Justification of modulation equations for hyperbolic systems via normal forms

The justification problem for the Nonlinear Schr?dinger equation as a modulation equation for almost spatial periodic wavetrains of small amplitude is considered. We show exact estimates between solutions of the original system and their approximations which are obtained by the solutions of the Nonlinear Schr?dinger equation. By a normal form transform the a priori dangerous quadratic terms of the considered hyperbolic systems are eliminated. Then the transformed systems start with cubic terms. This allows to justify the Nonlinear Schr?dinger equation by a simple application of Gronwall's inequality. Moreover, the influence of resonances is estimated. Received September 16, 1996     

Averaging principle and hyperbolic evolution equations

An averaging principle is derived for the abstract nonlinear evolution equation where the almost periodic right hand-side is a continuous perturbation of the time-dependent family of linear operators determining a linear evolution system. It generalizes classical Henry’s results for perturbations of sectorial operators on fractional spaces. It is also proved that the main hypothesis of the nonlinear averaging principle is satisfied for general hyperbolic evolution equations introduced by Kato.     

Justification of the Galerkin method for hypersingular equations

The paper presents a theoretical study of hypersingular equations of the general form for problems of electromagnetic-wave diffraction on open surfaces of revolution. Justification of the Galerkin is given. The method is based on the separation of the principal term and its analytic inversion. The inverse of the principal operator is completely continuous. On the basis of this result, the equivalence of the initial equation to a Fredholm integral equation of the second kind is proven. An example of numerical solution with the use of Chebyshev polynomials of the second kind is considered.     

Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.     

Open boundary conditions for hyperbolic equations

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10^?2 of the incident wave.     

Contact-equivalence problem for linear hyperbolic equations

We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. é. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 119–142, 2005.     

Active shielding model for hyperbolic equations

^** Email: s.utyuzhnikov{at}manchester.ac.uk The problem of active shielding (AS) in application to hyperbolicequations is analysed. According to the problem, two domainseffecting each other via distributed source terms are considered.It is required to implement additional sources nearby the commonboundary of the domains in order to "isolate" one domain fromthe action of the other domain. It is important to note thatthe total field of the original sources is only known. In thepaper, the theory of difference potentials is applied to thesystem of hyperbolic equations for the first time. It allowsone to obtain a one-layer AS not requiring any additional computations.Local one-layer and two-layer AS sources are obtained for anarbitrary hyperbolic system. The solution does not require eitherthe knowledge of the Green's function or the specific characteristicsof the sources and medium. The optimal one-layer AS solutionis derived in the case of free space. In particular, the resultsare applicable to the system of acoustics equations. The questionsrelated to a practical realization including the mutual situationof the primary and secondary sources, as well as the measurementpoint, are discussed. The active noise shielding can be realizedvia a one-layer source term requiring the measurements onlyat one layer nearby the domain shielded.     

Dirichlet problem for third-order hyperbolic equations

We consider Dirichlet problem for third-order linear hyperbolic equations. We prove the existence and uniqueness of classical solution by means of an energy inequality and Riemann’s method. We reveal the influence of coefficients at lower derivatives on the well-posedness of the Dirichlet problem.     

Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity

In this paper we study the gauge invariance of the time-dependent Ginzburg-Landau equations through the introduction of a model which uses observable variables. We observe that the various choices of gauge lead to a different representation of such variables and therefore to a different definition of the weak solution of the problem. With a suitable decomposition of the unknown fields, related to the choice of London gauge, we examine the Ginzburg-Landau equations and deduce some energy estimates which prove the existence of a maximal attractor for the system.     

Interior and boundary difference equations for hyperbolic differential equations

Interior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second-order accurate, two time-level, explicit, and stable (for one-dimensional, time-dependent problems) for cΔt/Δx ? 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.