Some Notes on Singular Perturbation Theory
Physics 313
H. S. Greenside
November, 2008
Singular Perturbation Theory For Spatially Extended Systems
The following should give you a flavor for one of the few analytical methods available to analyze the complex
nonlinear partial differential equations that arise in many nonequilibrium pattern forming systems. Since
there are so few analytical methods and because analytical results are valuable when working with a subject
as rich and subtle as nonequilibrium pattern formation, it is important that you have some first-hand
knowledge of how such methods are used, the kinds of conclusions that one can make, and the weaknesses
of such methods. You should not expect to become expert in these methods with this brief exposure, nor
should you feel compelled at this point to truly master the technique. For all except the simplest model
equations, the analytical method can be hard, tedious and subtle to apply and to interpret. Even among
theorists working on nonequilibrium physics, only a few have become experts in singular perturbation theory.
To give a quick overview before getting into details, we note that analytical progress in sustained spatially
extended nonequilibrium systems has been achieved in three different regimes:
Amplitude Equations Near Onset: One regime is near the onset of a supercritical bifurcation, usually
from a uniform to nonuniform state. Sufficiently close to but above onset, there is a small parameter
² = (p−pc )/pc measuring the distance to the onset pc of some bifurcation and one can make systematic
perturbation expansions in this parameter using singular (also called multiscale or two-scale) perturbation theory. This leads to the amplitude equation formalism, in which one solves for slow modulations
of a “fast” pattern.
Phase Equations Away From Onset: Further from onset when ² = O(1) is no longer small, one can
make some analytical progress by assuming that the dynamics takes place in a large cell, with aspect
ratio Γ À 1. One then obtains a small parameter 1/Γ and can do perturbation theory in slow
spatial changes of patterns. This leads to so-called “nonlinear phase equations” which are described
in Section 9.1.2 of the Cross-Greenside book. We will not have time to discuss this approach in a
one-semester course.
Kolmogorov Theory of High Reynolds Turbulence: For still larger driving from equilibrium, at least
for fluid systems, the medium develops strong fluctuations in space and time and can no longer be
treated as a small perturbation (small amplitude or slow distortions) of ideal periodic patterns. This
is a regime for which little analytical progress has been achieved except in the limit of strong driving,
when the Rayleigh or Reynolds number R becomes so large that one can try to make an expansion
in the small parameter 1/R. In 1941, the Russian mathematician Kolmogorov had an inspired insight
that the statistical dynamics of the fluid would be isotropic and self-similar (no preferred length or
time scales) and was able to develop an elegant theory that predicted almost correctly the scaling
of wavenumber spectrum and energy with wavenumber k, a great achievement in fluid dynamics and
nonequilibrium physics. (I say almost, because experiments show an irregular statistical behavior
called “intermittency” that prevents Kolmogorov’s theory from being accurate.) We will not discuss
this theory in the course, you can get a quick glimpse from Section 33 in the Landau and Lifshitz book
Fluid Dynamics, 2nd Edition.
Finally, some progress has been achieved by a non-perturbative method called “renormalization” in which
one tries to integrate out presumably irrelevant noise-like details at short wavelengths and fast time scales
and recover renormalized equations that describe the dynamics at longer wavelengths and slower time scales
(rather similar in spirit to how the Navier Stokes equations of fluid dynamics are derived from the molecular
1
dynamics of colliding molecules). The renormalization approach has succeeded in a few cases (the KuramotoSivashinsky equation is one of them) and so is really a frontier for research rather than a technique that can
be taught at the 313 level. Although not yet useful in nonequilibrium systems, renormalization represents an
important theoretical insight and technique in condensed matter physics, where it unified and clarified much
experimental work on second-order phase transitions (equilibrium equivalent of supercritical bifurcations).
Here we will have time only to discuss the amplitude equation approach since it has the broadest applicability
and has achieved the greatest quantitative success in comparison with experiment. Before getting into details
of the mathematics through some simple examples, you should appreciate the essential insight that makes
singular perturbation work: that often the dynamics of a system can be described by at least two different
length scales (or time scales if thinking about time-dependent problems), in which there is some kind of
fast oscillatory behavior on the short length scale and this fast behavior is modulated slowly on the much
longer second length scale. One can then try to average analytically over the fast local behavior and obtain
an equation for the slowly changing modulation. This new equation is called an “amplitude equation” or
“envelope equation” and is often substantially simpler than the original equations of interest, e.g., it often
involves only a single complex-valued field that depends on two spatial variables while the original equations
may involve many fields that all depend on three spatial variables. Further, the form of the lowest-order
amplitude equation turns out to be universal and independent of the original equations, depending only on
the class of linear stability (type Is , IIIo , etc.).
You have actually seen the occurrence of two length or time scales in your introductory physics course, when
you were hopefully exposed to the idea of “beating”, which arises when two spatial oscillations of nearly
identical identical wavenumbers k1 ≈ k2 are added together:
c1 exp(Ik1 x) + c2 exp(Ik2 x) =
· µ
¶ ¸½
· µ
¶ ¸
· µ
¶ ¸¾
k1 + k2
k1 − k2
k1 − k2
exp I
x
c1 exp I
x + c2 exp −I
x .
