Constant-coefficient Systems Research Articles

Ring-like two-breather structures of a partially nonlocal NLS system with different two-directional diffractions under a parabolic potential

Via a relation of the hypothetic solution, a variable-coefficient partially nonlocal nonlinear Schrödinger(NLS) system with different diffractions under a parabolic potential is associated with the constant-coefficient system. Ring-like two-breather solution of the variable-coefficient partially nonlocal NLS system is constructed by mapping the related solution of the constant-coefficient system. The characteristics of two-breather in the constant-coefficient system influence the properties of ring-like two-breather in the variable-coefficient partially nonlocal NLS system in xyt plane. In the xyz plane, the layer number of the ring-like two-breather along the z-direction augments with the add of the Hermite parameter.

On the form of Lie symmetries of systems with three pdes: The examples of two variable coefficient Hirota Satsuma systems

We consider a general class of systems of three partial differential equations and we provide restrictions on the form of Lie symmetry operators admitted by such systems. When these restrictions are known in advance, the symmetry analysis becomes simpler. Special cases of this class are two generalizations of Hirota–Satsuma systems with variable coefficients. We derive the equivalence groups for these two systems. With the aid of the equivalence groups and the restricted form of the Lie symmetry operators, we present an enhanced Lie group classification for the two Hirota–Satsuma systems. For each class a specific system is single out which has the property to be mapped into a constant coefficient system.

The balanced proper orthogonal decomposition applied to a class of frequency-dependent damped structures

A model reduction technique aimed at computing efficiently Frequency Response Functions of damped structures is presented. The frequency-dependent complex moduli are approximated by a mini-oscillators model, known as the Golla–Hughes–MacTavish (GHM) model, which permits to recast the original problem as a more familiar second-order, constant-coefficient system of equations. The matrix system, although much larger, is then treated by application of the Balanced Proper Orthogonal Decomposition (BPOD) which aims at approximating the transfer function matrix, or equivalently the admittance matrix, connecting forces and displacements at a specified set of points of the vibrating structure. All the necessary ingredients of the reduction strategy as well as its efficiency measured in terms of data reduction, accuracy and computational cost are shown. Two illustrative examples of increasing complexity involving a clamped cantilever beam and a realistic windshield are presented. It is shown that the admittance matrix can be approximated by matrices of very small size which computation can be speeded up via diagonalization. It is concluded that the application of BPOD combined with the GHM decomposition of the frequency-dependent algebraic system proves extremely efficient for the modeling of vibrating structures made of different materials, either viscoelastic or purely elastic.

Endotrivial modules for finite groups via homotopy theory

Classifying endotrivialkGkG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite groupGG, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thévenaz, and others, it has been reduced to understanding the subgroup consisting of modular representations that split as the trivial modulekkdirect sum a projective module when restricted to a Sylowpp-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivialpp-subgroups with values in the unitsk×k^\times, viewed as a constant coefficient system. We then use homotopical techniques to give a number of formulas for this group in terms of the abelianization of normalizers and centralizers inGG, in particular verifying the Carlson–Thévenaz conjecture—this reduces the calculation of this group to algorithmic calculations in local group theory rather than representation theory. We also provide strong restrictions on when such representations of dimension greater than one can occur, in terms of thepp-subgroup complex andpp-fusion systems. We immediately recover and extend a large number of computational results in the literature, and further illustrate the computational potential by calculating the group in other sample new cases, e.g., for the Monster at all primes.

Reducibility and Analysis of Linear Quasi-Periodic Systems Via Normal Forms

Abstract This article introduces a technique to accomplish reducibility of linear quasi-periodic systems into constant-coefficient linear systems. This is consistent with congruous proofs common in literature. Our methodology is based on Lyapunov–Floquet transformation, normal forms, and enabled by an intuitive state augmentation technique that annihilates the periodicity in a system. Unlike common approaches, the presented approach does not employ perturbation or averaging techniques and does not require a periodic system to be approximated from the quasi-periodic system. By considering the undamped and damped linear quasi-periodic Hill-Mathieu equation, we validate the accuracy of our approach by comparing the time-history behavior of the reduced linear constant-coefficient system with the numerically integrated results of the initial quasi-periodic system. The two outcomes are shown to be in exact agreement. Consequently, the approach presented here is demonstrated to be accurate and reliable. Moreover, we employ Floquet theory as part of our analysis to scrutinize the stability and bifurcation properties of the undamped and damped linear quasi-periodic system.

