Table Of ContentAsymptotic Perturbation
Theory of Waves
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Asymptotic Perturbation
Theory of Waves
Lev Ostrovsky
Imperial College Press
ICP
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ASYMPTOTIC PERTURBATION THEORY OF WAVES
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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.
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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.
