New amplitude equation of single—mode laser

The white-gain model and the white-loss model of a single-mode laser are investigated in the presence of cross-correlations between the real and imaginary parts of quantum noise as well as pump noise. It was found that, like the white cubic model (2001 Chin. Phys. Lett. 18 370), the amplitude equations are all decoupled from the phase equations for the two models, and the same novel term appears in the amplitude equations of the two models. So we can put the amplitude equations of all the models into a general form, that is, the new amplitude equation. We further use this new amplitude equation to study specifically the stationary properties of the laser intensity for the white-gain model.     

Local electron mean energy profile of positive primary streamer discharge with pin-plate electrodes in oxygen nitrogen mixtures

Local electron mean energy (LEME) has a direct effect on the rates of collisional ionization of molecules and atoms by electrons. Electron-impact ionization plays an important role and is the main process for the production of charged particles in a primary streamer discharge. Detailed research on the LEME profile in a primary streamer discharge is extremely important for a comprehensive understanding of the local physical mechanism of a streamer. In this study, the LEME profile of the primary streamer discharge in oxygen-nitrogen mixtures with a pin-plate gap of 0.5 cm under an impulse voltage is investigated using a fluid model. The fluid model includes the electron mean energy density equation, as well as continuity equations for electrons and ions and Poisson’s electric field equation. The study finds that, except in the initial stage of the primary streamer, the LEME in the primary streamer tip tends to increase as the oxygen-nitrogen mole ratio increases and the pressure decreases. When the primary streamer bridges the gap, the LEME in the primary streamer channel is smaller than the first ionization energies of oxygen and nitrogen. The LEME in the primary streamer channel then decreases as the oxygen-nitrogen mole ratio increases and the pressure increases. The LEME in the primary streamer tip is primarily dependent on the reduced electric field with mole ratios of oxygen-nitrogen given in the oxygen-nitrogen mixtures.     

A deep learning method based on prior knowledge with dual training for solving FPK equation

The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based...     

The extended symmetry approach for studying the general Korteweg-de Vries-type equation

The extended symmetry approach is used to study the general Korteweg-de Vries-type (KdV-type) equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation.     

Compact implicit integration factor methods for some complex-valued nonlinear equations

The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear Schro¨dinger (NLS) equation and the complex Ginzburg–Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.     

Stability Analysis of Solitary Wave Solutions for Coupled and(2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications

The searching exact solutions in the solitary wave form of non-linear partial differential equations(PDEs play a significant role to understand the internal mechanism of complex physical phenomena. In this paper, we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the(2+1)-dimensional cubic Klein-Gordon(K-G) equation. The Klein-Gordon equation are relativistic version of Schr¨odinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which severa solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions o PDEs arise in mathematical physics.     

Higher-order rogue waves with controllable fission and asymmetry localized in a (3 + 1)-dimensional generalized Boussinesq equation

The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations.Such a nonlinear model considered in this paper as the concrete example is the(3+1)-dimensional generalized Boussinesq(gB) equation,and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters.It is indicated by the(3+1)-dimensional gB equation that the embedded param...     

Analytical solution of the Boltzmann--Poisson equation and its application to MIS tunneling junctions

In order to consider quantum transport under the influence of an electron--electron (e--e) interaction in a mesoscopic conductor, the Boltzmann equation and Poisson equation are investigated jointly. The analytical expressions of the distribution function for the Boltzmann equation and the self-consistent average potential concerned with e--e interaction are obtained, and the dielectric function appearing in the self-consistent average potential is naturally generalized beyond the Thomas--Fermi approximation. Then we apply these results to the tunneling junctions of a metal--insulator--semiconductor (MIS) in which the electrons are accumulated near the interface of the semiconductor, and we find that the e--e interaction plays an important role in the transport procedure of this system. The electronic density, electric current as well as screening Coulombic potential in this case are studied, and we reveal the time and position dependence of these physical quantities explicitly affected by the e--e interaction.     

Propagation modeling of ocean-scattered low-elevation GPS signals for maritime tropospheric duct inversion

The maritime tropospheric duct is a low-altitude anomalous refractivity structure over the ocean surface,and it can significantly affect the performance of many shore-based/shipboard radar and communication systems.We propose the idea that maritime tropospheric ducts can be retrieved from ocean forward-scattered low-elevation global positioning system(GPS) signals.Retrieval is accomplished by matching the measured power patterns of the signals to those predicted by the forward propagation model as a function of the modified refractivity profile.On the basis of a parabolic equation method and bistatic radar equation,we develop such a forward model for computing the trapped propagation characteristics of an ocean forward-scattered GPS signal within a tropospheric duct.A new GPS scattering initial field is defined for this model to start the propagation modeling.A preliminary test on the performance of this model is conducted using measured data obtained from a 2009-experiment in the South China Sea.Results demonstrate that this model can predict GPS propagation characteristics within maritime tropospheric ducts and serve as a forward model for duct inversion.     