2
2
2
(1)
(2)
This formula can be interpreted as a rapidly oscillating wave with wavenumber k1 (the first factor on the right
side) multiplying the factors exp(±I4kx) which vary slowly in space specifically because |4k| = |k1 − k2 |/2
is tiny if k1 is sufficiently close to k2 . So there are two separate length scales, a short (fast) length scale 1/k1
and a long (slow) length scale 1/4k. The ratio of these two length scales defines a small parameter with
which perturbation theory can be attempted.
More generally, a similar situation arises near the onset of a type Is instability. Just above onset, the growth
rate σ(q) for a Fourier mode exp(Iqx) is positive only over a small interval of wavenumbers centered on the
critical wavenumber, q ∈ [q1 , q2 ] with |q2 − q1 | ¿ qc ≈ (q1 + q2 )/2. A time dependent pattern u(t, x) can
then be written as a superposition of the stable modes1
Z q2
u(t, x) =
u(q)eσ(q)t eIqx dq
(3)
q1
= eσ(qc )t eIqc x
Z
q2
u(q)e(σ(q)−σ(qc ))t eI(q−qc )x dq
(4)
q1
= eσ(qc )t+Iqc x A(t, x).
(5)
We again have a leading multiplicative factor representing “fast” growth in time and fast oscillations in space
(at the critical wavenumber qc ), which is multiplied by a function A(t, x) that involves a superposition of
Fourier modes with effective growth rates and effective wave numbers that are all close to zero in magnitude
and so are “slow”. One could then hope to derive a equation specifically for the slow modulations of the
amplitude or envelope function A(t, x). This can indeed be done and is often quite useful as we now proceed
to discuss.
1 This superposition is a crude guess since, once nonlinearities become important, the time dependence is no longer exponential
in time so the eσt factor for each Fourier mode is not correct.
2
A Brief Introduction to Singular Perturbation Theory
Resonances of Perturbed Odes
Perturbation theory generally involves some kind of series expansion in some small quantity ². A famous
and useful example is Isaac Newton’s binomial expansion
(1 + ²)α = 1 +
α
α(α − 1) 2 α(α − 1)(α − 2) 3
²+
² +
² + O(²4 ),
1!
2!
3!
|²| < 1,
(6)
which holds for any number ² of magnitude less than one and for any real exponent α. It turns out that
such expansions break down for many ode and pde problems that involve oscillations in either space or time
because of resonances, in which a nonlinear term generates an external oscillation that resonates with, and
so strongly excites, an existing oscillation. The result is that low-order results obtained by perturbation
theory can grow in time or space and fail to be small for all time or over all of space. Singular perturbation
theory provides a systematic way to cancel dangerous resonances and so yield a perturbation expansion that
remains consistent over long time or space scales.
To see how nonlinear terms can generate dangerous resonances, let us first recall how resonances arise in a
simple harmonic oscillator that is driven by an external sinusoidal term of angular frequency ω:
d2 y
+ y = cos(ωt).
dt2
(7)
The oscillator without driving, y 00 + y = 0, has a natural frequency ω = 1. For off-resonance driving with
ω 6= 1, the general solution to Eq. (7) is
y(t) = A cos(t) + B sin(t) +
1
cos(ωt),
1 − ω2
(8)
where the constants A and B are determined by the two initial data for Eq. (7). This solution remains
bounded for all time. On the other hand, if the driving is on resonance with ω = 1, one has a rather different
solution (as you can verify directly)
y(t) = A cos(t) + B sin(t) +
t
sin(t),
2
(9)
which grows without bound as t → ∞. The term with the time-dependent coefficient is called a secular2
term and arises because the inhomogeneity cos(t) in Eq. (7) is itself a solution of the left side ode. More
generally, one will get a secular response to an inhomogeneous constant coefficient ode,
y (n) + c1 y (n−1) + · · · + cn y 0 = g(t),
(10)
if the right side g(t) has an additive piece that is a solution of the left side homogeneous ode.
Nonuniformity of Regular Perturbation Theory
We can now see how a secular term causes a standard perturbation method to break down at long times,
in which case the expansion is said to be “non-uniform” since it holds only over short times and so is not
uniform in time. As an example, let us consider the nonlinear Duffing equation
d2 y
+ y + ²y 3 = 0,
dt2
(11)
2 The word secular, which has the non-technical meaning of occurring once in an age or century, arose historically from that
fact that singular perturbation methods were first developed for astronomical problems such as the motion of the moon and
planets, in which case the time for secular terms to become significant could indeed be of order centuries.
3
with initial data
y 0 (0) = 0,
y(0) = 1,
(12)
for ² a small parameter. This is one of the simplest nonlinear odes to consider and arises naturally when
considering the lowest-order nonlinear correction to the restoring force of a nonlinear spring or oscillator.