Exact solutions and self-similar symmetries of a nonlocal nonlinear Schrödinger equation

An analytical consideration of a nonlocal nonlinear Schrodinger equation with distributed coefficients is presented. The analysis is twofold: first, exact solutions of the nonlocal equation with constant coefficients are found along with the appropriate conditions for their existence, and second, using a similarity transformation the distributed equation, i.e. the equations with varying coefficients, is reduced to the constant coefficients case; the solutions of the latter will be utilized in the construction of the solutions for the general case. This similarity transformation is also physically relevant as it allows for the physical parameters of the solution (amplitude, width, centre and phase) to vary in a specific way so that the constant coefficient system is a natural similarity reduction of the distributed case. Certain compatibility conditions needed for the consistency of this reduction are also deduced.

Effective Rankine–Hugoniot conditions for shock waves in periodic media

Solutions of first-order nonlinear hyperbolic conservation laws typically develop shocks in finite time even with smooth initial conditions. However, in heterogeneous media with rapid spatial variation, shock formation may be delayed or avoided. When shocks do form in such media, their speed of propagation depends on the material structure. We investigate conditions for shock formation and propagation in heterogeneous media. We focus on the propagation of plane waves in two-dimensional periodic media with material variation in only one direction. We propose an estimate for the speed of the shocks that is based on the Rankine-Hugoniot conditions applied to a leading-order homogenized (constant coefficient) system. We verify this estimate via numerical simulations using different nonlinear constitutive relations and layered and smoothly varying periodic media. In addition, we discuss conditions and regimes under which shocks form in this type of media.

Asymptotic Stability of Non-Autonomous Systems and a Generalization of Levinson’s Theorem

We study the asymptotic stability of non-autonomous linear systems with time dependent coefficient matrices { A ( t ) } t ∈ R . The classical theorem of Levinson has been an indispensable tool for the study of the asymptotic stability of non-autonomous linear systems. Contrary to constant coefficient system, having all eigenvalues in the left half complex plane does not imply asymptotic stability of the zero solution. Levinson’s theorem assumes that the coefficient matrix is a suitable perturbation of the diagonal matrix. Our objective is to prove a theorem similar to Levinson’s Theorem when the family of matrices merely admits an upper triangular factorization. In fact, in the presence of defective eigenvalues, Levinson’s Theorem does not apply. In our paper, we first investigate the asymptotic behavior of upper triangular systems and use the fixed point theory to draw a few conclusions. Unless stated otherwise, we aim to understand asymptotic behavior dimension by dimension, working with upper triangular with internal blocks adds flexibility to the analysis.

$\mathcal C$-symmetric Hamiltonian systems with almost constant coefficients

We consider a \mathcal C -Symmetric Hamiltonian System of differential equations on a half interval or the real line. We determine the spectrum and construct the resolvent for the system. The essential spectrum is found to be a subset of an algebraic curve \Sigma defined by a characteristic polynomial for the system. The results are first proved for a constant coefficient system and then for an almost constant coefficient system. The results are applied to a number of examples including the complex hydrogen atomand the complex relativistic electron.

Geometry effects in nodal discontinuous Galerkin methods on curved elements that are provably stable

We investigate three effects of the variable geometric terms that arise when approximating linear conservation laws on curved elements with a provably stable skew-symmetric variant of the discontinuous Galerkin spectral element method (DGSEM). We show for a constant coefficient system that the non-constant coefficient problem generated by mapping a curved element to the reference element is stable and has energy bounded by the initial value as long as the discrete metric identities are satisfied. Under those same conditions, the skew-symmetric approximation is also constant state preserving and discretely conservative, just like the original DGSEM.

Conservative Semidiscrete Difference Schemes for Timoshenko Systems

We present a parameterized family of finite-difference schemes to analyze the energy properties for linearly elastic constant-coefficient Timoshenko systems considering shear deformation and rotatory inertia. We derive numerical energies showing the positivity, and the energy conservation property and we show how to avoid a numerical anomaly known as locking phenomenon on shear force. Our method of proof relies on discrete multiplier techniques.

Reducibility of Periodic Quasi-Periodic Systems

In this work, the reducibility of quasi-periodic systems with strong parametric excitation is studied. We first applied a special case of Lyapunov-Perron (L-P) transformation for time periodic system known as the Lyapunov-Floquet (L-F) transformation to generate a dynamically equivalent system. Then, we used the quasi-periodicnear-identity transformation to reduce this dynamically equivalent system to a constant coefficient system by solving homological equations via harmonic balance. In this process, we obtained the reducibility/resonance conditions that needed to be satisfied to convert a quasi-periodic system in to a constant one. Assuming the reducibility is possible, we obtain the L-P transformation that can transform original quasi-periodic system into a system with constant coefficients. Two examples are presented that show the application of this approach.

