Small-amplitude nonlinear waves on a black hole background

Let $G(x)$ be a $C^0$ function such that $|G(x)|⩽K{|x|}^p$ for $|x|⩽c$, for constants $K,c>0$. We consider spherically symmetric solutions of ${\square }_g\varphi =G(\varphi )$ where g is a Schwarzschild or more generally a Reissner–Nordström metric, and such that ϕ and ∇ϕ are compactly supported on a complete Cauchy surface. It is proven that for $p>4$, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, $|\varphi |⩽C{(\max \{v,1\})}^{\mathrm{-1}}$, where v denotes an Eddington–Finkelstein advanced time coordinate.     

Hydrocarbon microtremors interpreted as nonlinear oscillations driven by oceanic background waves

Characteristic low frequency seismic signals have been observed in areas where hydrocarbon reservoirs are present. A possible interpretation is the excitation of hydrocarbon related resonances. Basic models of an oscillating liquid filled porous medium are investigated. Synthetic spectra of the ab initio Navier–Stokes equations and of basic linear and nonlinear one-dimensional oscillators show characteristic features of measured spectra when oceanic background waves around 0.1–0.2 Hz are assumed to be the external driving force.     

A note on nonlinear oscillations at resonance

Existence of 2π-periodic solutions to the equation \(\ddot x + g(x) = p(t)\) is proved, under sharp nonresonance conditions on the interaction, ofg(x)/x with two consecutive eigenvaluesm ² and(m+1) ²: touch with the eigenvalues is allowed.     

On the approximation of some nonlinear equations

We approximate the solutions of a polynomial equation in a Banach space using the solutions of finite rank polynomial equations. Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Numerical approximation of anisotropic geometric evolution equations in the plane

Harald Garcke Naturwissenchaftliche Fakultät I' Mathematik, Universität Regensburg, 93040 Regensburg, Germany Robert Nürnberg Department of Mathematics, Imperial College London, London SW7 2AZ, UK Received on 13 April 2006. Revised on 20 February 2007. We present a variational formulation of fully anisotropic motionby surface diffusion and mean curvature flow, as well as relatedflows. The proposed scheme covers both the closed-curve caseand the case of curves that are connected via triple junctionpoints. On introducing a parametric finite-element approximation,we prove stability bounds and report on numerical experiments,including regularized crystalline mean curvature flow and regularizedcrystalline surface diffusion. The presented scheme has verygood properties with respect to the distribution of mesh pointsand, if applicable, area conservation.     

Stability analysis based on nonlinear inhomogeneous approximation

The asymptotic stability of zero solutions for essentially nonlinear systems of differential equations in triangular inhomogeneous approximation is studied. Conditions under which perturbations do not affect the asymptotic stability of the zero solution are determined by using the direct Lyapunov method. Stability criteria are stated in the form of inequalities between perturbation orders and the orders of homogeneity of functions involved in the nonlinear approximation system under consideration.     

Parametric oscillations of nonlinear flutter systems

We consider a special class of abstract nonlinear equations of hyperbolic type that contains the main boundary value problems of panel flutter theory. For these equations, we study the existence and stability of parametric oscillations bifurcating from the zero equilibrium in the case of small damping.     

Forced nonlinear oscillations of elastic membranes

An initial-boundary-value problem is considered for the spatially two-dimensional damped Boussinesq equation with a forcing term. The problem in question is argued to serve as a model for describing small nonlinear oscillations of a circular membrane under the influence of acoustic pressure. Eigenfunction expansion method is used for constructing the global-in-time solution of the problem in question. Existence and uniqueness follow from the construction. Long-time asymptotics is computed on the basis of the obtained series representation. Numerical simulations are conducted. The algorithm shows excellent convergence properties.     

Stability of the Faedo-Galerkin approximation of nonlinear wave-equations

Present investigation analyses the Ljapunov stability of the systems of ordinary differential equations arising in then-th step of the Faedo-Galerkin approximation for the nonlinear wave-equation $$\begin{gathered} u_{tt} - u_{xx} + M(u) = 0 \hfill \\ u(0,t) = u(1,t) = 0 \hfill \\ u(x,0) = \Phi (x); u_t (x,0) = \Psi (x). \hfill \\ \end{gathered}$$ For the nonlinearities of the classM (u)=u ^{2 p+1} ,p ∈N, ann-independent stability result is given. Thus also the stability of the original equation is shown.     

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let be a Hilbert space, and let be a representation ofL ( ) on . LetR be a positive operator inL ( ) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on (bounded, but noncontractive) such that