As an example of how the Duffing equation arises from some nonlinear problem, consider the small-amplitude
limit of a nonlinear undamped pendulum
d2 θ
+ sin(θ) = 0.
dt2
(13)
If we assume small amplitude excursions so that |θ(t)| < θ0 ¿ 1 is small for all time, we can expand the sin
term to lowest nonlinear order:
1
sin(θ) = θ − θ3 + O(θ5 ),
(14)
3!
Substituting Eq. (14) into Eq. (13) gives
θ00 + θ −
1 3
θ + h.o.t.s = 0,
3!
(15)
where “h.o.t.s” stands for “higher order terms”. If we assume θ = ²y is a small quantity where y is some
new variable of order one, Eq. (15) becomes
y 00 + y −
²2 3
y = 0,
3!
(16)
which is the Duffing equation Eq. (11) upon redefining ² accordingly.
Given that the cubic term in the Duffing equation is a small perturbation to a standard harmonic oscillator,
let us try a standard perturbation solution to Eqs. (11) and (12) by seeking a solution of the form:
y(t) = y0 (t) + ²y1 (t) + ²2 y2 (t) + · · ·
∞
X
=
²n yn (t),
(17)
n=0
where we anticipate that the functions yn (t) will be of order one since their smallness is already taken into
account by the explicit powers of ². We can satisfy the initial values Eq. (12) by requiring
y0 (0) = 1;
y00 (0) = 0,
0
yn (0) = yn (0) = 0,
n ≥ 1.
(18)
(19)
Challenge: Is this the only assignment of values that will satisfy the initial data Eq. (12)?
Substituting Eq. (17) into Eq. (11), using Eq. (6), and equating coefficients of like powers of ² (why is this
justified?), gives a sequence of linear differential equations of which all but the first are inhomogeneous:
y000 + y0 = 0,
y100 + y1 = −y03 ,
(20)
(21)
y200 + y2 = −3y02 y1 ,
···
(22)
Challenge: What is the right side for the O(²3 ) equation, y300 + y3 =?
The solution to the first equation, Eq. (20), satisfying the initial data Eq. (18) is simply
y0 (t) = cos(t).
4
(23)
To solve the next equation, Eq. (21), we observe that the right side can be written
3
1
−y03 = − cos(t)3 = − cos(t) − cos(3t),
4
4
(24)
and we see that a resonant −(3/4) cos(t) term has appeared. I will leave it to you as a simple exercise to
show that the general solution of Eq. (21) is given by
y1 (t) = A cos(t) + B sin(t) +
1
3
cos(3t) − t sin(t).
32
8
(25)
The constants A and B are determined by the initial data Eq. (19), which gives the specific solution
y1 (t) = −
1
1
3
cos(t) +
cos(3t) − t sin(t).
32
32
8
(26)
These results tells us that the solution y(t) to the Duffing equation Eq. (11) can be written to lowest order
as:
y(t) = y0 + ²y1 + O(²2 )
½
¾
1
1
3
= cos(t) + ² − cos(t) +
cos(3t) − t sin(t) + O(²2 ).
32
32
8
(27)
(28)
However, the big-oh term O(²2 ) is correct for a fixed time t only! When the time becomes so large that
²t > 1
or
t>
1
,
²
(29)
the secular term becomes of order one and the first-order term in Eq. (28) is no longer small. In fact, the
solution Eq. (28) has the alarming property that it diverges as t → ∞. This can’t be right since one can
easily show that solutions to the Duffing equation Eq. (11) are bounded for all time.
Challenge: Prove that the solution to Eq. (11) with initial data Eq. (12) is bounded for
all time.
Hint: This can be done without explicit knowledge of the solution y(t) by an “energy”
method: multiply both sides of Eq. (11) by an integrating factor dy/dt and integrate once
to get a constant of motion in the form of an algebraic relation a(y 0 )2 + by 2 + cy 4 = C,
with positive constants a, b, c, and C. Then explain why this relation forces y and its
derivatives to be bounded for all time.
The divergence of the first-order term with increasing t is somewhat misleading as a sign of trouble. The
example
1
(−1)n n n
e−²t = 1 − ²t + ²2 t2 + · · · +
² t + ··· ,
(30)
2!
n!
shows that one can have a perturbative expansion in which the term at each order is secular and yet the sum
of infinitely many such secular terms yields a finite quantity, since e−²t → 0 as t → ∞ if ² > 0. Chapter 11 of
the Bender-Orszag book shows that a similar phenomenon occurs for the Duffing equation. Using induction,
one can find and sum analytically the most secular terms at each order ²n (the pieces with the highest power
of t and so the most rapidly growing as t becomes large) for the expansion Eq. (17) and find the answer
¶¸
· µ
3
(31)
y(t) ≈ cos t 1 + ² .
8
This is interesting: the unperturbed solution cos(t) for ² = 0 turns into a perturbed solution of the Duffing
equation in which the frequency is “renormalized” or changed by an amount of order ². And indeed Eq. (31)
is bounded for all time.
