Spots in the Swift–Hohenberg equation

Spots in the Swift–Hohenberg equation
Björn Sandstede
Division of Applied Mathematics
Brown University
Providence, RI 02912, USA
Scott McCalla
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA 90095, USA
June 20, 2012
Abstract
The existence of stationary localized spots for the planar and the three-dimensional Swift–Hohenberg
equation is proved using geometric blow-up techniques. The spots found in this paper have a much larger
amplitude than that expected from a formal scaling in the far field. One advantage of the geometric blow-up
methods used here is that the anticipated amplitude scaling does not enter as an assumption into the analysis
but emerges naturally during the construction. Thus, the approach used here may also be useful in other
contexts where the scaling is not known a priori.
1
Introduction
Despite their long history, localized structures remain of great interest in the pattern-formation community,
and many recent experimental and theoretical studies have, in fact, focused exclusively on localized patterns.
Examples of localized planar structures that have recently been studied experimentally are activation patterns
in chemical reactions [20], vegetation patches in deserts [19], and surface structures in ferro-magnetic fluids [16];
references to other experiments as well as to theoretical work can be found in the survey articles [7, 9]. In
three space dimensions, localized spherically-symmetric patterns have been found experimentally in Belousov–
Zhabotinsky reactions [3] and previously in numerical simulations, for instance in [12].
Some of the recent theoretical investigations in this area focused on proving the existence of localized patterns
with small amplitude near bifurcations from a homogeneous rest state. Among the bifurcation scenarios that
lead to localized structures are Turing bifurcations: it was shown in [18] that small-amplitude radial patterns
that emerge at planar Turing bifurcations are captured by the radial stationary Swift–Hohenberg equation
2
1
− ∂r2 + ∂r + 1 u − µu + νu2 − u3 = 0,
r
r>0
(1.1)
which is the normal form in this setting. This motivated the work [13] where localized solutions of (1.1) were
studied, and we now briefly review their results. First, for each fixed ν > 0, it was shown there that (1.1) admits
a spot (a localized solution whose amplitude is maximal at the core r = 0) for each 0 < µ 1. Furthermore,
p
it was shown in [13] that equation (1.1) has, for each fixed ν > ν∗ := 27/38, many ring solutions (localized
solutions whose amplitude is maximal away from the core) for each positive µ close to zero. The amplitude of
1
the spots and rings constructed in [13] scales like µ 2 as µ goes to zero: this scaling is expected as the far-field
behavior of solutions of (1.1), that is, the shape of their profile as r goes to infinity, is described to leading order
1
by a complex Ginzburg–Landau equation, whose solutions scale naturally with µ 2 . However, in the subsequent
work [14], we found a second family of spots that exists in (1.1) for ν > ν∗ and whose amplitude appears to
1
Figure 1: The left panel contains profiles of spot A and spot B solutions of (1.1) for µ = 0.005 and ν = 1.6. The right
3
panel illustrates the anticipated µ 8 scaling of the amplitude of spot B.
3
scale like µ 8 as µ goes to zero; in particular, their amplitude is larger than that expected from the asymptotic
Ginzburg–Landau equation. We remark that this second family was also found numerically in [10, Figure 11(c)]
but the amplitude scaling was not investigated there. To distinguish this family from the one found earlier in
1
[13], we refer to the spots with amplitudes of order O(µ 2 ) as spot A and to those whose amplitudes scale like
3
µ 8 as spot B: typical profiles of these solutions are shown in Figure 1.
In this paper, we shall prove the existence of spot B in the planar Swift–Hohenberg equation and illuminate
the origin of the amplitude scaling; in particular, the geometric blow-up analysis presented here will explain the
exponent 38 that appears in the amplitude scaling. We will also generalize these results to the Swift–Hohenberg
equation posed in three dimensions and establish the existence of both spot A and spot B solutions in this case.
First, we formulate our results for the planar Swift–Hohenberg equation given by
ut = −(1 + ∆)2 u − µu + νu2 − u3 ,
x ∈ R2 .
(1.2)
We are interested in stationary localized radial solutions u(x, t) = u(|x|) of (1.2) for 0 < µ 1 that are bounded
as |x| → 0, satisfy lim|x|→∞ u(|x|) = 0 and have small amplitude. Such solutions therefore satisfy (1.1) for r > 0.
We will always take ν > 0 as the case ν < 0 can be recovered upon taking u to −u. We need to make the
following assumption.
Hypothesis (H1) The equation
Ass = −
A
As
+ 2 + A − A3 ,
s
4s
A∈R
(1.3)
has a bounded nontrivial solution A(s) = q(s) on [0, ∞). In addition, the linearization of (1.3) about q(s) does
not have a nontrivial solution that is bounded uniformly on R+ .
