Exponentially Localized Solutions to the Klein–Gordon Equation

Exponentially localized solutions of the Klein–Gordon equation for two and three space variables are presented. The solutions depend on four free parameters. For some relations between the parameters, the solutions describe wave packets filled with oscillations whose amplitudes decrease in the Gaussian way with distance from a point running with group velocity along a ray. The solutions are constructed by using exact complex solutions of the eikonal equation and may be regarded as ray solutions with amplitudes involving one term. It is also shown that the multidimensional nonlinear Klein–Gordon equation can be reduced to an ordinary differential equation with respect to the complex eikonal. Bibliography: 12 titles.     

一类两参数非线性奇摄动边值问题解的渐近性态

本文研究了一类两参数非线性奇摄动边值问题的基本模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的三种不同情形下作了讨论,得到了该问题的渐近解并证明了在三种情形下不同的解的结构与渐近性态.     

A SINGULARLY PERTURBED PROBLEM OF THIRD ORDER EQUATION WITH TWO PARAMETERS

A singularly perturbed problem of third order equation with two parameters is studied. Using singular perturbation method, the structure of asymptotic solutions to the problem is discussed under three possible cases of two related small parameters. The results obtained reveal the different structures and limit behaviors of the solutions in three different cases. And in comparison with the exact solutions of the autonomous equation they are relatively perfect.     

Scale and time effects on mathematical models for transport in the environment

The purpose of this paper is to analyse mathematical models used in environmental modelling.Following a brief survey of the development in modelling scale-and time-dependent dispersion processes in the environment,this paper compares three similarity solutions,one of which is a solution of the generalized Feller equation(GF)with fractal parameters,and the other two for the newly-developed generalized Fokker-Planck equation(GFP).The three solutions are derived with parameters having physical significance.Data from field experiments are used to verify the solutions.The analyses indicate that the solutions of both GF and GFP represent the physically meaningful natural processes,and simulate the realistic shapes of tracer breakthrough curves.     

一类两参数半线性奇摄动问题解的渐近性态

本文研究了一类两参数半线性奇摄动问题的基本模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的三种不同情形下作了讨论.得到了该问题在三种不同情形下的渐近解并证明了在三种情形下解的结构与渐近性态.     

Existence of three solutions for a two point boundary value problem

The existence of an open interval of parameters so that an ordinary Dirichlet problem has at least three solutions is established. A completely novel assumption is emphasized. The approach is based on variational methods and critical points.     

Three-Phase Solutions of the Kadomtsev–Petviashvili Equation

The Kadomtsev–Petviashvili (KP) equation is known to admit explicit periodic and quasiperiodic solutions with N independent phases, for any integer N , based on a Riemann theta-function of N variables. For N =1 and 2, these solutions have been used successfully in physical applications. This article addresses mathematical problems that arise in the computation of theta-functions of three variables and with the corresponding solutions of the KP equation. We identify a set of parameters and their corresponding ranges, such that every real-valued, smooth KP solution associated with a Riemann theta-function of three variables corresponds to exactly one choice of these parameters in the proper range. Our results are embodied in a program that computes these solutions efficiently and that is available to the reader. We also discuss some properties of three-phase solutions.     

Constructing Solutions for the Generalized Hénon–Heiles System Through the Painlevé Test

The generalized Hénon–Heiles system is considered. New special solutions for two nonintegrable cases are obtained using the Painlevé test. The solutions have the form of the Laurent series depending on three parameters. One parameter determines the singularity-point location, and the other two parameters determine the coefficients in the Laurent series. For certain values of these two parameters, the series becomes the Laurent series for the known exact solutions. It is established that such solutions do not exist in other nonintegrable cases.     

Three dimensional axisymmetric problems in piezoelectric media: Revisited by a real fundamental solutions based new method

The purpose of the present work is to establish a set of real fundamental solutions for the differential governing equations of three dimensional axisymmetric problems in piezoelectric media. Firstly, conventional complex fundamental solutions are derived by analysis on the eigenvalue problem, and then, Euler’s formula is used to transform them into equivalent real fundamental solutions. As an example of application, the fracture problem of an axisymmetric penny-shaped crack in a piezoelectric layer is resolved by the real fundamental solutions based new method. Theoretical derivation and numerical computation are validated in the special case of a penny-shaped crack in an infinite piezoelectric body. Effects of geometrical parameters and electric-loading coefficient on energy release rates are surveyed and their agreement with the results of existing papers is also indicated. The advantage of such a real fundamental solutions based new method is that it can effectively help to avoid the difficult complex analysis in mixed boundary value problems.     

Higher nonlocal problems with bounded potential

The aim of this paper is to study a class of nonlocal fractional Laplacian equations depending on two real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci, we establish the existence of three weak solutions for nonlocal fractional problems exploiting an abstract critical point result for smooth functionals. We emphasize that the dependence of the underlying equation from one of the real parameters is not necessarily of affine type.     

Variational Approach to Fourth-Order Impulsive Differential Equations with Two Control Parameters

In this paper, we are concerned with the multiplicity of solutions for a fourth-order impulsive differential equation with Dirichlet boundary conditions and two control parameters. Using variational methods and a three critical points theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. We also provide an example in order to illustrate the main abstract results of this paper.     