5
The Method of Multiple Scales
Let us review where we are so far. We observed that the singular perturbation theory of nonlinear partial
differential equations is a general way to eliminate resonances that create secular terms when a straightforward perturbation theory is carried out. As a simple example, we looked at a simple nonlinear ode, the
undamped undriven Duffing equation
d2 y
+ y + ²y 3 = 0,
(32)
dt2
and observed that the cubic nonlinearity causes a secular term to appear at O(²) (and at all higher orders
also). Rather remarkably, adding up the most secular terms at each order in perturbation theory leads to a
uniformly convergent result (i.e., an answer whose accuracy does not change with time) that was also useful,
i.e., it gave a more accurate approximation to the analytical solution of the Duffing equation for given ics
y 0 (0) = 0.
y(0) = 1,
(33)
Finding and summing the most secular terms of a perturbed ode is generally too awkward and laborious.
Fortunately, there is an ingenious method—the method of multiple scales—that allows one to eliminate
secular terms at successive orders and to obtain a uniformly convergent perturbation expansion. For an
ordinary differential equation like the Duffing equation, the method produces a sequence of partial differential
equations ordered by powers of ². This might seem a step backwards since pdes are generally harder to solve
than odes but the pdes offer some increased flexibility to adjust the perturbation series so that secular terms
do not appear. If one carries out a singular perturbation expansion of some pde, one still ends up with
a sequence of pdes at successive orders but the pdes are often much simpler than the original pde. For
example, the sequence of pdes may involve fewer independent variables and involve simpler operators.
To see how this works, recall that our first attempt at a simple-minded perturbation theory for Eq. (32) with
ics Eq. (33) led to the expression Eq. (28) which was secular because of the combination ²t appearing at first
order. We can think of this product as being a “slow” time scale since, if ² is small, the quantity ²t increases
slowly (does not change much) when t changes by order one. Let us then introduce an explicit slow variable
T = ²t,
(34)
and try the crazy idea of treating t and T as independent variables when carrying out a perturbation expansion
for the solution y(t), which we now schizophrenically think of as a function of both t and T , y = y(t, T ).
Let us see where this might lead. We again write out the perturbation expansion Eq. (17) but now with the
variables t and T appearing in each term:
y(t) = Y0 (t, T ) + ²Y1 (t, T ) + · · · .
We also observe that time derivatives of y will now require invocation of the chain rule:
µ
¶
µ
¶
dy
∂Y0
dT ∂Y0
∂Y1
dT ∂Y1
=
+
+²
+
+ O(²2 )
dt
∂t
dt ∂T
∂t
dt ∂T
µ
¶
∂Y0
∂Y1
∂Y0
+²
+
+ O(²2 ).
=
∂t
∂T
∂t
(35)
(36)
Note the characteristic way that the first-order piece has a derivative of Y0 w.r.t. the slow variable T and the
derivative of the higher-order piece Y1 w.r.t. the fast variable t. Differentiating yet once more, you should
verify that:
µ 2
¶
∂ 2 Y0
∂ Y0
∂ 2 Y1
d2 y
=
+
²
2
+
+ O(²2 ).
(37)
dt2
∂t2
∂t∂T
∂t2
Substituting Eq. (37) and Eq. (35) into Eq. (32) and collecting like powers of ² gives again a sequence of terms
6
for each order of perturbation theory, but now the differential equations are partial rather than ordinary:
∂ 2 Y0
+ Y0 = 0,
∂t2
∂ 2 Y1
∂ 2 Y0
3
+
Y
=
−Y
,
−
2
1
0
∂t2
∂t∂T
···
(38)
(39)
Challenge: Derive the next O(²2 ) equation for Y2 (t, T ).
Here is where the craziness of treating t and T as separate variables pays off. Eq. (38) looks like our previous
zeroth-order harmonic-oscillator equation Eq. (20), y000 + y0 = 0, but for the crucial fact that it involves
partial derivatives w.r.t. the fast variable t. This means that the general solution to Eq. (38) has to allow
coefficients that involve the independent slow variable T and so has the form:
Y0 (t, T ) = A(T )eIt + A∗ (T )e−It ,
(40)
where A(T ) is some arbitrary complex-valued function of the slow variable T . (The use of a complex coefficient A(T ) and exponential form eIt is a convenient and concise way to represent the two linearly independent
harmonic solutions involving sin and cos; it is completely equivalent to writing Y0 = C(T ) cos(t)+S(T ) sin(t)
for real coefficients C(T ) and S(T ) determined by initial conditions.) But we can use the arbitrariness of
this A(T ) function to require that secular terms disappear in the next order equation Eq. (39), which will
make the perturbation expansion more well-behaved. In fact, you should be able to use Eq. (40) to show
that the right side of Eq. (39) can be written in the form:
½
¾
©
ª
∂ 2 Y0
dA
−Y03 − 2
= eit −2i
− 3|A|2 A + e3it −A3 + c.c.,
(41)
∂t∂T
dT
where as usual “c.c.” denotes the complex conjugate of the preceding terms (so “αeit + c.c.” is a short hand
for “αeit + α∗ e−it ”). The terms e±it are solutions of the constant-coefficient homogeneous pde of the left
side
∂ 2 Y1
+ Y1 = 0,
(42)
∂t2
and so will cause a resonance that generates a secular response and that makes the perturbation theory
nonuniform in time unless we can somehow eliminate the e±it terms in Eq. (41). But we can eliminate these
resonant terms by requiring the coefficient of eit in Eq. (41) to vanish
−2i
dA
− 3|A|2 A = 0.