We remark that [17, Propositions 1-3] asserts that Hypothesis (H1) is true but it appears as if the proof given
in [17] is flawed, and we therefore state the hypothesis as we were unable to verify it analytically; we refer to
Appendix A for a numerical verification using auto07p. The following theorem is our first main result.
p
Theorem 1 Fix ν > ν∗ := 27/38 and assume that Hypothesis (H1) is met; then there is a µ0 > 0 such that
3
equation (1.2) has a stationary localized radial solution u(x, t) = uB (|x|) of amplitude O(µ 8 ) for each µ ∈ (0, µ0 ).
More precisely, there is a constant d > 0 such that uB (|x|) has the expansion
3
√
uB (r) = −dµ 8 J0 (r) + O( µ)
uniformly on bounded intervals [0, r0 ] as µ → 0, where J0 is the Bessel function of the first kind of order zero.
2
Figure 2: The dynamics of the five-dimensional autonomous version of (1.5) are illustrated in this three-dimensional
cartoon. Details of the coordinate charts and the dynamical behavior are given in the main text.
The preceding result remains valid if we add terms of order O(u4 ) to the nonlinearity in (1.1) or (1.2): since
we consider small-amplitude solutions, the proof given here for the cubic nonlinearity carries over without any
change to the more general situation.
Next, we discuss the existence of localized radial spots for the three-dimensional Swift–Hohenberg equation
ut = −(1 + ∆)2 u − µu + νu2 − u3 ,
x ∈ R3
(1.4)
in the region 0 < µ 1. We have the following existence result for spots of (1.4) and refer to Theorems 3 and 4
in §4 for additional properties of these solutions.
Theorem 2 First, fix ν > 0; then there is a µ0 > 0 such that equation (1.4) has a stationary localized radial
1
solution u(x, t) = uA (|x|) of amplitude O(µ 2 ) for each µ ∈ (0, µ0 ): there is a constant d > 0 such that
1
uA (r) = dµ 2
sin r
+ O(µ)
r
p
uniformly on bounded intervals [0, r0 ] as µ → 0. Second, fix ν > ν∗ = 27/38; then there is a µ0 > 0 such that
1
equation (1.4) has a stationary localized radial solution u(x, t) = uB (|x|) of amplitude O(µ 4 ) for each µ ∈ (0, µ0 ):
there is a constant d > 0 such that
1 sin r
√
uB (r) = −dµ 4
+ O( µ)
r
uniformly on bounded intervals [0, r0 ] as µ → 0.
In the remainder of the introduction, we shall outline our strategy for proving Theorem 1 and comment on the
mechanism behind the unfamiliar amplitude scaling. The rest of the paper is then concerned with implementing
this strategy for the planar case and adapting it to the three-dimensional situation. We begin our discussion by
recalling the relevant equation
2
1
r>0
(1.5)
− ∂r2 + ∂r + 1 u − µu + νu2 − u3 = 0,
r
that we wish to solve. Throughout the following discussion, we restrict ourselves to small solutions of (1.5)
without always mentioning this restriction explicitly. Equation (1.5) can be rewritten as a five-dimensional
autonomous first-order system for (u, ur , urr , urrr ) upon adding the quantity α = 1/r with α0 = −α2 as an
3
additional dependent variable. Figure 2 contains a (necessarily imperfect) three-dimensional illustration of this
autonomous system, and we now explain the different components of this sketch step by step. In particular, we
shall argue that the region r > 0 can be divided naturally into three disjoint intervals in which solutions of (1.5)
behave qualitatively differently and that the dynamics in these regions can be captured by geometric blow-up
methods.
First, we need to identify small solutions of (1.5) that remain bounded as r → 0. Intuitively, these solutions
should satisfy (1.5) on r > 0 with Neumann boundary conditions ur (0) = urrr (0) = 0 at r = 0. This intuition is
correct, and there is indeed a two-dimensional family of solutions to (1.5) that, together with their derivatives,
stay bounded as r → 0: we will refer to them as the core solutions. On each bounded interval [0, r0 ], these
core solutions can, to leading order in their amplitude, be written as linear combinations of u1 (r) := J0 (r) and
2
u2 (r) = rJ1 (r), which satisfy the equation ∂r2 + 1r ∂r + 1 u = 0 obtained by linearizing (1.5) about u = 0 at
µ = 0. Though the representation of the family of core solutions as linear combinations of u1,2 breaks down in
1
1
the limit r → ∞, we note that u1 (r) = O(r− 2 ) decays and u2 (r) = O(r 2 ) grows algebraically as r → ∞: these
properties will become relevant below.
Next, we recall that we are interested in solutions of (1.5) for µ > 0 that decay as r → ∞. One way of constructing
√
√
such solutions is by deriving an amplitude equation via the ansatz u(r) = µA( µr) cos(r). Using this scaling,
√
it can be shown that A(s) with s = µr satisfies a second-order real Ginzburg–Landau equation. We will see
that solutions to this equation decay or grow exponentially with rates ± 21 as s tends to infinity. To extract
the decaying solutions, it is convenient to use the coordinates (A, z) with z = As /A: if A(s) = exp(± 12 s), then
z(s) = ± 12 , and we can therefore capture decaying solutions via the stable manifold of (A, z) = (0, − 21 ): we can
