Spectral analysis for periodic solutions of the Cahn�CHilliard equation on {\mathbb{R}}

We consider the spectrum associated with the linear operator obtained when the Cahn–Hilliard equation on \mathbbR{\mathbb{R}} is linearized about a stationary periodic solution. Our analysis is particularly motivated by the study of spinodal decomposition, a phenomenon in which the rapid cooling (quenching) of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions of different crystalline structure, separated by steep transition layers. In this context, a natural problem regards the evolution of solutions initialized by small, random (in some sense) perturbations of the pre-quenching homogeneous state. Solutions initialized in this way appear to evolve transiently toward certain unstable periodic solutions, with the rate of evolution described by the spectrum associated with these periodic solutions. In the current paper, we use Evans function methods and a perturbation argument to locate the spectrum associated with such periodic solutions. We also briefly discuss a heuristic method due to Langer for relating our spectral information to coarsening rates.     

On solutions of nonlinear Schrödinger equations with Cantor-type spectrum

The solvability of the Cauchy problem for the Nonlinear Nonfocusing Schrödinger equation (NNSE) with almost periodic initial data satisfying certain conditions is studied. It is shown that solutions are uniform almost periodic functions with respect to each variable. An example of initial data with Cantor-type spectrum is given. The Cauchy problem for NNSE is solved in the class of limit periodic functions which are well approximated by periodic ones.     

Stability of a critical nonlinear neutral delay differential equation

This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about the asymptotic stability of constant and periodic solutions. A complete analysis of the stability diagram is given in the linear homogeneous case. Under periodic forcing, existence of periodic solutions is discussed, involving a Diophantine condition on the period of the source.     

跳跃非线性项依赖于导数的二阶微分方程周期解的存在性

本文研究二阶微分方程χ"+aχ+-bχ-+f(χ)g(χ'=p(t)周期解的存在性,这里χ+=max{χ,0},χ-=max{-χ,0},a,6是正常数并且点(a,b)位于某一条Fucik谱曲线上.当g(χ)的极限lim.g(χ)=g(+∞),lim g(χ)=g(-∞)和f(χ)的极限lim,g(χ)=f(+∞),lim f(χ)=f(-∞)都存在且有限时,给出了此方程存在周期解的充分条件.     

The Periodic Solutions of a Nonhomogeneous String with Dirichlet-Neumann Condition

This paper deals with the existence of periodic solutions of a nonhomogeneous string with Dirichlet-Neumann condition. The authors consider the case that the period is irrational multiple of space length and prove that for some irrational number, zero is not the accumulation point of the spectrum of the associated linear operator. This result can be used to prove the existence of the periodic solution avoid using Nash-Moser iteration.     

The Stability Analysis of the Periodic Traveling Wave Solutions of the mKdV Equation

The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. In this paper, we derive all periodic traveling wave solutions of the focusing and defocusing mKdV equations. We show that in the defocusing case all such solutions are orbitally stable with respect to subharmonic perturbations: perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution. We do this by explicitly computing the spectrum and the corresponding eigenfunctions associated with the linear stability problem. Next, we bring into play different members of the mKdV hierarchy. Combining this with the spectral stability results allows for the construction of a Lyapunov function for the periodic traveling waves. Using the seminal results of Grillakis, Shatah, and Strauss, we are able to conclude orbital stability. In the focusing case, we show how instabilities arise.     

Boundedness and Almost Periodicity of Solutions of Partial Functional Differential Equations

We study necessary and sufficient conditions for the abstract functional differential equation x=Ax+Fx_t+f(t) to have almost periodic, quasi periodic solutions with the same structure of spectrum as f. The main conditions are stated in terms of the imaginary solutions of the associated characteristic equations and the spectrum of the forcing term f. The obtained results extend recent results to abstract functional differential equations.     

Spectral Analysis for Stationary Solutions of the Cahn–Hilliard Equation in ℝ d

We consider the spectrum associated with three types of bounded stationary solutions for the Cahn–Hilliard equation on ? ^d , d ≥ 2: radial solutions, saddle solutions (only for d = 2), and planar periodic solutions. In particular, we establish spectral instability for each type of solution. The important case of multiply periodic solutions does not fit into the framework of our approach, and we do not consider it here.     

Existence of periodic solutions for the Newtonian equation of motion with p‐Laplacian operator

In this paper, we study the existence of periodic solutions for the Newtonian equation of motion with p ‐Laplacian operator by asymptotic behavior of potential function, establish some new sufficient criteria of existence of periodic solutions for the differential system under the frame of Fuc?ik spectrum, generalize and improve some known works, and give an example to illustrate the application of the theorems. Copyright © 2017 John Wiley & Sons, Ltd.     

