Multiplicity and stability of time-periodic solutions of Ginzburg-Landau equations of superconductivity

In this article we shall show that the Ginzburg-Landau equations admit at least three time-periodic solutions. One of the time-periodic solutions describes the non-superconductive (or normal) state and the other one describes the superconductivity state. We will also show that the time-periodic solutions are exponentially stable. Furthermore, the method we use in this article can be used to find numerical approximations to the time-periodic solutions.     

Existence of periodic solutions and their asymptotic stability to the Navier–Stokes equations with the Coriolis force

We consider the time-periodic problem for the Navier–Stokes equations in the rotational framework. We prove the unique existence of time-periodic solutions for the prescribed external force. Furthermore, we also show the asymptotic stability of small time-periodic solutions provided the initial disturbance is sufficiently small.     

Time-periodic solution to a nonlinear parabolic type equation of higher order

In this paper, the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a class of parabolic type equation of higher order are proved by Galerkin method.     

Time-periodic Solutions to Systems of Conservation Laws with Ellipticity

We study the time-periodic solutions to systems of conservation laws with ellipticity. It is shown that under time-periodic boundary condition, the system admits at least one global time-periodic solution bounded uniformly with the same period.     

Time-periodic solutions of the Einstein's field equations Ⅰ:general framework

In this paper,we develop a new algorithm to find the exact solutions of the Einstein's field equations.Time-periodic solutions are constructed by using the new algorithm.The singularities of the time-periodic solutions are investigated and some new physical phenomena,such as degenerate event horizon and time-periodic event horizon,are found.The applications of these solutions in modern cosmology and general relativity are expected.     

抛物型时间周期问题的Schwarz波形松驰方法

周期稳态是科学和工程系统中一类重要的运行状态,其计算复杂度远高于相应的初值问题,因此有更迫切的并行计算需要.我们提出了计算抛物型方程时间周期解的并行方法—基于区域分解(又称Schwarz方法)的波形松驰方法,该方法只需在子区域上求解较低维的周期问题.我们分析了两种不同的传输条件下方法的收敛性,并用数值实验支持了理论结果.     

Non-existence of small-amplitude doubly periodic waves for dispersive equations

We formulate the question of the existence of spatially periodic, time-periodic solutions for evolution equations as a fixed point problem, for certain temporal periods. We prove that if a certain estimate applies for the Duhamel integral, then time-periodic solutions cannot be arbitrarily small. This provides a partial analogue in the spatially periodic case of scattering results for dispersive equations on the real line, as scattering implies the non-existence of small-amplitude traveling waves. Furthermore, it also complements small-divisor methods (e.g., the Craig–Wayne–Bourgain method) for proving the existence of small-amplitude time-periodic solutions (again, for frequencies in certain set).     

Time-periodic solutions of the Einstein’s field equations II: geometric singularities

In this paper, we construct several kinds of new time-periodic solutions of the vacuum Einstein’s field equations whose Riemann curvature tensors vanish, keep finite or take the infinity at some points in these space-times, respectively. The singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.     

Existence of periodic weak solutions to the three-dimensional nonlinear viscoelastic system

In this paper, we discuss the time-periodic solutions to a general three-dimensional nonlinear viscoelastic system with Dirichlet boundary condition. Employing the singularity of the integral kernel, we obtain the energy estimates in Sobolev spaces with fractional index and then show the existence of time-periodic solutions to the problem for viscoelastic solid and liquid models, respectively.     

Asymptotics of Small Exterior Navier-Stokes Flows with Non-Decaying Boundary Data

We prove the existence of unique solutions for the 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay in time. As a corollary, the existence of unique small time-periodic solutions is shown. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically self-similar, then the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity.     

Forced oscillations of nonlinear damped equation of suspended string

We shall study the existence of time-periodic solutions of nonlinear damped equation of suspended string to which a periodic nonlinear force works. We shall be conterned with weak, strong and classical time-periodic solutions and also the regularity of the solutions. To formulate our results, we shall take suitable weighted Sobolev-type spaces introduced by [M. Yamaguchi, Almost periodic oscillations of suspended string under quasiperiodic linear force, J. Math. Anal. Appl. 303 (2) (2005) 643-660; M. Yamaguchi, Infinitely many time-periodic solutions of nonlinear equation of suspended string, Funkcial. Ekvac., in press]. We shall study properties of the function spaces and show inequalities on the function spaces. To show our results we shall apply the Schauder fixed point theorem and the fixed point continuation theorem in the function spaces.     

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic L ⁿ -solution is unique within the class of solutions which have sufficiently small L ^∞(L ⁿ )-norm. In this paper, we show that a small time-periodic L ⁿ -solution is unique within the class of all time-periodic L ⁿ -solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L ⁿ space. The proof of the present uniqueness theorem is based on the method of dual equations.      

Energy method for multi-dimensional balance laws with non-local dissipation

In this paper, we are concerned with a class of multi-dimensional balance laws with a non-local dissipative source which arise as simplified models for the hydrodynamics of radiating gases. At first we introduce the energy method in the setting of smooth perturbations and study the stability of constants states. Precisely, we use Fourier space analysis to quantify the energy dissipation rate and recover the optimal time-decay estimates for perturbed solutions via an interpolation inequality in Fourier space. As application, the developed energy method is used to prove stability of smooth planar waves in all dimensions n?2, and also to show existence and stability of time-periodic solutions in the presence of the time-periodic source. Optimal rates of convergence of solutions towards the planar waves or time-periodic states are also shown provided initially L¹-perturbations.     

A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations

This paper is concerned with a time-periodic reaction-diffusion equation. It is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation. Such solutions are called subharmonic solutions. In this paper, for any domain Ω, time-period τ>0 and integer n?2, we construct an example of a time-periodic reaction-diffusion equation on Ω with a minimal period τ which possesses a stable solution of minimal period nτ.     

Oscillating solitons of the driven,damped nonlinear SchrÖdinger equation

We obtain time-periodic solitons of the parametrically driven, damped nonlinear Schrödinger equation as solutions of the boundary value problem on a two-dimensional domain. We classify the stability and bifurcations of singly and doubly periodic solutions.     

Existence and stability of periodic solutions for parabolic systems with time delays

This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.     

The viscous Cahn�CHilliard equation with periodic potentials and sources

We investigate the existence, uniqueness and asymptotic behavior of solutions to the viscous Cahn–Hilliard equation with time-periodic potentials and sources.     

Breathers in a damped-driven nonlinear Schrödinger equation

We construct time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation as solutions of a boundary value problem for the space-dependent Fourier coefficients.     

Existence of Time Periodic Solutions to Boundary Value Problem of One-dimensional Semilinear Viscoelastic Dynamic Equation with Memory

In this paper, we prove the existence of time-periodic solutions to the boundary value problem of semilinear one-dimensional dynamical equation for viscodastic materials.     

Time-periodic Solutions to the Ginzburg-Landau-BBM Equations

In this paper, the existence and uniqueness of the time-periodic solutions to the Ginzburg-Landau-BBM equations are proved by using a priori estimates and Leray-Schauder fixed point theorem.