Completely Integrable Equation for the Quantum Correlation Function of Nonlinear Schrödinger Equation

Correlation functions of exactly solvable models can be described by differential equations [1]. In this paper we show that for the non-free fermionic case, differential equations should be replaced by integro-differential equations. We derive an integro-differential equation, which describes a time and temperature dependent correlation function of the penetrable Bose gas. The integro-differential equation turns out to be the continuum generalization of the classical nonlinear Schr?dinger equation. Received: 15 January 1997 / Accepted: 21 March 1997     

Note on solutions to a class of nonlinear singular integro-differential equations

Certain classes of nonlinear singular integro-differential equations are considered. These equations are mapped, via explicit transformations, to either ordinary differential equations or to linearizable partial differential equations.     

简支梁大挠度非线性振动中的突变和混沌

本文研究由于大挠度引起的曲率非线性对简支梁振动的影响。用Галёркин法可导得一组非线性的积分一微分方程。计算表明,这一非线性系统具有尖点突变和混沌等有趣的动力学行为。     

Separation of variables in the two-point BBGKY equations

Under two particular closure conditions, the two-point BBGKY equation is shown to be separable into equations for one- point turbulent fluctuations, yielding, respectively, a linear equation anda nonlinear integro-differential equation of convolution type. Analogy with Schrödinger's equation is discussed.     

Coulomb quantum kinetics in pulse-excited semiconductors

The quantum kinetic equations of the electron and hole densities and the interband polarization are derived for a laser-pulse-excited semiconductor with Coulomb interaction including renormalization effects, excitonic effects and scattering with memory kernels. Numerical solutions of this set of non-Markovian, nonlinear integro-differential equations are obtained for a statically screened Coulomb potential.     

A direct matrix method for computing analytical Jacobians of discretized nonlinear integro-differential equations

In this article, we present a simple direct matrix method for analytically computing the Jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integro-differential equations. The method is based on a formulation of the discretized equations in vector form using only matrix-vector products and component-wise operations. By applying simple matrix-based differentiation rules, the matrix form of the analytical Jacobian can be calculated with little more difficulty than required to compute derivatives in single-variable calculus. After describing the direct matrix method, we present numerical experiments demonstrating the computational performance of the method, discuss its connection to the Newton–Kantorovich method and apply it to illustrative 1D and 2D example problems from electrochemical transport.     

Theoretical investigation of an electron beam propagating through a wide gap cavity

This paper presents a self-consistent nonlinear theory of the current and energy modulations when an electron beam propagates through an inductively-loaded wide gap cavity. The integro-differential equations axe obtained to describe the modulation of the beam current and kinetic energy. A relativistic klystron amplifier （RKA） model is introduced, which uses an inductively-loaded wide gap cavity as an input cavity. And a numerical code is developed for the extended model based on the equations, from which some relations about the modulated current and modulated energy are numerically given.     

A new fully nonisothermal coupled model for the simulation of ice sheet flow

A shallow ice thermocoupled model for the complex nonlinear polythermal ice sheet dynamics is proposed and solved by means of efficient numerical methods. A novelty is the obstacle problem formulation associated to a nonlinear integro-differential equation (with nonlocal temperature dependent coefficients) for the ice sheet profile. This formulation is motivated by the free boundary feature and the influence of the temperature on the profile (fully nonisothermal model). Concerning the temperature equation, a dynamically prescribed surface temperature, obtained from an Energy Balance model corrected by the altitude effect, is posed. As the profile and temperature equations are fully coupled, a nonlinear PDE system governing the upper ice sheet profile, the velocity field, the temperature and the basal stress is stated. In addition to the numerical difficulties associated to the new profile equation, several techniques have been considered for the numerical solution of the temperature, velocity and basal magnitudes. Discussions concerning the nonlinear dynamics of the different involved magnitudes and the improvement in their computed values with respect to previous works are also presented.     

Numerical solution of certain classes of transport equations in any dimension by Shannon sampling

A method is developed for computing solutions to some class of linear and nonlinear transport equations (hyperbolic partial differential equations with smooth solutions), in any dimension, which exploits Shannon sampling, widely used in information theory and signal processing. The method can be considered a spectral or a wavelet method, strictly related to the existence of characteristics, but allows, in addition, for some precise error estimates in the reconstruction of continuous profiles from discrete data. Non-dissipativity and (in some case) parallelizability are other features of this approach. Monotonicity-preserving cubic splines are used to handle nonuniform sampling. Several numerical examples, in dimension one or two, pertaining to single linear and nonlinear (integro-differential) equations, as well as to certain systems, are given.     

Noise signal propagation in soft biological tissues

Mathematical models are formulated that discribe linear and nonlinear wave propagation in biological tissues. The basis of the method is evolutionary integro-differential equations with a kernel that takes into account the specific properties of tissue. An equation is obtained for the correlation function of acoustic noise in a medium with memory. The procedure for calculating the correlation function by the given type of kernel and noise spectrum at the entrance to the medium is described. It is shown that in different tissue, there is a difference in the laws of decrease in full intensity of a wideband signal with distance. It is demonstrated that the nonlinear equation in the limiting cases of ??short-?? and ??long-term?? memory reduces to equations that have been well studied in statistical nonlinear acoustics.     

Numerical Solutions of the System of Singular Integro-Differential Equations in Classical Hölder Spaces

New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integro-differential equations that are defined on arbitrary smooth closed contours of the complex plane. We carry out the convergence analysis in classical Hölder spaces. A numerical example is also presented.     