dT
(43)
This is an amplitude equation for A, that is an ode or pde (here an ode) for a slowly evolving envelope function A(T ) that represents the condition that a secular response is set to zero at some order in a perturbation
expansion. At lowest order in ², you should think of A(T ) as modulating the underlying fast oscillation eit
in Eq. (40).
You should appreciate that an amplitude equation such as Eq. (43) is not a physical equation that can be
derived from first-principle physical laws such as conservation of energy, momentum, or mass; the lowestorder amplitude equation rarely has a physical interpretation. On the other hand, the amplitude equation can
often be understood as a simple consequence of existing symmetries of the fundamental physical equations.
For example, the nonlinear pendulum and Duffing equation have a reflection symmetry y → −y and so the
lowest-order equation for a complex function A(T ) would have the form A0 + c|A|2 A + h.o.t.s = 0 where c
is some complex coefficient. You should also appreciate that the function A(T ) is not directly a physical
quantity. Further, this amplitude equation only removes the secular driving from the O(²) equation Eq. (39)
and does not do anything to remove secular responses at higher-order in the right sides of the Y2 or higher
equations. Thus we can expect that the perturbation expansion Eq. (35) to be valid only up to times t < 1/²2 ,
and not uniformly valid over all time.
7
Solution of the Amplitude Equation 0 = −2iA0 − 3|A|2 A
The lowest-order amplitude equation Eq. (43) turns out to be analytically solvable and so we can complete
the process of calculating an explicit O(²) approximation to the Duffing equation that is uniform over a
time scale t > 1/². An often useful gambit when dealing with complex amplitude equations is to write the
solution in polar complex form
A(T ) = a(T )eiθ(T ) ,
(44)
where a(T ) is a real-valued magnitude and θ(T ) is a real-valued phase. Substituting Eq. (44) into Eq. (43)
and collecting the linearly independent real and imaginary parts gives two coupled odes for the magnitude
and phase:
da
= 0,
dT
dθ
3
= a2 .
dT
2
Eq. (??) implies that a = a0 is a constant while Eq. (46) is easily integrated to give
(45)
(46)
3
(47)
θ(T ) = θ0 + a20 T,
2
where θ0 is another constant of integration. We conclude that the solution to the amplitude equation Eq. (43)
has the form
· µ
¶¸
3 2
iθ
A(T ) = ae = a0 exp i θ0 + a0 ,
(48)
2
and so the zeroth-order multiscale approximation to the Duffing equation solution is given by
Y0 (t, T ) = A(T )eit + c.c.
¶
µ
3 2
= 2a0 cos θ0 + a0 T + t .
2
(49)
At this point, we need to use the specified initial conditions, Eq. (33), to determine the values of the
constants a0 and θ0 . We can write
y(0) = 1
=⇒
Y0 (0, 0) = 1, Y1 (0, 0) = 0,
(50)
∂Y
∂Y
∂Y
0
1
0
y 0 (0) = 0 =⇒
(0, 0) = 0,
(0, 0) = −
(0, 0),
(51)
∂t
∂t
∂T
where the condition ∂t Y1 = −∂t Y0 follows from Eq. (36) above. (More explicitly, in the expansion Eq. (35),
it is often simplest to impose any nontrivial ics on the lowest-order term Y0 and then impose zero ics on the
higher-order terms.) From Eq. (49) and Eq. (50), we know that
Y0 (0, 0) = 2a0 cos(θ0 ) = 1,
(52)
and similarly combining Eq. (49) with Eq. (51) gives a second equation
∂t Y0 (0, 0) = −2a0 sin(θ0 ) = 0.
(53)
From Eq. (53), we deduce that θ0 = 0 and Eq. (52) then implies a0 = 1/2. We then have finished deriving
our zeroth-order multiscale solution to the Duffing equation which turns out to be
·
¸
· µ
¶¸
3
3
Y0 (t, T ) = cos t + T = cos t 1 + ² .
(54)
8
8
The lowest-order correction is a shift in angular frequency by the amount (3/8)², which turns out most
remarkably to be the same quantity that was derived by adding all of the most secular terms in the regular
perturbation theory. (By the way, I don’t understand this coincidence or the generality of such coincidences.)
The Mathematica code below generates plots which show that Eq. (54) is much more accurate than the
previous solution y0 = cos(t) but eventually becomes inaccurate (relative error larger than ²) on time scales
of order 1/².