expect to find expansions of this manifold over intervals of the form [s1 , ∞) for some s1 > 0.
√
So far, we have discussed the core region 0 ≤ r ≤ r0 and the far field r ≥ s1 / µ, which correspond to the upper
left and bottom right parts, respectively, of Figure 2; the far field is referred to as the rescaling chart in Figure 2.
√
Missing is therefore the intermediate regime in which r varies from r0 to s1 / µ. The dynamics in this interval
1
turn out to be related to the algebraic growth and decay of the solutions u1,2 (r) = O(r± 2 ) that originate in the
core region. Our coordinates in this region will be of the form z1 := rux /u so that solutions with algebraic decay
rates ± 12 correspond to the stationary solutions z1 (r) = ± 21 ; in particular, the core solutions u1 and u2 connect
to two different equilibria in the transition chart.
This completes our discussion of the rationale behind the core region and the transition and rescaling charts
pictured in Figure 2. It remains to discuss the dynamics in these charts and illustrate the construction of spots
and rings as indicated in Figure 2. The idea behind the construction is related to concepts from geometric
singular perturbation theory: first, construct solutions that exist in the various charts in the singular limit µ = 0
and then glue or match these solutions together to find solutions for the regular problem for µ > 0. Setting
µ = 0, we find that the core solution u1 connects to the equilibrium z1 = − 12 , while all other core solutions
connect the core to z1 = 12 as they grow algebraically as r increases. Next, we identify two heteroclinic orbits
that connect the equilibria z1 = ± 12 in the transition chart to the equilibrium z = − 12 in the rescaling chart and
therefore correspond to solutions that decay as r → ∞. The first heteroclinic orbit is found as a singular pulse
2
solution of the real Ginzburg–Landau equation that exists only when the cubic coefficient c3 = 43 − 19
18 ν in this
1
equation is negative: this orbit emerges from z1 = 2 . The second heteroclinic orbit emerges from z1 = − 12 and
exists for all ν > 0: it has A = 0 and z 6= 0 and can therefore be thought of as a solution in the tangent space of
the stable manifold of u = 0 in the full Swift–Hohenberg equation.
Ring solutions are now constructed by following the singular pulse from the rescaling to the transition chart,
passing near the equilibrium z1 = 12 , and matching with the core solutions near u2 ; since the singular pulse exists
only for ν > ν∗ , so do the rings. Spot A is found by following the singular tangent-space solution, passing near
the equilibrium z1 = − 21 in the transition chart, and matching with the core solutions near u1 . The idea for
finding spot B is now to initially follow the singular pulse, then follow the heteroclinic orbit in the transition
4
chart that connects z1 = 12 to z1 = − 12 before matching with the core solutions near u1 . We will see later that the
peculiar amplitude scaling of spot B arises through a combination of the eigenvalues at the equilibria z1 = ± 12
that are relevant because spot B passes near both. A second interesting outcome of this construction is that the
envelope of the spot B profile cannot be monotone in the intermediate region in the limit µ → 0 as it resembles
a singular pulse in this region. We will come back to this issue in §3.6 where we present numerical computations
for µ close to zero that illustrate that the envelope is indeed not monotone for sufficiently small µ.
From a technical viewpoint, making this picture precise requires a detailed understanding of the passage near
the equilibria in the transition chart and the verification of various transversality conditions that are needed to
match with the core. Some of the initial steps in this proof are drawn from the earlier work [13], and we begin
by reviewing the appropriate sections from that paper and setting up the transition and rescaling charts as well
as the connecting orbits between these charts. Afterwards, we present a formal calculation that predicts the
amplitude scaling of spot B and provides further intuition before completing the proof by analysing the passage
near the equilibria in the transition chart and matching with the core solutions. Finally, §4 contains the existence
proof of spots for the three-dimensional Swift–Hohenberg equation.
2
Geometric blow-up coordinates in the planar case
In this section, we introduce the details of the various coordinate charts we shall use in the construction of spots.
We focus exclusively on the planar Swift–Hohenberg equation.
2.1
The planar Swift–Hohenberg equation near the core
The structure of solutions of the radial planar Swift–Hohenberg equation near the core r = 0 is known from [13].
Writing u1 := u, equation (1.1) can be rewritten as the system
1
(∂r2 + ∂r + 1)u1
r
1
(∂r2 + ∂r + 1)u2
r
=
u2
(2.1)
=
−µu1 + νu21 − u31
or, equivalently, as the first-order system

