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Asymptotic Perturbation Theory of Waves p572hc_9781848162358_tp.indd 1 25/4/14 9:39 am May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Asymptotic Perturbation Theory of Waves Lev Ostrovsky Imperial College Press ICP p572hc_9781848162358_tp.indd 2 25/4/14 9:39 am Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Control Number: 2014949402 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ASYMPTOTIC PERTURBATION THEORY OF WAVES Copyright © 2015 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-1-84816-235-8 Printed in Singapore Catherine - Asymptotic Perturbaiton Theory.indd 1 27/8/2014 11:12:59 AM To the memory of David Crighton, Michail Miller, Alexander Potapov, Alwyn Scott, Frederick Tappert, and George Zaslavsky – remarkable scientists and good friends, who passed away in the 21st century. May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Preface As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality. Albert Einstein Using approximations to find solutions of differential equations has a long history, especially in the areas of applied science and engineering, where relatively simple ways to obtain quantitative results are crucial. Since the 18th century, perturbation methods have been actively used in celestial mechanics for describing planetary motion as a three-body interaction governed by gravitation. As was shown by Poincaré [19], the three-body problem is not completely integrable, and its solution was often based on finding a small perturbation to the solution of the integrable two-body problem, when the mass of at least one of the three bodies is small. One famous example is the prediction of an existing and the position of a new planet by Adams and Leverrier based on calculation of perturbations of Uranus, which resulted in the discovery of the planet Neptune by Galle in 1846. A more recent overview of this topic can be found, for example, in [6, 7]. In the 20th century the notion of a “theory of oscillations” as a unifying concept, meaning the application of similar equations and methods of their solution to quite different physical problems, came into being. In particular, the phase plane method and Poincaré mapping proved to be very efficient. For early work in this area, see the classic book by Andronov et al. [3]. vii viii Preface Regular perturbation schemes allow one to find a deviation from the basic, unperturbed solution, when the deviation remains sufficiently small indefinitely or at least at a time interval at which the solution should be determined. However, in many important cases, a small initial perturbation can strongly change the solution even if the basic equations are only slightly disturbed. In these cases the smallness of the expansion parameter is reflected in the slowness, but not necessarily smallness, of the deviation from the solution existing when that parameter is zero. The slowness means that the time of significant deviation from the unperturbed solution is much longer than the characteristic time scale of the process, such as the period of oscillations or the duration of a pulse. An adequate tool for solving such problems is the asymptotic perturbation theory, which constructs a series in a small parameter in which the main term of the expansion differs from the unperturbed solution in that it contains slowly varying parameters; their variation can be found from “compatibility,” or “orthogonality,” conditions which secure the finiteness of the higher-order perturbations. In the framework of an asymptotic method, the series may not even converge, but a sum of a finite number of terms in the series approaches the exact solution when the expansion parameter tends to zero, which is sufficient in most applications. Still more involved is singular perturbation theory, in which the asymptotic series does not converge to the unperturbed solution in the total interval in time or space when the small parameter tends to zero; in these cases the matched asymptotic expansions are used, for example, for the boundary layers in hydrodynamics. As an early example of such an approach we mention the Poincaré– Lindstedt method applied to a differential equation (Duffing equation) with a small nonlinear term in which the frequency of a basic harmonic solution should be perturbed to avoid a cumulative deviation from a harmonic solution of the unperturbed linear equation [19]. In the 20th century the area of application of perturbation theories has greatly broadened, due particularly to the development of radio electronics and radiophysics. The systematic development of asymptotic perturbation schemes can be attributed to the work of Krylov and Bogoluybov [14]. An excellent description of the main ideas and applications can be found Preface ix in the book by Bogolyubov and Mitropolsky [5]; for the later development see, for example, [4, 11, 15]. In spite of many variations of the asymptotic theory for ordinary differential equations (ODEs), its idea is rather general and consists of the following main steps. 1. A system of ODEs containing a small parameter µ<<1 is considered. It is supposed that at µ=0 this system has a family of periodic solutions, u =U(θ,A), where θ=ωt and ω and A are constants; in particular, but not necessarily, the function U can be a sinusoid so that ω is its frequency and A is its amplitude. 2. At µ≠0 the solution is expanded in a power series ofµ, and the parameters ω and A depend on the “slow” parameter T =µt. This series is substituted into the equations to be solved. 3. These equations are, in turn, expanded in a series of µ. As a result, in each order of µ we obtain linear, inhomogeneous equations, with a right-hand side (“forcing”) that is known from the solution obtained in previous approximations. 4. The solution of these equations is, in general, secularly (i.e., as a power of time) divergent, and to keep the perturbations finite, additional “orthogonality” conditions should be imposed, which are typically differential equations determining a slow time variation of the parameters, including the functions A(T) and ω(T). This defines first the variation of U and then the higher-order perturbations. This procedure can be complicated; for example, the solution can depend on more than one phase θ and there can be several parameters i A, but the concept of the method remains essentially the same. Note that due to the use of “two times,” t and T, such a scheme is also called the method of multiple scales. Various approximate approaches for waves described by partial differential equations (PDEs) have been used since very early as well; it suffices to mention the use of the ray concept in optics. Intensive use of perturbation methods in the theory of waves began in the 20th century, in relation to quantum mechanics methods such as the “quasi-classical” WKB theory used in linear boundary problems [4] and, for nonlinear waves, to physical oceanography, nonlinear optics, and many other areas.