Bounds for the spectrum of a matrix differential operator with a damping term

We consider a second-order matrix ordinary regular differential nonselfadjoint operator with a damping term and selfadjoint boundary conditions. An estimate for the resolvent and bounds for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.     

Bounds for the spectrum of a matrix differential operator with a damping term

We consider a second-order matrix ordinary regular differential nonselfadjoint operator with a damping term and selfadjoint boundary conditions. An estimate for the resolvent and bounds for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.     

The Relation and Evolution of Squeezing and Instability for Systems with Quadratic Hamiltonians

We propose a method that allows relating the quantum squeezing effect to the classical instability by establishing evolution equations for elements of the dispersion matrix directly in terms of elements of the stability matrix. The solution of these equations is written in terms of the evolution operator. Knowing this operator, we can analyze the system instability at finite times. Based on the developed formalism, we investigate two physical systems: the degenerate and nondegenerate parametric amplifiers with external -shaped pulses. We show that we can either amplify or, on the contrary, weaken both the squeezing effect and the system instability using -pulses.     

On the spectral instability of the Sturm-Liouville operator with a complex potential

Using the Sturm-Liouville operator with a complex potential as an example, we analyze the spectral instability effect for operators that are far from being self-adjoint. We show that the addition of an arbitrarily small compactly supported function with an arbitrarily small support to the potential can substantially change the asymptotics of the spectrum. This fact justifies, in a sense, the necessity of well-known sufficient conditions for the potential under which the spectrum of the operator is localized around some ray. For an operator with a logarithmic growth, we construct a perturbation that preserves the asymptotics of the spectrum but has infinitely many poles inside the main sector.     

Modulational instability in a full‐dispersion shallow water model

We propose a shallow water model that combines the dispersion relation of water waves and Boussinesq equations, and that extends the Whitham equation to permit bidirectional propagation. We show that its sufficiently small and periodic traveling wave is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value, like the Benjamin‐Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability to the leading order in the amplitude parameter. We discuss the effects of surface tension. The results agree with those from a formal asymptotic expansion and a numerical computation for the physical problem.     

发展方程的计算稳定性问题

一、演变过程方程及差分格式 在数值天气预报中以及求解非定常流体运动时,必须设计计算稳定的格式,所以关于计算稳定性问题的理论研究是很有意义的.在这一类问题中,所要求解的问题大都可以     

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 spectral stability in subspaces for difference schemes with nonlocal boundary conditions

We consider an initial-boundary value problem for the heat equation with nonlocal boundary conditions containing a parameter γ > 1. The spectrum of the main differential operator contains some number (depending on γ) of eigenvalues lying in the left complex half-plane, which results in the instability of the problem with respect to the initial data. For difference schemes approximating the original problem, we obtain a criterion for stability in the subspaces generated by stable harmonics.     

A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

We consider a simple reaction-diffusion system exhibiting Turing’s diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential operator, and a variational approach is used in a certain non-direct way.     

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

In this paper, we study the forward and the backward in time problems for a class of nonlinear diffusion equations with respect to the pseudo‐differential operator. Herein, we investigate the stability of the solution of the forward problem in relationship with parameters of the pseudo‐differential operator and initial data. Besides, as known, the backward in time problem is instability. Hence, we give a method to regularize the solution of the backward problem in the case of the parameters are perturbed.     

A remark on linearly unstable standing wave solutions to NLS

We obtain an optimal growth estimate of a semigroup generated by a linearized operator around a standing wave solution nonlinear Schrödinger equations in two-dimension. Using the growth estimate of the semigroup, we prove that a linearly unstable standing wave solution is orbitally unstable and that instability of the standing wave solution is mainly caused by a mode of an eigenfunction associated with the rightmost (or the leftmost) eigenvalues of the linearized operator. Our result is obtained by using the method of Yajima and Cuccagna that proved L^p$L^p$-boundedness of the wave operator.     

Modulational Instability in the Whitham Equation for Water Waves

We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long‐wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small‐amplitude parameter. We extend the result to related, nonlinear dispersive equations.     

A theorem of instability by linear approximation for a one-dimensional non-linearly elastic body

On the basis of and in a development of the ideas and results of A.A. Movchan (Sr.), that extend to continuous bodies the definitions and main fundamental theorems of Lyapunov on stability and instability, a criterion for instability of the equilibrium position of a one-dimensional non-linearly elastic body subject to potential external forces is established. For the specified simplest type of continuous elastic system (which possesses, however, a number of fundamental properties of continuous elastic systems including unboundedness of the operator of linear approximation and discreteness of its spectrum) a theorem of instability by linear approximation is stated and proved. The method of proof is a version of Persidskii's sector method.     

Funzioni regolari di più variabili quaternioniche

Summary In this paper the stability analysis of an incompressible toroidal Hall Current plasma with resistivity and viscosity on the basis of the linearization of the governing equations and boundary condition is rigorously justified. A nonlinear local existence theorem for an initial-boundary value problem is first proved, and local stable and unstable invariant manifolds of a nonlinear resolving operator are then constructed. It is shown that linear stability implies nonlinear stability and global existence, and linear instability implies nonlinear instability in some sense.     

Star product,discrete Wigner functions,and spin-system tomograms

We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.     

The free-parameter solution of the convection equations in a vertical cylinder with a volume heat source

An exact solution of the free-convection equations is constructed in the Oberbeck–Boussinesq approximation, describing the flow of a viscous heat-conducting fluid in a vertical cylinder of large radius when heated by radiation. The initial problem is reduced to an operator equation with an extremely non-linear operator, satisfying Schauder's theorem in C[0,1]. An iteration procedure is proposed for determining the independent parameter, that occurs in the solution, which enables three different values to be obtained and, correspondingly, three classes of solution of the initial problem. The linear stability of all the solutions obtained is investigated and it is shown that, for chosen values of the problem parameters, the most dangerous one is the plane wave mode and two instability mechanisms are present. The flow structure and the type of instability depend considerably on the values of the free parameter.     

Weighted generalized functions and hyperbolic Riesz B‐potential

The paper is devoted to the study of the fractional integral operator, which is a negative real power of the singular wave operator generated by Bessel operator using weighted generalized functions. We give conditions for this operator to be bounded in appropriate spaces, obtain formula for the Hankel transform of this operator, and get formula of connection between this operator and natural degree of singular wave operator generated by Bessel operator.     

On the spectral properties of odd-order differential operators with integrable potential

We consider a boundary value problem with irregular boundary conditions for a differential operator of arbitrary odd order. The potential in this operator is assumed to be an integrable function. We suggest a method for studying the spectral properties of differential operators with integrable coefficients. We analyze the asymptotic behavior of solutions of the differential equation in question for large values of the spectral parameter. The eigenvalue asymptotics for the considered differential operator is obtained.     

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.     

The second iterate for the Navier-Stokes equation

We consider the iterative resolution scheme for the Navier-Stokes equation, and focus on the second iterate, more precisely on the map from the initial data to the second iterate at a given time t. We investigate boundedness properties of this bilinear operator. This new approach yields very interesting results: a new perspective on Koch-Tataru solutions; a first step towards weak-strong uniqueness for Koch-Tataru solutions; and finally an instability result in , for q>2.