Ur = AU + F(U, µ),
0
0

A=
−1
0
0
0
1
−1
1
0
− 1r
0

0
1 

,
0 
− 1r

0


0


F(U, µ) = 



0
2
3
−µu1 + νu1 − u1

(2.2)
for U = (u1 , u2 , ∂r u1 , ∂r u2 ). We now recall from [13] the characterization of solutions of (2.2) that remain small
and bounded as r → 0. The linearized system Vr = AV admits the linearly independent solutions
V1 (r)
=
V2 (r)
=
V3 (r)
=
V4 (r)
=
√
√
√
√
2π (J0 (r), 0, −J1 (r), 0)t
2π (rJ1 (r), 2J0 (r), rJ0 (r), −2J1 (r))t
2π (Y0 (r), 0, −Y1 (r), 0)t
2π (rY1 (r), 2Y0 (r), rY0 (r), −2Y1 (r))t ,
where Jk and Yk denote the Bessel functions of the first and second kind, respectively. Note that V1,2 remain
bounded as r → 0, while V3,4 blow up like ln r. The following result shows that the set of small solutions of the
nonlinear system (2.2) that stay bounded as r → 0 is two-dimensional and can be parametrized by their V1,2
components.
5
Lemma 2.1 For each fixed r0 > 0, there is a constant δ0 > 0 such that the set W−cu (µ) of solutions U (r) of
(2.2) for which sup0≤r≤r0 |U (r)| < δ0 is, for each |µ| < δ0 , a smooth two-dimensional manifold. Furthermore,
each U ∈ W−cu (µ) can be written uniquely as
U (r0 )
=
d1 V1 (r0 ) + d2 V2 (r0 ) + V3 (r0 )O(|µ||d| + |d|2 )
1
1 νd21 + O(|µ||d| + |d1 |3 + |d2 |2 ) ,
+V4 (r0 ) √ + O √
r0
3
(2.3)
where d = (d1 , d2 ) ∈ R2 is small, and the right-hand side in (2.3) depends smoothly on (d, µ).
√
−1
Proof. This statement was proved in [13, Lemma 1] except that the coefficient [1/ 3 + O(r0 2 )] in front of νd21
√
was given there as [1/ 3 + o(1)] as r0 → ∞. This coefficient is given by the integral
Z ∞
Z
Z
Z ∞
π r0
π
1
π ∞
rJ0 (r)3 dr =
rJ0 (r)3 dr −
rJ0 (r)3 dr.
rJ0 (r)3 dr = √ −
4 0
4 0
4
3
r0
r0
q
R
3
∞
2
Using the expansion J0 (r) = πr
cos(r − π4 ) + O(r− 2 ) from [1, §9], the integral r0 . . . dr in the above expression
−1
can be shown to be O(r0 2 ) upon using the expansions given in [15, §7.5 and §7.12]; we omit the details.
2.2
The planar Swift–Hohenberg equation in the far field
Next, we review the normal-form calculations in the far field r 1 from [18] and [13] as the resulting coordinate
changes will also be used below. It will be convenient to use the variable α = 1r so that the far-field regime r 1
corresponds to 0 < α 1. Equation (2.2) for U = (u1 , u2 , u3 , u4 )t can then be recast as

