The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than , the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as . We investigate the case , ν∈N^?, which encompasses the gravitational fluctuations, ν=4, and the electromagnetic waves, ν=2. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as , and we get global Lp estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some L²−L^∞ estimates in suitable weighted Sobolev spaces.     

Justification of Asymptotic Formulas for the Fourth Painlevé Equation

A special case of the fourth Painlevé equation is studied. The existence theorem is proved, and asymptotic formulas for the two parametric family of solutions near negative infinity are obtained.     

Perturbation Solutions of the Whittaker-Hill Equation

When the Helmholtz equation ²V+k²V = 0 is separated in the generalparaboloidal co-ordinate system, the three ordinary differentialequations obtained each take, after a suitable change of variable,the form of the Whittaker-Hill equation. For the case k²<0,a considerable amount is known about the periodic solutionsof this equation. For k²>0, however, very little is so farknown. In this paper solutions of the Whittaker-Hill equationfor small positive k² are derived. These are the first explicitsolutions to be obtained for the case k²>0, and they couldbe employed to solve the Dirichlet or Neumann problem for ageneral paraboloid when k² is small. Three limiting cases arenoted, involving the reduction of the solutions to Mathieu functionsand the reduction of the co-ordinate system to the rotation-paraboloidalsystem.     

Compact Difference Schemes for the Multidimensional Klein–Gordon Equation

Differential Equations - Stable compact difference schemes of $$4+2$$ and $$4+4 $$ approximation orders are considered and studied on standard stencils for the multidimensional Klein–Gordon...     

The Klein–Gordon Equation in a Domain with Time‐Dependent Boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.     

Properties of a Semi-discrete Approximation to the Beam Equation

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx = 0, 0 < x < L, t > 0, where it is assumed that w and p are positive on the interval[O, L], is approximated by using the method of straight lines.The resulting approximation is a linear system of differentialequations with coefficient matrix S. The matrix S is studiedunder very general boundary conditions which result in a conservativesystem. In all cases the matrix S is either an oscillation matrixor possesses nearly all the properties of an oscillation matrix.     

An Approximation for the Zakai Equation

In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. 60F25, 60H10.} Accepted 23 April 2001. Online publication 14 August 2001.     

Approximation to Nonlinear Schrodinger Equation of the Complex Generalized Ginzburg-Landau Equation

In this paper, we prove that in the inviscid limit the solution of the generalized derivative Ginzburg-Landau equations converges to the solution of derivative nonlinear Schrödinger equation, we also give the convergence rates for the difference of the solution.     

Justification of Dipole Approximation in the Problem of Nonlinear Wave Generation by a Submerged Sphere

Some nonlinear dipole approximation is constructed for the nonstationary problem of a solid sphere motion under a free surface. The approximation is justified in the class of analytic functions decaying at infinity.Original Russian Text Copyright © 2005 Pyatkina E. V.The author was supported by the State Maintenance Program for the Leading Scientific Schools (Grant NSh-440.2003.1) and the Russian Foundation for Basic Research (Grant 05-01-00250).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 907–927, July–August, 2005.     

Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, that is, stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular nondegeneracy condition, spectral stability near the origin is equivalent with the local well‐posedness of the Whitham system.     

非线性分数阶微分方程的奇摄动

研究了—类奇摄动非线性分数阶微分方程Cauchy问题．在适当的条件下,首先求出了原问题的外部解,然后利用伸长变量、合成展开法和幂级数展开理论构造出解的初始层项,并由此得到解的形式渐近展开式．最后利用微分不等式理论,讨论了问题解的渐近性态,得到了原问题解的一致有效的渐近估计式．     

A Spectral Approximation of the Barotropic Vorticity Equation

A spectral scheme is considered for solving the barotropic vorticity equation. The error estimates are proved strictly. The technique used in this paper is also useful for other nonlinear problems defined on a spherical surface.     

Approximation of an Inverse Initial Problem for a Biparabolic Equation

In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous and nonlinear biparabolic equation. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous problem. Then, we also give a regularized solution and consider the convergence between the regularized solution and the sought solution. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Hilbert space and Hilbert scale space.     

Approximation of Boundary Conditions at Infinity for a Harmonic Equation

Starting from the canonical boundary reduction, this paper studies an approximate differential boundary condition and an approximate integral boundary condition on an artificial boundary for the exterior problem of a harmonic equation, and gives an error estimate for the latter. This estimate reveals the relationship between the error and the approximate grade boundary conditions as well as the radius of the artificial boundary.     

一类具对流项的奇异扩散方程解的逼近性质

讨论了一类带对流项的奇异扩散方.程的Neumann边值问题,证明了整体解的存在唯一性;讨论了带对流项非线性间题解的线性逼近,得到了逼近的显式表示式;同时还对││u-(u)││L2(0,1)进行了估计,得到了解关于时间t充分大时的渐近性态,其中(u)＝∫o/1udx.     

Perturbation Method for a Parabolic Equation with Drift on a Riemannian Manifold

We construct a fundamental solution of a parabolic equation with drift on a Riemannian manifold of nonpositive curvature by the perturbation method on the basis of a solution of an equation without drift. We establish conditions for the drift field under which this method is applicable.     

半李普希兹算子方程的逼近可解性

本文先在（π）1空间的闭集上证明了凝聚映射必为A-proper映射,运用此性质证明了型如f（x）-λx=0方程当f为弱内向、半李普希兹映射时是弱逼近可解的,若f为李普希兹型映射,方程还是强逼近可解的.     

Justification of the Method of Approximation of Solutions to Contact Problems of Filtration Theory

We suggest a new convergent algorithm for numerical solution of the nonlinear problem of finding the parameters of conformal mappings describing fluid filtration flows with free (contact) boundaries in porous media.