Periodic projectile linear theory for aerodynamically asymmetric projectiles

A new analytical tool is proposed to aid in the design and performance evaluation of advanced guided projectile concepts. A projectile linear theory applicable to aerodynamically asymmetric configurations is created, leading to a linear, periodic dynamic system. Utilizing concepts in Floquet theory, stability of asymmetric projectile configurations is explored. While stability of many asymmetric projectile configurations can be accurately predicted using averaged linear, constant coefficient system dynamics, there are some configurations where the use of linear periodic systems theory is required for accurate stability prediction. This fact is shown by comparing stability boundaries of an example projectile configuration using the conventional projectile linear theory model and the new periodic projectile linear theory model.

Spectral theory of Hamiltonian difference systems with almost constant coefficients

The spectral theory for a general linear Hamiltonian difference system with asymptotically constant coefficients is developed with rather mild smoothness conditions for the coefficients. It is shown that the absolutely continuous spectrum is that of the limiting constant coefficient system. The main techniques are asymptotic summation of the difference equations and estimates of the Titchmarsh–Weyl M-matrix.

The Two-Dimensional Power Spectral Density: A Connection Between 2-D Rational Functions and Linear Systems

A derivation of the power spectral density (PSD) matrix of a continuous-time, linear, constant-coefficient system with white Gaussian noise input is given. The assumption of wide-sense stationarity is not made. By transforming the two-dimensional (2-D) autocorrelation function (ACF), a 2-D spectral density is obtained that takes the system's transient behavior into account. The 2-D PSD is a non-separable rational function. The results are applicable to second-order Volterra series representations of nonlinear systems. Simulations are presented for a fourth order system that compare the 2-D PSD amplitude (both theoretical and empirical) with its 1-D counterpart on 50 realizations of a short data segment containing an initial transient.

On a Mixed Problem for a Constant Coefficient Second-Order System

The paper is devoted to the study of an initial boundary value problem for a linear second-order differential system with constant coefficients. The first part of the paper is concerned with the existence of the solution to a boundary value problem for the second-order differential system, in the strip , where is a suitable positive number. The result is proved by means of the same procedure followed in a previous paper to study the related initial value problem. Subsequently, we consider a mixed problem for the second-order constant coefficient system, where the space variable varies in and the time-variable belongs to the bounded interval , with sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution , by studying two related mixed problems, whose solutions exist due to the results proved for the Cauchy problem in a previous paper and for the boundary value problem in the first part of this paper.

Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems

In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with high-contrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the degrees of freedom into “high” and “low” permeability regions which may be of arbitrary geometry. This partition yields a corresponding blocking of the stiffness matrix and hence a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. The structure of the required sub-block inverses in the high contrast case is revealed by a singular perturbation analysis (with the contrast playing the role of a large parameter). This shows that for high enough contrast each of the sub-block inverses can be approximated well by solving only systems with constant coefficients. The second phase of the algorithm involves the approximation of these constant coefficient systems using multigrid methods. The result is a general method of algebraic character which (under suitable hypotheses) can be proved to be robust with respect to both the contrast and the mesh size. While a similar performance is also achieved in practice by algebraic multigrid (AMG) methods, this performance is still without theoretical justification. Since the first phase of our method is comparable to the process of identifying weak and strong connections in conventional AMG algorithms, our theory provides to some extent a theoretical justification for these successful algebraic procedures. We demonstrate the advantageous properties of our preconditioner using experiments on model problems. Our numerical experiments show that for sufficiently high contrast the performance of our new preconditioner is almost identical to that of the Ruge and Stuben AMG preconditioner, both in terms of iteration count and CPU-time.

A new method for solving first-order constant coefficient systems of linear differential equations

In this paper, we introduce a new method for solving first-order systems of linear differential equations with constant coefficient.

Modelling a small internet cafe charging policy

In this paper we use a continuous time Markov chain to model a simple optimisation problem with stochastic elements. The prerequisites are a knowledge of linear constant coefficient systems and some elementary probability—including the negative exponential distribution, which is briefly reviewed. The problem addressed has wide appeal and has the advantage of being based on modelling with differential equations. It offers an alternative to the classic mechanics and population models. The problem has easy generalizations with which to create variations. Special consideration is given to looking at some of the assumptions of the model and questioning the model solution.

New Stability Criteria for Constant-Coefficient Linear Systems

New coefficient stability criteria are derived from spectral radius estimates and spectral abscissa using the Gershgorin–Ostrovskii localization theorems. That critical eigenvalues have only prime elementary divisors is demonstrated through the use of recently derived estimates for the elements of inverse matrices under the Hadamard and Ostrovskii regularity criteria.