Goal programming using multiple objective hybrid metaheuristic algorithm

In this paper, a Goal Programming (GP) model is converted into a multi-objective optimization problem (MOO) of minimizing deviations from fixed goals. To solve the resulting MOO problem, a hybrid metaheuristic with two steps is proposed to find the Pareto set's solutions. First, a Record-to-Record Travel with an adaptive memory is used to find first non-dominated Pareto frontier solutions preemptively. Second, a Variable Neighbour Search technique with three transformation types is used to intensify every non dominated solution found in the first Pareto frontier to produce the final Pareto frontier solutions. The efficiency of the proposed approach is demonstrated by solving two nonlinear GP test problems and three engineering design problems. In all problems, multiple solutions to the GP problem are found in one single simulation run. The results prove that the proposed algorithm is robust, fast and simply structured, and manages to find high-quality solutions in short computational times by efficiently alternating search diversification and intensification using very few user-defined parameters.     

Parameter-less algorithm for evolutionary-based optimization

The development of a simple, adaptive, parameter-less search algorithm was initiated by the need for an algorithm that is able to find optimal solutions relatively quick, and without the need for a control-parameter-setting specialist. Its control parameters are calculated during the optimization process, according to the progress of the search. The algorithm is intended for continuous and combinatorial problems. The efficiency of the proposed parameter-less algorithm was evaluated using one theoretical and three real-world industrial optimization problems. A comparison with other evolutionary approaches shows that the presented adaptive parameter-less algorithm has a competitive convergence with regards to the comparable algorithms. Also, it proves algorithm’s ability to finding the optimal solutions without the need for predefined control parameters.     

Some exact solutions for fractional generalized Burgers’ fluid in a porous space

This work is concerned with deriving the equation for describing the magnetohydrodynamic (MHD) flow of a fractional generalized Burgers’ fluid in a porous space. Modified Darcy's law has been taken into account. Closed form solutions for velocity are obtained in three problems. The solutions for Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the obtained solutions. A parametric study of some physical parameters involved in the problems is performed to illustrate the influence of these parameters on the velocity profiles.     

利用改进的(G′/G)-展开法求广义的(2+1)维Boussinesq方程的精确解

利用改进的(G′/G)-展开法,求广义的(2+1)维Boussinesq方程的精确解,得到了该方程含有较多任意参数的用双曲函数、三角函数和有理函数表示的精确解,当双曲函数表示的行波解中参数取特殊值时,便得到广义的(2+1)维Boussinesq方程的孤立波解.     

A CLASS OF SINGULARLY PERTURBED EQUATIONS FOR LARGE PARAMETER WITH THREE TURNING POINTS

A class of singularly perturbed problem of third order equation with two para-meters is studied. Using singular perturbation method, the structure of solutions to the problem is discussed in three different cases about two small parameters. The asymptotic solutions to the problem are given. The structure of solutions and the different limit behaviors are revealed. And the solutions are compared with the exact solutions to the equation in which the coefficients are constants and a relatively more perfect res...     

具有双参数拟线性微分方程的奇摄动Robin边值问题(英文)

主要研究一类具有双参数的拟线性微分方程的奇摄动Robin边值问题.利用微分不等式理论,对两参数分三种不同情形对解的构造进行分析.并得到相应问题在各情形下的渐近解和余项估计.     

Existence of positive solution for a sixth–order differential system with variable parameters

This paper investigates the existence of positive solutions for a sixth-order differential system with three variable parameters. Using a fixed point theorem and an operator spectral theorem we give some new existence results.     

Analytic rogue wave solutions for a generalized fourth‐order Boussinesq equation in fluid mechanics

A bilinear transformation method is proposed to find the rogue wave solutions for a generalized fourth‐order Boussinesq equation, which describes the wave motion in fluid mechanics. The one‐ and two‐order rogue wave solutions are explicitly constructed via choosing polynomial functions in the bilinear form of the equation. The existence conditions for these solutions are also derived. Furthermore, the system parameter controls on the rogue waves are discussed. The three parameters involved in the equation can strongly impact the wave shapes, amplitudes, and distances between the wave peaks. The results can be used to deeply understand the nonlinear dynamical behaviors of the rogue waves in fluid mechanics.     

圆柱形容器中自由表面波的三波内共振相互作用

由Luke变分原理导出了圆柱形容器中自由毛细重力表面波的基本方程.首先对速度势和自由面波高作Galerkin展开,用多重尺度法导出了控制方程前二阶摄动方程,在此基础上讨论了三个波二阶内共振的非线性相互作用,导出了三波内共振的非线性耦合作用方程和守恒律.针对非退化的作用方程,分析了相平面上平衡点的位置,研究了参数对应共振与非共振的各种情况,在不同参数情况下求出二阶作用方程的稳态解并分析了解的稳定性态,还讨论了只在有限时间内有效的解.分析表明,在非退化情况下,由于初始条件不同,3个波之间能量传递的模式不尽相同,有可能能量在3波之间周期性传递,亦可能单波的能量有衰减或增长.