Mapping deformation method and its application to nonlinear equations

An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein－Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.     

Wronskian Form Solutions for a Variable Coefficient Kadomtsev–Petviashvili Equation

Starting from a simple transformation, and with the aid of symbolic computation, we establish the relation- ship between the solution of a generalized variable coefficient Kadomtsev-Petviashvili （vKP） equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover, we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.     

Analysis of the wave functions for accelerating Kerr-Newman metric

As is well-known, it is very difficult to solve wave equations in curved space-time. In this paper,we find that wave equations describing massless fields of the spins s≤2 in accelerating KerrNewman black holes can be written as a compact master equation. The master equation can be separated to radial and angular equations, and both can be transformed to Heun's equation,which shows that there are analytic solutions for all the wave equations of massless spin fields.The results not only demonstrate that it is possible to study the similarity between waves of gravitational and other massless spin fields, but also it can deal with other astrophysical applications, such as quasinormal modes, scattering, stability, etc. In addition, we also derive approximate solutions of the radial equation.     

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.     

Validation of a RANS transition model using a high-order weighted compact nonlinear scheme

A modified transition model is given based on the shear stress transport (SST) turbulence model and an intermittency transport equation. The energy gradient term in the original model is replaced by flow strain rate to saving computational costs. The model employs local variables only, and then it can be conveniently implemented in modern computational fluid dynamics codes. The fifth-order weighted compact nonlinear scheme and the fourth-order staggered scheme are applied to discrete the governing equations for the purpose of minimizing discretization errors, so as to mitigate the confusion between numerical errors and transition model errors. The high-order package is compared with a second-order TVD method on simulating the transitional flow of a flat plate. Numerical results indicate that the high-order package give better grid convergence property than that of the second-order method. Validation of the transition model is performed for transitional flows ranging from low speed to hypersonic speed.     

Backlund transformation and variable separation solutions for the generalized Nozhnik—Novikov—Veselov equation

Using the extended homogeneous balance method, the B?cklund transformation for a (2+1)-dimensional integrable model, the generalized Nizhnik－Novikov－Veselov (GNNV) equation, is first obtained. Also, making use of the B?cklund transformation, the GNNV equation is changed into three equations: linear, bilinear and trilinear form equations. Starting from these three equations, a rather general variable separation solution of the model is constructed. The abundant localized coherent structures of the model can be induced by the entrance of two variable-separated arbitrary functions.     

Anisotropic Bianchi Type Ⅰ Cosmological Models with Generalized Chaplygin Gas and Dynamical Gravitational and Cosmological Constants

This paper deals with study of generalized Chaplygin gas model with dynamical gravitational and cosmological constants. In this paper a new set of exact solutions of Einstein field equations for spatially homogeneous and anisotropic Bianchi type Ⅰ space-time have been obtained. The solutions of the Einstein's field equations are obtained by considering(i) the power law relation between Hubble parameter H and scale factor R and(ii) scale factor of the form R =-1/t + t~2, t 1. The assumptions lead to constant and variable deceleration parameter respectively. The physical and dynamical behaviors of the models have been discussed with the help of graphical representations. Also we have discussed the stability and physical acceptability of solutions for solution type-Ⅰ and solution type-Ⅱ.     

Calculation of Free Energy of the Integrable Landau—Lifshitz model

The generalized Bethe-ansatz method of thermodynamic analysis of integrable systems was employed to compute the free energy of a classical integrable model,i.e.the Landau-Lifshitz model.Using the action-angle variables of the model and by imposing a periodic boundary condition.we derive a phase-shifted density of states for the excitations of the system.The free energy,in the thermodynamic limit,can be expressed analytic in terms of two coupled nonlinear integral equations of the finie temperature excited energy for effective phonons and kinks (antikinks).we solve these equations iteratively for a special case that the model is in the limit of anisotropic strong yz coupling.     

A comprehensive approach to an equation of state for hard spheres and Lennard-Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way.We will guess equations of state by using suitable axiomatic functional forms (n=1,2,3,4,5) for surface tension S n m (r),r ≥ d/2 with intermolecular separation r as a variable,where m is an arbitrary real number (pole).Among the equations of state obtained in this way are Percus-Yevick,scaled particle theory and Carnahan-Starling equations of state.In addition,we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately.It is found that for both hard sphere fluids as well as Lennard-Jones fluids,with m=3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results.Furthermore,this equation of state gives the Percus-Yevick (pressure) EOS for the m=0,the Carnahan-Starling EOS for m=4/5,while for the value of m=1 it corresponds to a scaled particle theory EOS.