Lam函数和非线性演化方程的多级准确解的不变性

利用小扰动方法对非线性演化方程作展开得到原始方程的各级近似方程.并在Lam方程和Lam函数的基础上，应用 Jacobi椭圆函数展开法求出了非线性演化方程的多级准确解，从而得到了多级准确解中存在的守恒形式. 关键词： Lam函数 Jacobi椭圆函数 多级准确解 非线性演化方程 扰动方法 不变性     

Approximation of the soliton solution for the generalized Vakhnenko equation

A class of generalized Vakhnemko equation is considered. First, we solve the nonlinear differential equation by the homotopic mapping method. Then, an approximate soliton solution for the original generalized Vakhnemko equation is obtained. By this method an arbitrary order approximation can be easily obtained and, similarly, approximate soliton solutions of other nonlinear equations can be acquired.     

ELECTROSTATIC POTENTIAL OF STRONGLY NONLINEAR COMPOSITES: HOMOTOPY CONTINUATION APPROACH

The homotopy continuation method is used to solve the electrostatic boundary-value problems of strongly nonlinear composite media, which obey a current-field relation of J=σ E+χ|E|²E. With the mode expansion, the approximate analytical solutions of electric potential in host and inclusion regions are obtained by solving a set of nonlinear ordinary different equations, which are derived from the original equations with homotopy method. As an example in dimension two, we apply the method to deal with a nonlinear cylindrical inclusion embedded in a host. Comparing the approximate analytical solution of the potential obtained by homotopy method with that of numerical method, we can obverse that the homotopy method is valid for solving boundary-value problems of weakly and strongly nonlinear media.     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

A new method to obtain approximate symmetry of nonlinear evolution equation from perturbations

A novel method for obtaining the approximate symmetry of a partial differential equation with a small parameter is introduced. By expanding the independent variable and the dependent variable in the small parameter series, we obtain more affluent approximate symmetries. The method is applied to two perturbed nonlinear partial differential equations and new approximate solutions are derived.     

长水波近似方程的多孤子解

利用直接而简单的齐次平衡方法，给出了长水波近似方程的多孤子解.本方法可进一步推广研究一大类非线性波动方程. 关键词：     

Studies in nonlinear stochastic processes. IV. A comparison of statistical linearization,diagrammatic expansion,and projection operator methods

We compare the methods of statistical linearization, perturbation expansions, and projection operators for the approximate solution of nonlinear multimode stochastic equations. The model equations we choose for this comparison are coupled, nonlinear, first-order, one-dimensional complex mode rate equations. We show that the method of statistical linearization is completely equivalent to the neglect of certain well-defined diagrams in the perturbation expansion resulting in the first Kraichnan-Wyld approximation, and to the retention of only Markovian terms in the projection operator method, i.e., those terms that are local in time.     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

Martensitic phase transition from cubic to tetragonal V<Subscript>3</Subscript>Si: an electronic structure study

We analyze the epidemic spread via a contact infection process in an immobile population within the Susceptible-Infected-Removed (SIR) model. We present both the results of stochastic simulations assuming different numbers of individuals (degrees of freedom) per cell as well as the solution of the corresponding deterministic equations. For the last ones we show that the appropriate system of nonlinear partial differential equations (PDE) allows for a complete separation of variables and present the approximate analytical expressions for the infection wave in different ranges of parameters. Comparing these results with the direct Monte-Carlo simulations we discuss the domain of applicability of the PDE models and their restrictions.     

High-Order Finite Volume Methods for Aerosol Dynamic Equations

Aerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this paper we consider numerical methods for the nonlinear aerosol dynamic equations on time and particle size. The finite volume element methods based on the linear interpolation and Hermite interpolation are provided to approximate the aerosol dynamic equation where the condensation and removal processes are considered. Numerical examples are provided to show the efficiency of these numerical methods.     

Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method

This paper presents the coupled version of a previous work on nonlinear Schrödinger equation [23]. It focuses on the construction of approximate solutions of nonlinear Schrödinger equations. In this paper, we applied the differential transformation method (DTM) to solving coupled Schrödinger equations. The obtained results show that the technique suggested here is accurate and easy to apply.     

The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems. In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form. Received 26 February 1999     

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra-Fredholm integral equations

While the approximate solutions of one-dimensional nonlinear Volterra-Fredholm integral equations with smooth kermels are now well understood,no systematic studies of the numerical solutions of their multi-dimensional counterparts exist.In this paper,we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra-Fredholm integral equations based on the multi-variate Legendre-collocation approach.Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view.Consequently,rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors.The existence and uniqueness of the numerical solution are established.Numerical experiments are provided to support the theoretical convergence analysis.The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.     

Solitary Waves of a Perturbed sine—Gordon Equation

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves,different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained.All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.     

Solution of a mixed parity nonlinear oscillator: Harmonic balance

The method of harmonic balance is used to calculate first-order approximations to the periodic solutions of a mixed parity nonlinear oscillator. First, the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction. Then the two auxiliary equations, where the restoring force functions are odd, are solved by using a first harmonic term (without a constant). Therefore, we obtain the two approximate solutions to the mixed parity nonlinear oscillator. One is expressed in terms of the exact amplitude in the negative direction, the other in terms of the approximate amplitude. These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincaré method for large amplitudes.     

布拉格光栅中微扰光声孤子

研究了布拉格光栅中光波和声波的相互作用. 利用多重尺度方法,将光声耦合方程约化为非线性薛定谔方程,并给出了单光声孤子和双光声孤子近似解析解. 进一步讨论了光声孤子抑制光速的机理和双光声孤子的相互作用性质. 关键词： 布拉格光栅 光声耦合 多重尺度方法 光声孤子     

A deep learning method for solving high-order nonlinear soliton equations

We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higher-order nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.     

A New Asymptotic Approach to Solve Nonlinear Evolution Equations

A conformal-invariant asymptotic expansion approach to solve any nonlinear integrable and nonintegrable models with any dimensions is proposed. Taking the compound KdV-Burgers (cKdVB) equation and the KdV-Burgers (KdVB) equation as concrete examples,we obtain many new conformal-invariant models with Painleve' property and the approximate solutions of the cKdVB and KdVB equations. In some special cases, the approximate solutions become exact.     

On spectral methods for solving nonlinear acoustics equations

Two very efficient methods for obtaining approximate solutions to nonlinear acoustics equations are discussed. I proposed these methods earlier, but they are still little known. The first method is based on expanding an unknown function into a Taylor series with respect to the coordinate (evolution variable) and on approximate summation of the terms of this series in all orders up to the infinite order. This series can be summed completely only in particular cases, e.g., for a simple wave. It has been noted that the partial summation technique is implemented more easily if all the terms of the series are represented as corresponding topological diagrams. The second method is based on introducing a “nonlinear” phase delay (proportional to the wave amplitude) for the temporal variable in linear solutions of the problem. The application technique of these methods is illustrated by obtaining approximate solutions of the Burgers equation.