Global well-posedness for the KP-II equation on the background of a non-localized solution

Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in R×T and perturbations that are square integrable in R². In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.     

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.     

Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the bounds arising from using the optimal basis. We also present explicit bounds for SDP using the Monteiro-Zhang family of search directions and specialize them to the AHO, H..K..M, and NT directions. Received: December 1999 / Accepted: January 2001?Published online March 22, 2001     

The relative frame bundle of an infinite-dimensional flag variety and solutions of integrable hierarchies

We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular ZxZ matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker—Akhiezer function.     

Memetic Algorithms for Multiple Interference Cancellations of Linear Array Based on Phase-Amplitude Perturbations

A novel method based on the memetic algorithm for the design of multiple interference cancellations of a linear array antenna by phase-amplitude perturbations is proposed. The adaptive array antenna is capable of sensing the presence of interference sources and suppressing the interferences in the interfering directions. This technique can increase the signal-to-interference ratio. The memetic algorithm is applied to find the weighting vector which makes the pattern nulling optimization of the proposed adaptive antenna. This technique is also able to do the cancellation of multiple interferences for different incident directions.     

Differentiation of analytic functions of sectorial operators in noncommuting directions

We find expressions and estimates for integer powers of the differentiation operator in noncommuting directions for analytic functions of sectorial operators on a complex Banach space. We study the exponential of such a differentiation operator and give an application to the theory of perturbations of evolution equations.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 70–74.     

Spectra of Anderson type models with decaying randomness

In this paper we consider some Anderson type models, with free parts having long range tails and with the random perturbations decaying at different rates in different directions and prove that there is a.c. spectrum in the model which is pure. In addition, we show that there is pure point spectrum outside some interval. Our models include potentials decaying in all directions in which case absence of singular continuous spectrum is also shown.     

Wave interactions and stability of the Riemann solutions for the chromatography equations

We prove that the Riemann solutions are stable for the chromatography system under the local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis. It is noteworthy that both the propagation directions of the shock wave S and rarefaction wave R are unchanged when they interact with the contact discontinuity J. Moreover, the global structures and large time asymptotic behaviors of the perturbed Riemann solutions are constructed and analyzed case by case.     

Regularizations of Differential-Algebraic Equations Revisited

The present paper deals with quasilinear differential-algebraic equations with index 2. These equations are approximated by regularization methods. Such methods lead to singularly perturbed differential-algebraic equations. Using a geometric theory of singular perturbations convergence of the solutions of the regularized problems towards that of the index 2 problem is proved. The limits of the present theory are discussed and directions of future research are proposed.     

Propagation of perturbations in fluids excited by moving sources

A large series of A.A. Dorodnicyn’s works deals with rigorous mathematical formulations and development of efficient research techniques for mathematical models used in inhomogeneous fluid dynamics. Numerous problems he studied in these directions are closely related to stratified fluid dynamics, which were addressed in a series of works having been published in this journal by this paper’s authors and their coauthors since 1980. This paper describes the results of a series of works analyzing the propagation of small perturbations in various stratified and/or uniformly rotating inviscid fluids. It is assumed that each of the fluids either occupies an unbounded lower half-space with a free surface or is a semi-infinite two-component fluid layer. The perturbations are excited by a moving source specified as a periodic plane wave traveling along the interface of the fluids. Problems for five mathematical fluid models are formulated, their explicit analytical solutions are constructed, and their existence and uniqueness are discussed. The asymptotics of the solution as t → +∞ are studied, and the long-time wave patterns developing in five fluid models are compared.     

Center manifolds for infinite delay

We obtain a C¹ center manifold theorem for perturbations of delay difference equations in Banach spaces with infinite delay. Our results extend in several directions the existing center manifold theorems. Besides considering infinite delay equations, we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable, unstable and central behaviors with respect to arbitrary asymptotic rates ecρ(n) for some diverging sequence ρ(n). This includes as a very special case the usual exponential behavior with ρ(n)=n.     

