Dynamics in coupled oscillators with recurrent/random forcing, a PDE approach

The current paper is devoted to the study of coupled oscillators with recurrent/random forcing. Special attention is given to the solutions having the same recurrence/randomness as that of the forcing (recurrent/random solutions for short). By embedding coupled oscillators into coupled parabolic equations, it establishes a general theorem on the existence of recurrent/random solutions. It also finds conditions under which such solutions are unique. When the recurrent forcing is actually quasi-periodic or almost periodic, recurrent solutions are refereed to as quasi-periodic or almost periodic solutions in a weak sense and they are quasi-periodic or almost periodic in the classical sense under the uniqueness conditions. In addition, applications of the general theory to coupled Duffing type oscillators and Josephson junctions are considered and the results obtained extend several existing ones for quasi-periodic Duffing oscillators.     

关于高阶整函数系数微分方程解的增长性的进一步结果

本文研究一类高阶整函数系数微分方程的增长性问题，当存在某个系数对方程的解的性质起主要支配作用时，得到了齐次与非齐次方程解的超级的精确估计及方程的解与小函数的关系。     

Exact solutions for a class of nonlinear evolution equations: A unified ansätze approach

In this paper, we propose a modified generalized transformation for constructing analytic solutions to nonlinear differential equations. This improved unified ansätze is utilized to acquire exact solutions that are general solutions of simpler equations that are either integrable or possess special solutions. The ansätze is constructed via the choice of an integrable differential operator or a basis set of functions. The technique is implemented to obtain several families of exact solutions for a class of nonlinear evolution equations with nonlinear term of any order. In particular, the Klein–Gordon, the Sine–Gordon and Landau–Ginburg–Higgs equations are chosen as examples to illustrate the method.     

Tronquée solutions of the painlevé II equation

We study special solutions of the Painlevé II (PII) equation called tronquée solutions, i.e., those having no poles along one or more critical rays in the complex plane. They are parameterized by special monodromy data of the Lax pair equations. The manifold of the monodromy data for a general solution is a twodimensional complex manifold with one- and zero-dimensional singularities, which arise because there is no global parameterization of the manifold. We show that these and only these singularities (together with zeros of the parameterization) are related to the tronquée solutions of the PII equation. As an illustration, we consider the known Hastings-McLeod and Ablowitz-Segur solutions and some other solutions to show that they belong to the class of tronquée solutions and correspond to one or another type of singularity of the monodromy data.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Explicit periodic travelling waves for a discrete lambda–omega reaction–diffusion system

In 1973, Kopell and Howard introduced a λ–ω reaction–diffusion system and found an explicit family of periodic travelling wave solutions lying on circles with radius less than 1. Since λ–ω systems represent universal models for studying chemical processes, and onset of turbulent behaviour, etc., explicit solutions of λ–ω systems with delays or discrete λ–ω systems can be of further help when the only method for obtaining other solutions is through numerical computation. There are now much investigations of various λ–ω systems. However, it is of interest to note that none attempts to find explicit travelling wave solutions. In this paper, we investigate the existence of explicit solutions for the simplest Euler scheme of a λ–ω system with delays or advancements which is described as a coupled pair of partial difference equations. We are able to provide necessary as well as sufficient conditions for the existence of numerical periodic travelling wave solutions. Additionally, we also provide some examples to show that our explicit solutions are qualitatively different from those found by Kopell and Howard and hence they may be of interests for specialists in the area of reaction–diffusion systems.     

Positive Solutions for a Lotka–Volterra Prey–Predator Model with Cross-Diffusion of Fractional Type

In this paper we consider a Lotka–Volterra prey–predator model with cross-diffusion of fractional type. The main purpose is to discuss the existence and nonexistence of positive steady state solutions of such a model. Here a positive solution corresponds to a coexistence state of the model. Firstly we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system. Secondly we derive some necessary conditions to ensure the existence of positive solutions, which demonstrate that if the intrinsic growth rate of the prey is too small or the death rate (or the birth rate) of the predator is too large, the model does not possess positive solutions. Thirdly we study the sufficient conditions to ensure the existence of positive solutions by using degree theory. Finally we characterize the stable/unstable regions of semi-trivial solutions and coexistence regions in parameter plane.     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

Derivation of the particle dynamics from kinetic equations

The microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov are considered. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, the so called approximate microscopic solutions are constructed. These solutions are continuous and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the timereversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.     

Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions

This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution     

One-Dimensional Pressureless Gas Systems with/without Viscosity

A general global existence result for one-dimensional pressureless Euler/Euler-Poisson systems with or without viscosity is obtained by employing the “sticky particles” model. We first construct entropy solutions for some appropriate scalar conservation laws, then we show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. Another novel contribution is the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric.     

具有Neumann边界的抛物型方程组的反馈能稳性

该文研究了一类半线性抛物型方程组平衡解的有限维反馈能稳性, 控制器由对应于平衡解的线性化系统的LQ问题所导出. 该文发现在受控系统中如何施加控制是与平衡解有关的, 对于不同的平衡解, 在受控系统中应施加不同的控制.     

向量优化问题的一类非线性标量化定理

利用Gertewitz泛函研究向量优化问题的一类非线性标量化问题. 证明了向量优化问题的(C, \varepsilon)-弱有效解或(C, \varepsilon)-有效解与标量化问题的近似解或严格近似解间的等价关系, 并估计了标量化问题的近似解．     

Roles of weight functions in a nonlinear nonlocal parabolic system

This paper deals with a semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. We establish the conditions for global and non-global solutions. It is interesting to observe that the weight functions for the nonlocal Dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or non-global, but also whether (for the non-global solutions) the blowing up occurs for any positive initial data or just for large ones.     

DISCONTINUOUS FORCING OF PERIODIC SOLUTIONS IN C1 VECTOR FIELDS WITH APPLICATIONS TO POPULATION MODELS

Averaging methods are used to compare solutions of two-dimensional systems of ordinary differential equations with constant or periodic forcing. The asymptotic separation of solutions of the periodically forced equations from the solutions of the constantly forced equations is proportional to the L¹ norm of the periodic forcing terms. This result is applied to population models of Kolmogorov-type with climax fitness functions where forcing represents stocking or harvesting of a population. The asymptotic behavior of such systems may be controlled, to some extent, by varying the period and/or amplitude of the forcing functions.     

Analysis of the heterogeneous multiscale method for elliptic homogenization problems

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

     

Exact solutions of nonlinear generalizations of the wave equation

Exact solutions of nonlinear generalizations of the wave equation are constructed. In some cases these solutions are solitary waves or solitions. Thus, by explicit construction solitons or solitary waves are shown to exist in dispersionless systems. In contrast to previous solitary wave solutions, these solutions are limiting cases of solutions of nonlinear partial differential equations with dispersion.     

A unified ansätze for the solution of nonlinear differential equations

The aim of the paper is to propose a generalized ansätze for constructing exact solutions to nonlinear ordinary differential equations. This unified transformation is manipulated to acquire analytical solutions that are general solutions of simpler linear or nonlinear systems of ordinary differential equations that are either integrable or possess special solutions. The method is implemented to obtain several families of traveling wave solutions for a class of nonlinear evolution equations and for higher order wave equations of KdV type (I).     

Asymptotic behavior of solutions of a periodic diffusion system of plankton allelopathy

This paper is concerned with the existence and asymptotic behavior of periodic solutions for a periodic reaction diffusion system of a planktonic competition model under Dirichlet boundary conditions. The approach to the problem is by the method of upper and lower solutions and the bootstrap argument of Ahmad and Lazer. It is shown under certain conditions that this system has positive or semi-positive periodic solutions. A sufficient condition is obtained to ensure the stability and global attractivity of positive periodic solutions.     

Regular and extremal solutions for difference equations with generalized phi-Laplacian

Non-oscillatory solutions for second-order difference equations with generalized phi-Laplacian are studied. Solutions are classified according to the asymptotic behaviour as regular or extremal solutions. Their existence and possible coexistence are investigated as well. In particular, the existence of infinitely many extremal solutions for equations with the discrete mean curvature operator is proved by means of an iterative method. This paper is completed by examples and some open problems.