Parametrische Wechselwirkung in Medien mit hoher Nichtlinearität

Summary In the electromagnetic formulation of the parametric interaction two coupled differential equations of second order have to be solved in the case of nonlinear media with a large second power term of the polarisation. For reflection free amplification the solution of the complete system of equations differs only slightly from the solution of the approximate first order differential equations [2], even for very high nonlinearities. However backward waves of signal or idler frequency (propagating opposed to the pump wave) can cause a significant change of the amplification of the corresponding forward wave already for a smaller nonlinear term. In lossy nonlinear media even a cross coupling of a reflected signal on the forward idler wave (and vice versa) takes place.The solutions of the coupled inhomogeneous differential equations are deduced and some numerically calculated examples are given to illustrate the magnitude of the discussed effects.

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Gewidmet Herrn Professor Dr. K. P. Meyer zu seinem 60. Geburtstag     

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

Exact Boundary Controllability for a Coupled System of Wave Equations with Neumann Boundary Controls

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls.In order to establish the corresponding observability inequality,the authors introduce a compact perturbation method which does not depend on the Riesz basis property,but depends only on the continuity of projection with respect to a weaker norm,which is obviously true in many cases of application.Next,in the case of fewer Neumann boundary controls,the non-exact boundary controllability for the initial data with the same level of energy is shown.     

Exact Boundary Controllability for a Coupled System of Wave Equations with Neumann Boundary Controls(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next,in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.     

Über elastische Wellen in porösen Medien

Summary The expansion process of elastic waves in porous media is described by a system of differential equations which is a generalisation of the wave equation for homogeneous continua. For the plane problem four parameters allow to comprehend the interactions of longitudinal and transversal waves due to the porosity.

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Durchgeführt am Institut für Techn. Mechanik der Universität Saarbrücken.     

Onde di discontinuità ed equazioni costitutive nei corpi elastici isotropi sottoposti a deformazioni finite

Summary We consider an elastic isotropic material with finite deformations. The existence of a double wave (therefore exceptional because the field equations are in the conservative form) for all the deformations and the discontinuity propagation direction is required. This because in the linear theory there exists a double wave, furthermore, because in many non-linear theories of Mathematical physics there exists at least one exceptional wave (this wave doesn’t produce shocks). This request implies conditions for the response function in the constitutive equations. Furthermore, under these assumptions, we can determine explicitly all the possible propagation speeds. Therefore we can find theorems generalizing (in the case of the imposed conditions) those ones obtained by Truesdell and Green for the principal waves (whose unit normal has the direction of the eigenvectors of the deformation matrix). In the last part of this work we examine the case of a hyperelastic material and we determine some classes of possible thermodynamic potentials.

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Entrata in Redazione il 18 febbraio 1976.

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Lavoro eseguito nell’ambito dei contratti del C.N.R. - Gruppo Nazionale per la Fisica Matematica.     

THE LONG WAVE LIMIT FOR A GENERAL CLASS OF 2D QUASILINEAR HYPERBOLIC PROBLEMS

ABSTRACT

This paper is dedicated to the study of long wave approximation for quasilinear symmetric hyperbolic systems. We use ansatz with multiple scales in order to capture both dispersive and nonlinear effects, occuring at the same time scale as a relatively small transverse correction of the unidirectional propagation. Using the techniques developped by Joly, Métivier and Rauch for nonlinear geometric optics, we prove that the exact solutions split up into two wave packets, one moving to the left and one to the right, and whose amplitude satisfy a system of Kadomtsev-Petviashvili equations. Whether the limit system is coupled or not is critical to the error estimate obtained. In the last part, we formally apply these techniques to derive both uncoupled and coupled KP systems from the Euler equations with free surface in the context of water waves.     

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACT

We study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.     

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT

We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.     

The approximately synchronizable state for a kind of coupled system of wave equations

This paper deals with the approximately synchronizable state by groups for a kind of coupled system of wave equations with Dirichlet boundary controls. So far, the approximate boundary synchronization for a kind of coupled system of wave equations has already been deeply studied; however, the study on the approximately synchronizable state still needs to be done in details. In this paper, we will give some results on the determination of approximately synchronizable state by groups and the attainable set of them.     

Algebraic travelling waves for the generalized Burgers-Fisher equation

Abstract

Using the definition of algebraic travelling wave solution for general 2-th order differential equations given in [9], we provide the only possible algebraic travelling wave solutions for the celebrated generalized Burgers-Fisher equation.     

Stabilization of transmission coupled wave and Euler–Bernoulli equations on Riemannian manifolds by nonlinear feedbacks

We consider the transmission system of coupling wave equations with Euler–Bernoulli equations on Riemannian manifolds. By introducing nonlinear boundary feedback controls, we establish the exponential and rational energy decay rate for the problem. Our proofs rely on the geometric multiplier method.     

Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings

Based on the theory of semi‐global piecewise C² solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd.     

Simultaneous observability and stabilization of some uncoupled wave equations

We consider uncoupled wave equations with different speed of propagation in a bounded domain. Using a combination of the Bardos–Lebeau–Rauch observability result for a single wave equation and a new unique continuation result for uncoupled wave equations, we prove an observability estimate for that system. Applying Lions? Hilbert uniqueness method (HUM), one may derive simultaneous exact controllability results for the uncoupled system; the controls being locally distributed, with their supports satisfying the geometric control condition of Bardos, Lebeau and Rauch. Afterwards, we discuss the related simultaneous stabilization problem; this latter problem is solved by a combination of the new observability inequality, and a result of Haraux establishing an equivalence between observability and stabilization for second order evolution equations with bounded damping operators. Our observability and stabilization results generalize to higher space dimensions some earlier results of Haraux established in the one-dimensional setting.     

Controllability of classical solutions implies controllability of weak solutions for a coupled system of wave equations and its application

In this paper, for a coupled system of one‐dimensional wave equations with Dirichlet boundary controls, we show that the controllability of classical solutions implies the controllability of weak solutions. This conclusion can be applied in proving some results that are hardly obtained by a direct way in the framework of classical solutions. For instance, we strictly derive the necessary conditions for the exact boundary synchronization by two groups in the framework of classical solutions for the coupled system of wave equations. Copyright © 2015 John Wiley & Sons, Ltd.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方 法,即改进的广义射影Ricccati方程方法,求解(2 1)维色散长波方程,得到该方程的新的 更一般形式的行波解,包括扭状孤波解,钟状解,孤子解和周期解．并对部分新形式孤波解画 图示意．     

A note on the exact synchronization by groups for a coupled system of wave equations

By means of a non‐exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the exact synchronization by groups for a coupled system of wave equations

By means of a non‐exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2014 John Wiley & Sons, Ltd.     

Navier–Stokes equations: an analysis of a possible gap to achieve the energy equality

The paper is concerned with the IBVP of the Navier–Stokes equations. The goal is to evaluate the possible gap between the energy equality and the energy inequality deduced for a weak solution. This kind of analysis is new and the result is a natural continuation and improvement of a result obtained by the same authors in Crispo et al. (Some new properties of a suitable weak solution to the Navier–Stokes equations. arXiv:1904.07641).