8
eps = 0.1 ;
(* Set parameter to some value *)
time = 300. ;
(* Integration interval *)
nsoln = NDSolve[
(* MMa fn for numerical solution of odes *)
{
y’’[t] + y[t] + eps y[t]^3 == 0 ,
y[0] == 1 ,
y’[0] == 0
} ,
y ,
{t, 0, time} ,
(* Time range of integration *)
MaxSteps -> 5000 ,
PrecisionGoal -> 6 , (* Relative error of six digits *)
AccuracyGoal -> 6
(* Absolute error of six digits *)
] ;
g1 = Plot[
(* Plot the numerical Duffing solution, presumably exact *)
Evaluate[ y[t] /. nsoln ] ,
{t, 0, time} ,
PlotPoints -> 200,
AspectRatio -> 1/3
]
g2 = Plot[
(* Plot the secular approx y0 + eps y1 *)
Cos[t] + eps (
(-1./32.) Cos[t] + (1./32.) Cos[3t] - (3./8.) t Cos[t]
) ,
{t, 0, time} ,
PlotPoints -> 200,
AspectRatio -> 1/3
]
g3 = Plot[
(* Plot the multiple-scales solution *)
Cos[(1. + (3./8) eps) t ] ,
{t, 0, time} ,
PlotPoints -> 200,
AspectRatio -> 1/3.
]
g4 = Plot[
(* Plot the zeroth-order solution *)
Cos[t] ,
{t, 0, time} ,
PlotPoints -> 200,
AspectRatio -> 1/3.
]
g5 = Plot[
(* Compare Y0 with y(t) *)
Cos[(1. + (3./8) eps)t ] - Evaluate[ y[t] /. nsoln ] ,
{t, 0., time} ,
PlotPoints -> 200
]
g6 = Plot[
(* Compare y0 with y(t) *)
9
Cos[t] - Evaluate[ y[t] /. nsoln ] ,
{t, 0., time} ,
PlotPoints -> 200
]
g7 = Plot[
(* Compare secular expansion Eq.(28),
Cos[t] + eps (
(* y0 + eps y1, with y(t) *)
(-1./32.) Cos[t] + (1./32.) Cos[3t] - (3./8.) t Cos[t]
) - Evaluate[ y[t] /. nsoln ],
{t, 0, time} ,
PlotPoints -> 200
]
*)
Challenge: Run the above Mathematica commands and include the output in your homework assignment.
Please run these commands on your own and think about what the plots mean. In particular, for the
comparision plots, look at the scale of the vertical axis to appreciate how big or small the error is. All
the perturbation approximations look cosine-like but the phase error grows more rapidly for the standard
perturbation method compared to the multiscale method. On time scales of order 1/²3 ≈ 1000, one finally
sees relative errors of O(1) in the difference between Y0 (t, T ) and the numerical Duffing solution.
To see if you understand the above arguments, please solve the following problem.
Challenge: For the Duffing-like nonlinear oscillator
y 00 + y + ²(y 0 )3 = 0,
with ics
y 0 (0) = 0,
y(0) = 1,
(55)
(56)
show that the O(²) multiscale approximation is given by
cos(t)
Y0 (t, T ) = q
,
1 + 34 ²t
(57)
and compare this solution with a numerical “exact” solution for the two cases ² = 0.1
and ² = −0.1.
Amplitude Equation For Type-Is Instabilities
Now that you have some experience with deriving and solving an amplitude equation for a perturbed nonlinear ode, let us discuss (but not fully carry out) applying this technique to the harder case of deriving an
amplitude equation for a pattern-forming partial differential equation such as the Swift-Hohenberg equation3 .
¡
¢
∂t u(t, x, y) = r − (4 + 1)2 u − u3 .
(58)
The singular perturbation analysis works quite generally for even more difficult pdes such as the five threedimensional Boussinesq equations or the CDIMA reaction-diffusion equations but the algebra is much simpler
for the model equation Eq. (58) and so better illustrates what is going on.
One has to proceed differently from our thinking for the Duffing equation y 00 + y 0 + ²y 3 = 0. In that case, we
recognized that the cubic term was a perturbation of an underlying harmonic oscillator equation and it was
fairly easy to identify the unperturbed dynamics as a cos(t) function which was then slowly modulated in
3 The notation 4 = ∇2 = ∂ 2 + ∂ 2 denotes the two-dimensional Laplacian operator and we will postpone for now prescribing
x
y
appropriate boundary conditions on the field u.
10
phase by the envelope A(T ). If we followed this reasoning for the SH equation, we would set the perturbation
parameter r to zero and look for an analytical solution for the remaining equation but this leads to nonsense.
