattice,breathers and istantons are constructed.     

Finite-dimensional even and odd nonlinear pair coherent states and their some nonclassical properties

In this paper a new class of finite-dimensional even and odd nonlinear pair coherent states （EONLPCSs）, which can be realized via operating the superposed evolution operators D±（τ） on the state ｜q, 0）, is constructed, then their orthonormalized property, completeness relations and some nonclassical properties are discussed. It is shown that the finite-dimensional EONLPCSs possess normalization and completeness relations. Moreover, the finite-dimensional EONLPCSs exhibit remarkably different sub-Poissonian distributions and phase probability distributions for different values of parameters q, η and ξ.     

Fractional soliton dynamics of electrical microtubule transmission line model with local M-derivative

In this paper, two integrating strategies namely exp[-Φ(X)] and G'/G~2-expansion methods together with the attributes of local-M derivatives have been acknowledged on the electrical microtubule(MT) model to retrieve soliton solutions. The said model performs a significant role in illustrating the waves propagation in nonlinear systems. MTs are also highly productive in signaling, cell motility, and intracellular transport. The proposed algorithms yielded solutions of bright, dark, singular, and combo fractional soliton type. The significance of the fractional parameters of the fetched results is explained and presented vividly.     

Elastic collisions between Si and D atoms at low temperatures and accurate analytic potential energy function and molecular constants of the SiD（χ^2П） radical

Interaction potential of the SiD(X2Π) radical is constructed by using the CCSD(T) theory in combination with the largest correlation-consistent quintuple basis set augmented with the diffuse functions in the valence range. Using the interaction potential, the spectroscopic parameters are accurately determined. The present D0, De, Re, ωe, αe and Be values are of 3.0956 eV, 3.1863 eV, 0.15223 nm, 1472.894 cm-1, 0.07799 cm-1 and 3.8717 cm-1, respectively, which are in excellent agreement with the measurements. A total of 26 vibrational states is predicted when J=0 by solving the radial Schro¨dinger equation of nuclear motion. The complete vibrational levels, classical turning points, initial rotation and centrifugal distortion constants when J=0 are reported for the first time, which are in good accord with the available experiments. The total and various partial-wave cross sections are calculated for the elastic collisions between Si and D atoms in their ground states at 1.0×10-11–1.0×10-3 a.u. when the two atoms approach each other along the SiD(X2Π) potential energy curve. Four shape resonances are found in the total elastic cross sections, and their resonant energies are of 1.73×10-5, 4.0×10-5, 6.45×10-5 and 5.5×10-4 a.u., respectively. Each shape resonance in the total elastic cross sections is carefully investigated. The results show that the shape of the total elastic cross sections is mainly dominated by the s partial wave at very low temperatures. Because of the weakness of the shape resonances coming from the higher partial waves, most of them are passed into oblivion by the strong s partial-wave elastic cross sections.     

Nonlinear Interaction of S-Polarized Incident Radiation and Surface Waves in Magnetoactive Plasma

The effect of an external magnetic field on the nonlinear interaction of S-polarized electromagnetic radiation incident on a S-polarized surface wave in a plasma layer was studied analytically. We have calculated the amplitudes of generated waves at combination frequencies. The generated waves are of P-polarization and can be either electromagnetic or surface waves, depending on the signal of the value=\documentclass{article}\pagestyle{empty}\begin{document}$ ^{\chi '^2 = \frac{{k'^2 }}{{\varepsilon '}} - \frac{{\omega '^2 }}{{c^2 }} + k'\frac{\partial }{{\partial x}}\frac{{\varepsilon '_2 }}{{\varepsilon '\varepsilon '_1 }}} $\end{document}.     

Oscillatory stability of quantum droplets in $\text{}{ \mathcal P }{ \mathcal T }$-symmetric optical lattice

We study stability and collisions of quantum droplets (QDs) forming in a binary bosonic condensate trapped in parity-time (${ \mathcal P }{ \mathcal T }$)-symmetric optical lattices. It is found that the stability of QDs in the ${ \mathcal P }{ \mathcal T }$-symmetric system depends strongly on the values of the imaginary part W₀ of the ${ \mathcal P }{ \mathcal T }$-symmetric optical lattices, self-repulsion strength g, and the condensate norm N. As expected, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs are entirely unstable in the broken ${ \mathcal P }{ \mathcal T }$-symmetric phase. However, the ${ \mathcal P }{ \mathcal T }$-symmetric QDs exhibit oscillatory stability with the increase of N and g in the unbroken ${ \mathcal P }{ \mathcal T }$-symmetric phase. Finally, collisions between ${ \mathcal P }{ \mathcal T }$-symmetric QDs are considered. The collisions of droplets with unequal norms are completely different from that in free space. Besides, a stable ${ \mathcal P }{ \mathcal T }$-symmetric QDs collides with an unstable ones tend to merge into breathers after the collision.     

