Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

Convergence theorems for solutions and energy functionals of boundary value problems in thick multilevel junctions of a new type with perturbed neumann conditions on the boundary of thin rectangles

Boundary value problems for the Poisson equation are considered in a multilevel thick junction consisting of a junction body and a lot of alternating thin rectangles of two levels depending on their lengths. Rectangles of the first level have a finite length, whereas rectangles of the second level have a length ε ^α , 0 < α < 1, where ε is the alternation period. On the boundary of thin rectangles, an inhomogeneous Neumann boundary condition involving additional perturbation parameters is imposed. We prove convergence theorems for solutions and energy integrals. Regarding the convergence of solutions of the original problem to solutions of the homogenized problem, we establish some (auxiliary) estimates necessary for obtaining the convergence rate. Bibliography: 48 titles. Illustrations: 3 figures. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 113–132.     

An asymptotic derivation of weakly nonlinear ray theory

Using a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an errorO(ε²) where ε is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhat’s theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory.     

Random evolutions with locally independent increments on increasing time intervals

Three main schemes of limit theorems for random evolutions are discussed: averaging, diffusion approximation, and the asymptotics of large deviations. Markov stochastic evolutions with locally independent increments on increasing time intervals T _ε  = t/ε → ∞, ε → 0, are considered. The asymptotic behavior of random evolutions is investigated with the use of solutions of the singular perturbation problems for reducibly invertible operators.     

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionu _t=εDu _xx+(A+εA ₁)u+F(u),u ∈ ℝ². These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0. One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε ^−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.     

Affine properties and injectivity of quasi-isometries

We approximateε-quasi-isometries between finite-dimensional Banach spaces by linear near-isometries. In this way we improve and extend a theorem of John. We also improve results of Gevirtz on injectivity criteria for quasi-isometries. Our approach is to show thatε-quasi-isometries almost satisfy the Jensen functional equation and to use then known facts about linear approximation of approximate solutions of Jensen’s equation.     

Shape Analysis of Bounded Traveling Wave Solutions and Solution to the Generalized Whitham-Broer-Kaup Equation with Dissipation Terms

This paper deals with the problem of the bounded traveling wave solutions'shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equati...     

Almost global wellposedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/ε] for initial data of the form ε Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,e ^T/ε ]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation. Financial support provided in part by NSF grant DMS-0400643.     

On a biharmonic equation involving nearly critical exponent

This paper is concerned with a biharmonic equation under the Navier boundary condition , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in , n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P _−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x ₀ ∈Ω as ε → 0, moreover x ₀ is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x ₀ of the Robin’s function, there exist solutions of (P _−ε) concentrating around x ₀ as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P _−ε), the supercritical problem (P _+ε) has no solutions which concentrate around a point of Ω as ε → 0. Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena Pacella for their constant support.     

THE ASYMPTOTIC THEORY OF INITIAL VALUEPROBLEMS FOR SEMILINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

This paper deals with the asymptotic theory of initial value problems for semilinear waveequations in three space dimensions. The well-posedness and validity of formal approximations ona long time scale of order |ε|^-1 are discussed in the classical sense of C^2 This result describes aceu-ratively the approximations of solutions. At the end of this paper an application of the asymptotictheory is given to analyze a special model for a perturbed wave equation,     

Null-controllability of a Hyperbolic Equation as?Singular Limit of Parabolic Ones

This article considers a hyperbolic equation perturbed by a vanishing viscosity term depending on a small parameter ε>0. We show that the resulting parabolic equation is null-controllable. Moreover, we provide uniform estimates, with respect to ε, for the parabolic controls and we prove their convergence to a control of the limit hyperbolic equation. The method we use is based on Fourier expansion of solutions and the analysis of a biorthogonal sequence to a family of complex exponential functions.     