A Full-Newton Step Infeasible Interior-Point Algorithm Based on Darvay Directions for Linear Optimization

We present a full-Newton step primal-dual infeasible interior-point algorithm based on Darvay’s search directions. These directions are obtained by an equivalent algebraic transformation of the centering equation. The algorithm decreases the duality gap and the feasibility residuals at the same rate. During this algorithm we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main iteration of the algorithm consists of a feasibility step and some centering steps. The starting point in the first iteration of the algorithm depends on a positive number ξ and it is strictly feasible for a perturbed pair, and feasibility steps find strictly feasible iterate for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterate close to the central path of the new perturbed pair. The algorithm finds an ?-optimal solution or detects infeasibility of the given problem. The iteration bound coincides with the best known iteration bound for linear optimization problems.     

Existence Results for Some Partial Integro-Differential Equations

In this note, we deal with semilinear integro-differential equations subject to homogeneous Dirichlet boundary conditions given on the boundaries of the sections. Even if the differentiation will be taken only in some directions, it is not possible to see the main problem parameterized by the other coordinates because of the non-local terms which also obliged the problem to be degenerate. We establish the existence of solutions by employing the singular perturbations method as a natural tool. The perturbed problems are classical, non-local, semilinear elliptic problems and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. Some improvement, concerning the existence of the solutions and the convergence results depending on the weights, will be established. The paper also gives an idea about the study of the anisotropic singular perturbations in the framework of weighted spaces.     

Transverse spectral instabilities in Konopelchenko–Dubrovsky equation

We study the transverse spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky (KD) equation. We show that these waves are transversely unstable with respect to two-dimensional perturbations that are periodic in both directions with long wavelength in the transverse direction. We also show that these waves are transversely stable with respect to perturbations which are either mean-zero periodic or square-integrable in the direction of the propagation of the wave and periodic in the transverse direction with finite or short wavelength. We discuss the implications of these results for special cases of the KD equation—namely, KP-II and mKP-II equations.     

Stability of Front Solutions of the Bidomain Equation

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.© 2016 Wiley Periodicals, Inc.     

On the linearization of infinite-dimensional random dynamical systems

We present a new version of the Grobman–Hartman's linearization theorem for random dynamics. Our result holds for infinite-dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the nonlinear perturbations, we do not require for the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides requiring the existence of stable and unstable directions) allow for the existence of a third (central) direction on which we do not prescribe any behavior for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are Hölder continuous when restricted to bounded subsets of the space.     

Evolution of flexure-gravitational stimulated vibrations of a floating nonuniformly compressed elastic plate

We consider the evolution of vibrations of a thin nonuniformly compressed elastic plate that floats on a surface of a homogeneous ideal incompressible fluid of finite depth, by using the method of integral transformations. Waves are caused by pressures of the type of traveling waves on the surface of the plate, which is harmonic in one of the directions. An analysis of conditions of perturbations of waves that form oscillations of the plate and the wave motion of the fluid is given. A study is given of the dependence of the structure and the number of waves from stressing forces and characteristics of perturbed pressures.Translated from Dinamicheskie Sistemy, No. 7, pp. 58–62, 1988.     

On asymptotic stability in 3D of kinks for the model

We add to a kink, which is a 1 dimensional structure, two transversal directions. We then check its asymptotic stability with respect to compactly supported perturbations in 3D and a time evolution under a Nonlinear Wave Equation (NLW). The problem is inspired by work by Jack Xin on asymptotic stability in dimension larger than 1 of fronts for reaction diffusion equations. The proof involves a separation of variables. The transversal variables are treated as in work on Nonlinear Klein Gordon Equation (NLKG) originating from Klainerman and from Shatah in a particular elaboration due to Delort et al. The longitudinal variable is treated by means of a result by Weder on dispersion for Schrödinger operators in 1D.

     

Theory and Computation of Covariant Lyapunov Vectors

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.     

Levitin–Polyak well-posedness by perturbations of inverse variational inequalities

The purpose of this paper is to investigate Levitin–Polyak type well-posedness for inverse variational inequalities. We establish some metric characterizations of Levitin–Polyak α-well-posedness by perturbations. Under suitable conditions, we prove that Levitin–Polyak well-posedness by perturbations of an inverse variational inequality is equivalent to the existence and uniqueness of its solution. Moreover, we show that Levitin–Polyak well-posedness by perturbations of an inverse variational inequality is equivalent to Levitin–Polyak well-posedness by perturbations of an enlarged classical variational inequality.