一类半线性方程周期边值问题

本文运用整体反函数理论证明了一类半线性方程边值问题周期解的存在唯一性，推广和改进了已有的一些结果．     

On the inverse spectral theory in a non-homogeneous interior transmission problem

We study the inverse spectral problem in an interior transmission eigenvalue problem. The Cartwright’s theory in value distribution theory gives a connection between the distributional structure of the eigenvalues and the asymptotic behaviours of its defining functional determinants. Given a sufficient quantity of transmission eigenvalues, we prove a uniqueness of the refraction index in inhomogeneous medium as an uniqueness problem in entire function theory. The asymptotically periodical structure of the zero set of the solutions helps to locate infinitely many eigenvalues of infinite degree of freedom.     

Bifurcation modes in a nonlinear optical system with distributed field rotation

We consider a model of a nonlinear optical system with distributed field rotation described by a functional-differential diffusion equation. An existence theorem is proved for periodical spatially nonhomogeneous traveling-wave solutions, which are generated from a spatially homogeneous stationary solution by an Andronov-Hopf (cycle-generating) bifurcation. A series expansion of the solution in powers of a small parameter is obtained and a stability condition is given. Simulation results are used to discuss the properties of the model. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 89–99.     

A Method for Constructing Traveling Wave Solutions to Nonlinear Evolution Equations

In this paper, an analytical method is proposed to construct explicitly exact and approximate solutions for nonlinear evolution equations. By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation are obtained explicitly. These solutions include solitary wave solutions, singular traveling wave solutions and periodical wave solutions. These results indicate that in some cases our analytical approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations. It can also be applied to some related nonlinear dynamical systems.     

An impulsive ratio-dependent predator–prey system with diffusion

We investigate the predator–prey system with diffusion, when biological and environmental parameters are assumed to change in periodical manner over time. The system is affected by impulses which can be considered as a control. Conditions for the permanence of the predator–prey system and for the existence of a unique globally stable periodic solutions are obtained.     

Dynamics of a neutral delay equation for an insect population with long larval and short adult phases

We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.     

Hopf bifurcation direction of a delayed bacteria-immunity system with quorum sensing mechanism

Considering the mechanism of quorum sensing, we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells on the basis of Zhang’s model (see [9] for more details). A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In the sequel, the length of delay which preserves the stability of the positive equilibrium is estimated, and the existence of Hopf bifurcation when the delay crosses through a critical value is investigated. Further, by using the normal form theory and center manifold theory, the explicit formulae are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.     

Nonlinear Schrödinger equation containing the time-derivative of the probability density: A numerical study

A nonlinear Schrödinger equation that contains the time-derivative of the probability density is investigated, which is motivated by the attempt to include the recoil effect of radiation. This equation has the same stationary solutions as its linear counterpart, and these solutions are the eigen-states of the corresponding linear Hamiltonian. The equation leads to the usual continuity equation and thus maintains the normalization of the wave function. For the non-stationary solutions, numerical calculations are carried out for the one-dimensional infinite square-well potential (1D ISWP) and for several time-dependent potentials that tend to the former as time increases. Results show that for various initial states, the wave function always evolves into some eigen-state of the corresponding linear Hamiltonian of the 1D ISWP. For a small time-dependent perturbation potential, solutions present the process similar to the spontaneous transition between stationary states. For a periodical potential with an appropriate frequency, solutions present the process similar to the stimulated transition. This nonlinear Schrödinger equation thus presents the state evolution similar to the wave-function reduction.     

具有初挠度的柔韧圆板的振动问题

本文推导出具有初挠度柔韧圆板的振动方程,在相平面上讨论了运动稳定性.用Galerkin法和Lindstedt-Poincaré摄动法求得具有初挠度圆板非线性振动的周期解,讨论了初挠度对柔韧圆板的动力特性的影响.     

Solutions for the fractional Landau-Lifshitz equation

This article considers the dynamic equation of a reduced model for thin-film micromagnetics deduced by A. DeSimone, R.V. Kohn and F. Otto in [A. DeSimone, R.V. Kohn, F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1-53]. To derive the existence of weak solutions under periodical boundary condition, the authors first prove the existence of smooth solutions for the approximating equation, then prove the convergence of the viscosity solution when the viscosity term vanishes, which implies the existence of solutions for the original equation.     

各向异性材料界面周期刚性线夹杂的反平面问题

研究了两种各向异性材料界面含周期分布刚性线夹杂的反平面剪切问题。运用复变函数方法，获得了封闭形式解答，并给出了刚性线尖端应力场公式，从该文解答的特殊情形，可直接导出各向同性材料界面以及均匀各向异性材料中相应问题的公式，其极限情形与已有的结果吻合。     

Periodical states and marching groups in a closed owari

Owari is an old African game that consists of cyclically ordered pits that are filled with pebbles. In a sowing move all the pebbles are taken out of one pit and distributed one by one in subsequent pits. Repeated sowing will give rise to recurrent states of the owari. Bouchet studied such periodical states in an idealised setup, where there are infinitely many pits. We characterise periodical states in owaris with finitely many pits. Our result implies Bouchet's result.     