Analysis of grazing bifurcation from periodic motion to quasi-periodic motion in impact-damper systems

A peculiar discontinuous bifurcation phenomenon that the periodic solution directly jumps to quasi-periodic attractor through grazing bifurcation is reported in this paper. This phenomenon is revealed in the impact damper system by the spectrum of the largest Lyapunov exponent in parameter plane. The origin of the quasi-periodic attractor and coexistence of solutions are analyzed. And the MDCM (multi-DOF cell mapping) method is used to reveal the variety of attraction basins of solutions.     

Periodic solutions for second order equations with time-dependent potential via time map

In this paper the approach via time-map is used to investigate the existence of periodic solutions for second order equation with time-dependent potential. Our theorem generalized the results obtained by Ding, Habets, Omari and Zanolin which are for time-independent potential and relate the behavior of the primitive of the nonlinearity with respect to the Fu?ik spectrum of the periodic problem.     

Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations

The Navier-Stokes-α equations subject to the periodic boundary conditions are considered.An-alyticity in time for a class of solutions taking values in a Gevrey class of functions is proven.Exponentialdecay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by theexponential decay are also obtained.     

Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.     

Semiclassical dynamics and coherent soliton condensates in self-focusing nonlinear media with periodic initial conditions

The semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of ICs, referred to as “periodic single-lobe” potentials, share the same qualitative features, which also coincide with those of solutions arising from localized ICs. The spectrum of the associated scattering problem in each of these cases is then numerically computed, and it is shown that such spectrum is confined to the real and imaginary axes of the spectral variable in the semiclassical limit. This implies that all nonlinear excitations emerging from the input have zero velocity, and form a coherent nonlinear condensate. Finally, by employing a formal Wentzel-Kramers-Brillouin expansion for the scattering eigenfunctions, asymptotic expressions for the number and location of the bands and gaps in the spectrum are obtained, as well as corresponding expressions for the relative band widths and the number of “effective solitons.” These results are shown to be in excellent agreement with those from direct numerical computation of the eigenfunctions. In particular, a scaling law is obtained showing that the number of effective solitons is inversely proportional to the small dispersion parameter.     

Operator Hamiltonians and anticanonic equations with periodic coefficients

In a Hilbert space we study Hamiltonians and anticanonic equations with periodic coefficients. We prove existence theorems for the solutions of ill-posed Cauchy problems for the given equations. Following Krein we define the notion of the genus of the spectrum points of the monodromy operator of an equation of the class being studied. We formulate existence and uniqueness theorems for the solutions when determining the reflected and the transmitted waves for a specified incident wave. The theory developed is applied to the study of cylindrical waveguides with a periodic filling.Translated from Problemy Matematicheskogo Analiza. No. 4: Integralnye i Differentsial'nye Operatory. Differentsial'nye Uraveniya, pp. 9–36, 1973.     

Boundary feedback stabilization of Boussinesq equations

The aim of this work is to design oblique boundary feedback controller for stabilizing the equilibrium solutions to Boussinesq equations on a bounded and open domain in R~2. Two kinds of such feedback controller are provided, one is the proportional stabilizable feedback control, which is obtained by spectrum decomposition method, while another one is constructed via the Ricatti operator for an infinite time horizon optimal control problem.An example of periodic Boussinesq flow in 2-D channel is also given.     

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.     

A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations

In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vector fields that are asymptotically additive in a well-defined sense. This result is used to obtain multibump solutions for two basic types of periodic stationary Schrödinger equations with superlinear nonlinearity. The nonlinear term may be of convolution type. If the superquadratic term in the energy functional is convex, our results also apply in certain cases if 0 is in a gap of the spectrum of the Schrödinger operator.     

D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

Non-smooth modal analysis is an extension of modal analysis to non-smooth systems, prone to unilateral contact conditions for instance. The problem of a one-dimensional bar subject to unilateral contact on its boundary has been previously investigated numerically and the corresponding spectrum of vibration could be partially explored. In the present work, the non-smooth modal analysis of the above system is reformulated as a set of functional equations through the use of both d’Alembert solution to the wave equation and the method of steps for Neutral Delay Differential Equations. The system features a strong internal resonance condition and it is established that irrational and rational periods of vibration should be carefully distinguished. For irrational periods, it was previously proven that the displacement field of the non-smooth modes of vibration is characterized with piecewise-linear functions in space and time and such a motion is unique for a prescribed energy. However, for rational periods, which are the subject of this work, new periodic solutions are found analytically. Findings consist of families of iso-periodic solutions with piecewise-smooth displacement fields in space and time and continua of piecewise-smooth periodic solutions of the same energy and frequency.     

离散大系统在结构扰动下周期解的存在性

对于离散系统稳定性的研究,近年来受到人们的重视,但对于周期解的研究,在文献中还很少看到。本文首先讨论了离散系统解的有界性,并且得到了若一个具有周期系数的差分方程的解为最终有界的,则存在周期解的结果。然后利用李雅普诺夫函数方法研究了离散大系统在结构扰动之下周期解的存在性和离散大系统的平稳振荡。