Iteration methods and algorithms for dielectric-resonator integral equations of the field

Quasi-eigenmodes of open cylindrical and rectangular dielectric resonators (DRs) are determined by the method of iterative solution of the volume integral and integro-differential equations with corresponding functionals. New forms of equations and iteration algorithms for the nonlinear input of the desired complex parameter are proposed. Frequencies and Q-factors of the H_01δ and H₀₁₁ modes of a cylindrical DR and the H mode of a rectangular DR for the uniform and nonuniform cases are obtained numerically. The influence of a thin semiconductor layer located at the ends of the DR and irradiated by high-power laser pulses on the frequencies and Q-factors of the DR modes is examined. It is shown that an up to ten or more percent tuning of resonant frequencies can be reached by transformation of a low conducting state to a high conducting state.     

Absorbing boundary conditions for nonlinear Euler and Navier–Stokes equations based on the perfectly matched layer technique

Absorbing boundary conditions for the nonlinear Euler and Navier–Stokes equations in three space dimensions are presented based on the perfectly matched layer (PML) technique. The derivation of equations follows a three-step method recently developed for the PML of linearized Euler equations. To increase the efficiency of the PML, a pseudo mean flow is introduced in the formulation of absorption equations. The proposed PML equations will absorb exponentially the difference between the nonlinear fluctuation and the prescribed pseudo mean flow. With the nonlinearity in flux vectors, the proposed nonlinear absorbing equations are not formally perfectly matched to the governing equations as their linear counter-parts are. However, numerical examples show satisfactory results. Furthermore, the nonlinear PML reduces automatically to the linear PML upon linearization about the pseudo mean flow. The validity and efficiency of proposed equations as absorbing boundary conditions for nonlinear Euler and Navier–Stokes equations are demonstrated by numerical examples.     

Microscopic Behavior of the Classical Electron in the Absence of External Forces

A system of nonlinear integro-differential equations is derived for the motion of the classical electron with a rigid and spherically symmetric 3D gaussian distribution of charge. The equations are analyzed for stability around the state of rest and of uniform rectilinear motion with velocity small with respect to the velocity of light. The extremely high-frequency and radiationless micro-oscillations that the electron executes when disturbed from the equilibrium states show the inconsistency of the Abraham-Lorentz equation and of all concepts associated with this equation, like the notion that the electron may have a mass of electromagnetic origin.     

Compact and noncompact structures for nonlinear fractional evolution equations

This Letter deals with compact and noncompact solutions for nonlinear evolution equations with time-fractional derivatives. We present a reliable approach of the homotopy perturbation method to handle nonlinear fractional evolution equations. The validity of the approach is verified through illustrative examples. New exact solitary wave and compacton solutions are developed. The proposed technique could lead to a promising approach for a wide class of nonlinear fractional evolution equations.     

On the spectral theory of the integro-differential operator a generating nonlinear evolution equations

The main steps of constructing the spectral theory of the integro-differential operator A generating a class of exactly soluble nonlinear evolution equations (NLEE) related to the linear matrix first-order problem L, are outlined. Compact expressions for the diagonal of the resolvent of operator L through A are obtained.     

Soliton multidimensional equations and integrable evolutions preserving Laplace's equation

The KP equation, which is an integrable nonlinear evolution equation in 2+1, i.e., two spatial and one temporal dimensions, is a physically significant generalization of the KdV equation. The question of constructing an integrable generalization of the KP equation in 3+1, has been one of the central open problems in the field of integrability. By complexifying the independent variables of the KP equation, I obtain an integrable nonlinear evolution equation in 4+2. The requirement that real initial conditions remain real under this evolution, implies that the dependent variable satisfies a nonlinear evolution equation in 3+1 coupled with Laplace's equation. A reduction of this system of equations to a single equation in 2+1 contains as particular cases certain singular integro-differential equations which appear in the theory of water waves.     

Coarse-grained computation for particle coagulation and sintering processes by linking Quadrature Method of Moments with Monte-Carlo

The study of particle coagulation and sintering processes is important in a variety of research studies ranging from cell fusion and dust motion to aerosol formation applications. These processes are traditionally simulated using either Monte-Carlo methods or integro-differential equations for particle number density functions. In this paper, we present a computational technique for cases where we believe that accurate closed evolution equations for a finite number of moments of the density function exist in principle, but are not explicitly available. The so-called equation-free computational framework is then employed to numerically obtain the solution of these unavailable closed moment equations by exploiting (through intelligent design of computational experiments) the corresponding fine-scale (here, Monte-Carlo) simulation. We illustrate the use of this method by accelerating the computation of evolving moments of uni- and bivariate particle coagulation and sintering through short simulation bursts of a constant-number Monte-Carlo scheme.     

非线性欧拉方程组的完美匹配层吸收边界条件

对于非线性Euler方程,提出一类基于完美匹配层(PML)技术的吸收边界条件。首先对线性化的Euler方程设计出PML公式,然后将线性化Euler方程中的通量函数替换成相对应的非线性通量函数,得到非线性的PML方程。考虑到PML方程中包含有一个刚性的源项,文中采用一种隐显Runge-Kutta方法来求解空间半离散后得到的ODE系统。数值实验表明设计的非线性PML吸收边界条件优于传统的特征边界条件。