Synchronization of time-delay coupled pulse oscillators

We present a detailed study of the dynamics of pulse oscillators with time-delayed coupling. We get the return maps, obtain strict solutions and analyze their stability. For the case of two oscillators, a periodical structure of synchronization regions is found in parameter space, and the regions corresponding to in-phase and antiphase regimes alternate with growth of time delay. Two types of switching between in-phase and antiphase regimes are studied. We also show that for different parameters coupling delay may have synchronizing or desynchronizing effect. Another novel result is that phase locked regimes exist for arbitrary large values. The specificity of system dynamics with large delay is studied.     

Computations of critical groups and applications to asymptotically linear wave equation and beam equation

In this paper, by using the Morse index theory for strongly indefinite functionals developed in [Nonlinear Anal. TMA, in press], we compute precisely the critical groups at the origin and at infinity, respectively. The abstract theorems are used to study the existence (multiplicity) of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero.     

Hopf bifurcation of a bacteria-immunity system with delayed quorum sensing

On the basis of Zhang’s model (see [P. Fergola, J. Zhang, M. Cerasuolo, Z.E. Ma, On the influence of quorum sensing in the competition between bacteria and immune system of invertebrates, in: Collective Dynamic: Topics on Competition and Cooperation in the Biosciences: A Selection of Papers in the Proceedings of the BIOCOMP2007 International Conference, AIP Conference Proceedings, vol. 1028, 2008, pp. 215-232] for more details), we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells with bacterial quorum sensing mechanism. A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. Subsequently, the length of delay which preserves the stability of the positive equilibrium is estimated and Hopf bifurcation occurs when time delay crosses through a critical value are researched. Further, by using the normal form theory and center manifold theory, the explicit formulaes are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.     

Group classification and symmetry reduction of a (2+1) dimensional diffusion-advection equation

Based on classical Lie group method, we consider the continuum problem of the driven diffusive flow of particles past an impenetrable obstacle (rod) of length L. The infinitesimals of the diffusion-advection equation in (2+1) dimensions were found for an arbitrary nonlinear advection. The symmetries corresponding to different forms of the nonlinear advection are obtained. Three models are studied in details. The results show that the presence of an obstacle, whether stationary or moving, in a driven diffusive flow with nonlinear drift will distort the local concentration profile to a state which divided the (x, y)-plane into two regions. The concentration is relatively higher in one side than the other side, apart from the value of where D is the diffusion coefficient and υ is the drift velocity. This problem has relevance for the size segregation of particulate matter which results from the relative motion of different-size particles induced by shaking. Also, the obtained solutions include soliton, periodical, rational and singular solutions. Received: November 10, 2003; revised: February 10, 2004     

ON A MIN-MAX THEOREM

In this paper we prove a nonvariational form of a rain-max problem in [l] and give its application to a periodical problem.     

Fiber Buckling in a Viscoelastic Matrix

Within the framework of the three-dimensional linearized theory of stability, an approach for investigating fiber buckling in the structure of unidirectional fibrous viscoelastic composites is developed. For simplicity, a small fiber concentration is considered, and the buckling problem for a single elastic fiber in an infinite viscoelastic matrix is investigated. In this case, it is assumed that the fiber has an insignificant initial periodical imperfection, and the growth of this imperfection with time is studied. The state where this imperfection starts to grow indefinitely is taken as a fiber-buckling criterion, and the critical time is determined from this criterion.     

On a Synthetic Jet Flow

Velocity vector field of a synthetic jet has been investigated experimentally. The synthetic jet is generated by an electrodynamical system designed for control of a free shear layer. To catch unsteady periodical structures the phase average method is applied to time series of instantaneous velocities. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A second-order reduced multiscale approach for non-linear axisymmetric structures with periodic configurations

An effective second-order reduced multiscale (SORM) approach is proposed for axisymmetric inelastic heterogeneous structures with periodic configurations. The axisymmetric structures studied in this work are periodical in radial and axial directions and homogeneous in circumferential directions. At first, the high-order linear and non-linear local solutions at microscale are gotten by solving distinct multiscale auxiliary functions. Further, the homogenized parameters are introduced, and the related non-linear homogenization equations defined on global domain are given. The significant feature of the presented approaches are (i) high-order homogenization solutions that do not require high-order continuities of the coarse-scale solutions, (ii) a novel reduced-model form based on transformation field analysis (TFA) to analyze non-linear local cell problem with less computing time compared with direct numerical simulations and (iii) a new SORM algorithm derived for simulating the axisymmetric inelastic structures. Finally, by some representative examples, the effectiveness and accuracy of the presented algorithm are confirmed.