The reason is that what is being perturbed is not a piece of the equation Eq. (58) but the distance r from
the onset of a bifurcation, where the state at r = 0 consists in one-space dimension of a stripe solution at
the critical wavenumber4 qc with vanishing amplitude. As r increases to just above zero, the stripe solution
can develop some modulations in amplitude and phase (the latter corresponding to small adjustments in the
local wavenumber and orientation) and so we might guess that we could write:
u(t, x, y) = A(T, X, Y )eiqc x + c.c.,
(59)
where A(T, X, Y ) is some complex-valued envelope or amplitude function that evolves slowly in time and
space as a function of the “slow” variables T , X, and Y whose scaling with the small parameter r we
have yet to determine. From our earlier analysis of √
steady-state one-dimensional solutions of the SwiftHohenberg
equation,
we
would
also
guess
that
A
=
O(
r) near onset since√the stripe√solution itself is of the
√
form r sin(qx) for wavenumbers q in the neutral stability band q ∈ [1 − r/2, 1 + r/2].
Note that Eq. (59) explicitly singles out the x-direction as the orientation of the stripes and the slow
modulations then occur about this direction; the amplitude formalism can not handle patterns involving
rolls that curve substantially, over angles of order one. This singling out of a particular stripe orientation
is not consistent with the rotational invariance of Eq. (58) and so is philosophically undesirable. However,
there is no convenient way to calculate the distortions of some underlying ideal pattern without specifying
the orientation. The phase diffusion equation approach (expansion in inverse aspect ratio) mentioned earlier
can be made into a rotationally invariant formalism that can handle substantial bending of stripes but can
not handle defects or boundaries, which amplitude equations can.
Using multiscale perturbation theory, the two lowest-order terms of the expansion would look like this
u(t, x, y) = ²1/2 Y0 (x; T, X, Y ) + ² Y1 (x; T, X, Y ) + O(²3/2 ).
(60)
√
Note how successive terms are smaller by ², unlike the case for the Duffing equation in which successive
terms were smaller by the first power of the perturbation parameter. Note also the fact that the Yn depend
only on the fast variable x since we are perturbing around a static stripe pattern aligned perpendicular to
the x-axis.
Section A2.3.2 in Appendix 2 of the Cross-Greenside book shows that the “solvability condition”, which
removes the resonance driving that could cause a secular term to appear at lowest nontrivial order in the
perturbation expansion, yields the following amplitude equation for A:
µ
¶2
i 2
2
τ0 ∂T A = A + ξ0 ∂X −
∂
A − g0 |A|2 A,
(61)
2qc Y
where the three parameters τ0 , ξ0 , and g0 have the values
τ0 = 1,
ξ02 = 4qc2 ,
g0 = 3.
(62)
Quite remarkably, Eq. (61) turns out to be the lowest-order amplitude equation for any type-Is instability of
a uniform to non-uniform state, e.g., it also holds for the five three-dimensional Boussinesq equations that
quantitatively describe convection of an incompressible fluid. Only the coefficients change from one system
to another, e.g., for the Boussinesq equations with free-slip bcs on the horizontal plates, the parameter values
in Eq. (62) change to
8
8
2
,
ξ02 =
,
g0 =
.
(63)
τ0 =
2
2
3π
3π
3π 2
For the Boussinesq equations with no-slip bcs on the horizontal plates (appropriate for most laboratory
experiments), the coefficients are again different
τ0 =
19.656
,
σ + 0.5117
ξ02 = 0.148,
and
g0 = 0.6995 − 0.0047σ −1 + 0.0083σ −2 ,
(64)
4 The critical wavenumber is q = 1 for Swift-Hohenberg because of the way various lengths and times have been scaled. For
c
now, it is useful to retain qc explicitly to illustrate the more general case.
11
where σ is the Prandtl number.
The fact that the same amplitude equation appears at lowest order for all type-Is instabilities is a kind of
universality: the dynamics of stripe patterns on slow time scales and over large length scales are independent
of the underlying equations. This means that a detailed analysis of Eq. (61) can answer once and for all
certain kinds of details of nonequilibrium dynamics near onset. We will see later on that solutions of the
amplitude equation Eq. (61) go unstable to Eckhaus and zig-zag instabilities which therefore themselves
have a universal nature for stripe patterns. It also turns out that Eq. (61) is variationally derived with a
Lyapunov functional and so, to lowest order, all dynamics decay to time-independent states (which are not
necessarily of the form Eq. (??)).
Another striking consequence of this universality is that a two-dimensional model like the Swift-Hohenberg
equation can have the same slow dynamics as the three-dimensional Boussinesq equations. Historically,
the amplitude equation for the Boussinesq equations was derived before Swift-Hohenberg. When it was
found to depend only on two space variables (X and Y ), researchers realized that one might be able to find
two-dimensional model equations5 that that yield the same two-dimensional amplitude equation, and the
Swift-Hohenberg model is indeed the simplest rotationally invariant and translationally invariant example
that does this. (Many other cubic terms also lead to the same amplitude equation.) The fact that SwiftHohenberg has the same dynamics as the Boussinesq equations on slow time scales and long wavelengths
helps to explain why Swift-Hohenberg is successful at capturing much of the pattern formation and dynamics
of a real convecting fluid near onset.
This insight also suggests a strategy for finding interesting model equations: derive an amplitude equation to
some order for “real” equations like Boussinesq and reaction-diffusion and then see if one can find equations
of reduced dimensionality (fewer spatial variables) that produce the same amplitude equation.