Stability conditions for singularly perturbed delay differential equations

Singular perturbation problems containing a small positive parameter ε occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. A uniformly valid, reliable interpretable approximation of such problems is required. This paper provides sufficient conditions to ensure the exponential stability of the analytical and numerical solutions of the singularly perturbed delay differential equations with a bounded time-lag for suf.ciently small ε > 0. The Halanay inequality is used to prove the main results of the paper. A numerical example is provided to illustrate the methodology and clarify the need for a stiff solver for numerical solutions of these problems.     

Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term

In this paper, we study the asymptotic behavior of solutions u _ε of the initial boundary value problem for parabolic equations in domains W_e ì \mathbbRⁿ {\Omega_\varepsilon } \subset {\mathbb{R}^n} , n ≥ 3, perforated periodically by balls with radius of critical size ε ^α , α = n/(n − 2), and distributed with period ε. On the boundary of the balls a nonlinear third boundary condition is imposed. The weak convergence of the solutions u _ε to the solution of an effective equation is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and a corrector result with respect to the energy norm is proved.     

Scaling Methods and Approximative Equations for Homogeneous Reaction—Diffusion Systems and Applications to Epidemics

Summary. We consider a reaction-diffusion equation that is homogeneous of degree one. This homogeneity is a symmetry. The dynamics is factorized into trivial evolution due to symmetry and nontrivial behavior by a projection to an appropriate hypermanifold. The resulting evolution equations are rather complex. We examine the bifurcation behavior of a stationary point of the projected system. For these purposes we develop techniques for dimension reduction similar to the Ginzburg-Landau (GL) approximation, the modulation equations. Since we are not in the classical GL situation, the remaining approximative equations have a quadratic nonlinearity and the amplitude does not scale with ε but with ε ² where ε ² denotes the bifurcation parameter. Moreover, the symmetry requires that not only one but two equations are necessary to describe the behavior of the system. We investigate traveling fronts for the modulation equations. This result is used to analyze an epidemic model. Received April 9, 1996; second revision received January 3, 1997; final revision received October 7, 1997; accepted January 19, 1998     

The central connection problem at turning points of linear differential equations

A system of linear differential equations of the vectorial form εdy/dx=A (x, ε) y is considered, where ε is a positive parameter, and the matrixA (x, ε) is holomorphic in |x|⩽x ₀, 0 < ε ⩽ ε₀ , with an asymptotic expansionsA (x, ε) ∼ ∑ _r=0 ^∞ A _r (x) ε ^r , as ε→0. The eigenvalues ofA ₀(x) are supposed to coalesce atx=0 so as to make this point a simple turning point. With the help of refinements of the representations for the inner and outer asymptotic solutions, as ε→0, that were introduced in the articles [9] and [10] by the author (see the references at the end of the paper), explicit connection formulas between these solutions are calculated. As part of this derivation it is shown that only the diagonal entries of the connection matrix are asymptotically relevant.     

Penalization techniques in L ∞ optimization problems with unbounded horizon

In this work we present a numerical procedure for the ergodic optimal minimax control problem. Restricting the development to the case with relaxed controls and using a perturbation of the instantaneous cost function, we obtain discrete solutions U _ε ^k that converge to the optimal relaxed cost U when the relation ship between the parameters of discretization k and penalization ε is an appropriate one. This paper aims to be a tribute to Prof. Roberto L.V. González who died after we finished this work. This paper was supported by grant PIP 5379 CONICET.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

An approximate method of computing the magneto-acoustic heating of electrically conducting bodies

We discuss a method of approximate computation of the noncontact magneto-acoustic heating of nonmagnetic viscoelastic matter, based on the asymptotic separation of the initial equations of the theory of magnetoelasticity and the expansion of the solutions of the resulting sequence of problems in a series of eigenfunctions of the classical problems of electrodynamics and the dynamic theory of elasticity. The asymptotic parameter ε<1 for the problem was taken to be the criterion ε=Co. Rm forRm≤1 and ε=Co forRm≥1. We obtain expressions for the average power of the Joule losses and losses due to internal friction. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37. 1994. pp. 70–73.