Multiple-Scale Analysis of the Quantum Anharmonic Oscillator

Conventional weak-coupling perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.Comment: 12 pages, Revtex, no figures, available through anonymous ftp from ftp://euclid.tp.ph.ic.ac.uk/papers/ or on WWW at http://euclid.tp.ph.ic.ac.uk/Papers/papers_95-6_.htm

Multiple-scale analysis and renormalization for pre-asymptotic scalar transport

Pre-asymptotic transport of a scalar quantity passively advected by a velocity field formed by a large-scale component superimposed to a small-scale fluctuation is investigated both analytically and by means of numerical simulations. Exploiting the multiple-scale expansion one arrives at a Fokker--Planck equation which describes the pre-asymptotic scalar dynamics. Such equation is associated to a Langevin equation involving a multiplicative noise and an effective (compressible) drift. For the general case, no explicit expression for both the effective drift and the effective diffusivity (actually a tensorial field) can be obtained. We discuss an approximation under which an explicit expression for the diffusivity (and thus for the drift) can be obtained. Its expression permits to highlight the important fact that the diffusivity explicitly depends on the large-scale advecting velocity. Finally, the robustness of the aforementioned approximation is checked numerically by means of direct numerical simulations.Comment: revtex4, 12 twocolumn pages, 3 eps figure

Gravitational Waveforms for Precessing, Quasicircular Compact Binaries with Multiple Scale Analysis: Small Spin Expansion

We obtain analytical gravitational waveforms in the frequency-domain for precessing, quasi-circular compact binaries with small spins, applicable, for example, to binary neutron star inspirals. We begin by calculating an analytic solution to the precession equations, obtained by expanding in the dimensionless spin parameters and using multiple-scale analysis to separate timescales. We proceed by analytically computing the Fourier transform of time-domain waveform through the stationary phase approximation. We show that the latter is valid for systems with small spins. Finally, we show that these waveforms have a high overlap with numerical waveforms obtained through direct integration of the precession equations and discrete Fourier transformations. The resulting, analytic waveform family is ideal for detection and parameter estimation of gravitational waves emitted by inspiraling binary neutron stars with ground-based detectors.Comment: 37 pages, 14 figures, final published versio

Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur

Integrability Test for Discrete Equations via Generalized Symmetries

In this article we present some integrability conditions for partial difference equations obtained using the formal symmetries approach. We apply them to find integrable partial difference equations contained in a class of equations obtained by the multiple scale analysis of the general multilinear dispersive difference equation defined on the square.Comment: Proceedings of the Symposium in Memoriam Marcos Moshinsk