Type IIIo and Io Amplitude Equations
For instabilities other than type-Is , one perturbs around a basic “ideal” pattern that is no longer a stationary
stripe pattern. Type IIIo instabilities correspond to global oscillations with critical frequency ωc of an
otherwise homogeneous state and only a fast time variable t appears in the fast part of an amplitude
representation:
U(t, x, y, z) = Y0 (t; T, X, Y ) + O(²)
= U0 A(T, X, Y )eiωc t + c.c. + O(²).
(65)
(66)
Here the vector U0 is possibly dependent on variables transverse to the x − y plane in which patterns form.
To lowest order, one can show that the amplitude A satisfies a universal amplitude equation known as the
complex Ginzburg-Landau equation (often abbreviated CGLE) which (after rescaling of time, length, and
amplitude) can be written in terms of fast variables in the form
∂t A(t, x, y) = ²A + (1 + ic1 )4A − (1 − ic3 )|A|2 A,
(67)
with appropriate bcs that are not universal and that depend on the specific problem.
Challenge: Show that Eq. (67) on an infinite domain has two kinds of analytical solutions,
an oscillatory solution of the form
√
A(t) = ²ei(c3 ²)t ,
(68)
and traveling wave solutions of the form
p
Ak (t, x, y) = ² − k2 ei(k·x−ωk t) ,
ωk = −c3 ² + (c1 + c3 )k2 .
(69)
5 Such model equations are often called “fast” or “microscopic” dynamical equations compared since all variables evolve on
time scales and length scales that are fast compared to the amplitude equations
12
Upon study, the CGLE has turned out to be an immensely interesting dynamical model in its own right,
with an amazingly rich variety of periodic and chaotic dynamics as the parameters c1 , c3 , Γ, and boundary
conditions are varied. The CGLE, Swift-Hohenberg, and Kuramoto-Sivashinsky equations are among the
most widely and intensely studied model partial differential equations in nonequilibrium physics because of
their simplicity, universality, and rich dynamics.
As an aside, I can finally explain where the KS equation comes from. If one writes the one-dimensional
CGLE amplitude in polar complex form:
A(t, x) = a(t, x)eθ(t,x) ,
(70)
and substitutes Eq. (70) into Eq. (67), and if one choose the parameters c1 and c3 to be just above the
so-called Newell line c1 c3 > 1 beyond which all traveling wave solutions Eq. (69) become linearly unstable6 ,
then it turns out that a → 1 as c1 c3 − 1 → 0 and so the only dynamical variable near this onset is the
phase θ(t, x). But then a perturbation expansion in gradients of the phase yields none other than the
Kuramoto-Sivashinsky equation, with the identification u = ∂x θ. So the KS equation can be derived as a
limiting case of the universal CGLE and so itself has a universal significance for many type IIIo instabilities
of a uniform state. Further, we now understand the KS field as being related to the phase of CGLE solutions.
For a type-Io instability in one-space dimension, there are two fast variables t and x since the typical physical
solution is a traveling wave. In one space dimension, it turns out that the amplitude representation of a
slowly modulated traveling wave needs two amplitudes to represent right- and left-traveling waves:
U(t, x, y, z) = Y0 (t, x; T, X, Y ) + O(²)
h
i
= U0 AR (T, X, Y )ei(qc x−ωc t) + AL (T, X, Y )e−i(qc x+ωc t) + O(²).
(71)
(72)
The two amplitudes can be shown to obey coupled CGL equations similar to Eq. (67) and are given by
Eqs. (10.82) and (10.83) on page 394 of the book. Again, the equations have a universal form with details
of the original equations arising only in various coefficients of various terms. For type-Io instabilities, the
boundary conditions are now especially important because the waves reflect at finite boundaries and the
reflected waves interact nonlinearly with the original wave.
Further Reading
The references
1. Chapter 11 of the book Advanced Mathematical Methods for Scientists and Engineers by Carl Bender
and Steven Orszag (McGraw-Hill, 1978).
2. Chapter 9-11 of the book Mathematics Applied to Deterministic Problems in the Natural Sciences by
C. Lin and L. Segel (SIAM,1988).
have some useful background information on singular perturbation theory, with more detail than provided
in the notes that follow. Applications of the theory to pattern formation evolution equations can be found in
Appendix 2 of the Cross-Greenside book and in pattern formation books by Paul Manneville and Rebecca
Hoyle. The calculations are complicated enough, even for Swift-Hohenberg, that we will not work through
the details for patterns near onset.
6 Traveling waves become linearly unstable to a long-wavelength Eckhaus-like instability called the “Benjamin-Feir” instability
named after British researchers who first discovered this instability in fluid flows. Alan Newell later discovered that this is a
universal instability that occurs in all type-IIIo instabilities. The criterion c1 c3 > 1 known as the “Newell line” was first derived
by Newell as the condition for all traveling wave solutions to be linearly unstable.
13