Theoretical study of collision and collision-free radiative lifetimes of {{\text{S}}_2}\left( {{{\text{B}}^3}\Sigma_{\text{u}}^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right)

We calculate multireference configuration-interaction wavefunctions and the potential-energy curves for the $ {B^3}\Sigma_u^{-} $ and $ {X^3}\Sigma_g^{-} $ states of the collision-free S₂ molecule and the T-shape collision complex S₂?CHe using cc-pVQZ basis sets. We obtain the transition dipole moments of the $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \to {X^3}\Sigma_g^{-} } \right) $ and the Franck?CCondon factors between the vibrational levels of this two states. We evaluate the radiative lifetimes of $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right) $ levels of the collision complex and the collision-free molecule and compare them with the experiments. The collision provides little change in the radiative lifetimes of $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right) $ according to the previous calculations. We obtain excellent agreement between the theoretical results and the experiments. The data calculated are very useful in the study of the microwave-driven high-pressure sulfur lamp and an S₂ laser pumped by a transverse fast discharge.     

Elastic collisions of sulfur and hydrogen in their ground states at low temperatures and spectroscopic parameters of SH（X^2∏） radical

This paper constructs the interaction potential of the SH（X^2∏） radical by using the coupled-cluster singlesdoubles-approximate-triples theory combining the correlation-consistent quintuple basis set augmented with the diffuse functions, aug-cc-pV5Z, in the valence range. Employing the potential, it accurately determines the spectroscopic parameters. The present De, Re, ωe, ωeχe, ae and Be values are of 3.7767eV, 0.13424nm, 2699.846 cm^-1, 47.7055 cm^-1, 0.2639cm^-1 and 9.4414 cm^-1, respectively, which are in excellent agreement with those obtained from the measure- ments. A total of 19 vibrational states has been found when J = 0 by solving the radial SchrSdinger equation of nuclear motion. The complete vibrational levels, classical turning points, initial rotation and centrifugal distortion constants when J = 0 are reported for the first time, which are in good accord with the experimental results. The total and various partial-wave cross sections are computed for the elastic collisions of sulfur and hydrogen in their ground states at low temperatures when two atoms approach each other along the SH（X^2∏） potential energy curve. Over the impact energy range from 1.0×10^-11 to 1.0×10^-4 a.u., eight shape resonances have been found in the total elastic cross sections. For each shape resonance, the resonant energy is accurately calculated. Careful investigations have pointed out that these resonances result from the 1 = 0, 1, 2, 3, 4, 6, 7, 8 partial-wave contributions.     

Fifth-Order Alice-Bob Systems and Their Abundant Periodic and Solitary Wave Solutions*

The study on the nonlocal systems is one of the hot topics in nonlinear science. In this paper, the well-known fifth-order integrable systems including the Sawada-Kotera (SK) equation, the Kaup-Kupershmidt (KK) equation and the fifth-order Koterweg-de Vrise (FOKdV) equation are extended to a generalized two-place nonlocal form, the generalised fifth-order Alice-Bob system. The Lax integrability of two sets of Alice-Bob systems for all the SK, KK and FOKdV type systems are explicitly given via matrix Lax pairs. The $\hat{P}\hat{T}$ symmetry breaking and symmetry invariant periodic and solitary waves for one set of nonlocal SK, KK and FOKdV system are investigated via a special travelling wave solution ansatz.     

Einfluß des Randes auf das effektive Verhalten heterogener Körper - Anwendung auf Rayleighwellen

Effective Elastic Properties of Finite Heterogeneous Media - Application to Rayleigh-waves Rayleigh waves in a heterogeneous material (multiphase mixtures, composite materials, polycrystals) are governed by integrodifferential equations derived by the aid of known methods for infinite heterogeneous media. According to this wave equation the velocity depends on the frequency, and the waves are damped. After some simplifications (isotropy, nonrandom elastic constants) the following is obtained: if the fluctuations of the mass density are restricted to the vicinity of the boundary, the frequency dependent part of the velocity behaves like \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^3 \omega ^3}}{{{\mathop c\limits^\circ} _t^3}} $\end{document} and the damping is proportional to \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^4 \omega ^5}}{{{\mathop c\limits^\circ} _t^5}} $\end{document}, whereas \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^2 \omega ^2}}{{{\mathop c\limits^\circ} _t^2}} $\end{document} respectively \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^3 \omega ^4}}{{{\mathop c\limits^\circ} _t^4}} $\end{document} is found if the fluctuations are present in the whole half-space. From this it is seen, what assumptions are necessary to describe the waves by differential equations with frequenc y-dependent mass density.     

Modified KdV equation for solitary Rossby waves with β effect in barotropic fluids

This paper uses the weakly nonlinear method and perturbation method to deal with the quasi-geostrophic vorticity equation, and the modified Korteweg-de Vries(mKdV) equations describing the evolution of the amplitude of solitary Rossby waves as the change of Rossby parameter β(y) with latitude y is obtained.