  
u1
u3

  

u2  
u4




d   
,
=
(2.4)
u
−u
−
αu
+
u
3
1
3
2



dr   
2
3
u4  −u2 − αu4 − µu1 + νu1 − κu1 
α
−α2
which the normal-form coordinates
Ã
B̃
!
1
=
4
!
2u1 − i(2u3 + u4 )
−u4 − iu2
from [8, 13, 18] transform further into
α
α
à + B̃ + ï + O (|µ| + |Ã| + |B̃|)(|Ã| + |B̃|)
Ãr =
i−
2
2 α
α¯
B̃r =
i−
B̃ − B̃ + O (|µ| + |Ã| + |B̃|)(|Ã| + |B̃|)
2
2
αr = −α2 .
(2.5)
(2.6)
Since we are interested in the regime µ > 0, we set µ = ε2 from now on. The theory developed in [18] can now
be used to transform the far-field equation into a more useful form.
Lemma 2.2 ([13, Lemma 2]) Write µ = ε2 , then there is a change of coordinates
!
!
A
Ã
−iφ(r)
=e
[1 + T (α)]
+ O (|ε|2 + |Ã| + |B̃|)(|Ã| + |B̃|)
B
B̃
(2.7)
such that (2.6) becomes
Ar
Br
αr
α
= − A + B + RA (A, B, α, ε)
2
ε2
α
= − B + A + c3 |A|2 A + RB (A, B, α, ε)
2
4
= −α2 ,
6
(2.8)
2
where c3 := 34 − 19ν
18 . The transformation (2.7) is polynomial in (Ã, B̃, α) and smooth in ε. The function
T (α) = O(α) is linear and upper triangular for each α, while φ(r) satisfies
φr = 1 + O(ε2 + |α|3 + |A|2 ),
φ(0) = 0.
(2.9)
The remainder terms are of the form


2
X
RA (A, B, α, ε) = O 
|Aj B 3−j | + |α|3 |A| + |α|2 |B| + (|A| + |B|)5 + ε2 |α|(|A| + |B|)
(2.10)
j=0
RB (A, B, α, ε)
=


1
X
O
|Aj B 3−j | + |α|3 |B| + ε2 (ε2 + |α|3 + |A|2 )|A| + (|A| + |B|)5 + ε2 |α||B| .
j=0
For later use, we will transform the core manifold W−cu (ε) evaluated at α = α0 := 1/r0 into the (A, B)-coordinates.
As in [13, (3.23)], we obtain
!
2
2
2
A
W−cu (ε)|α=α0 :
= ei[−π/4+O(α0 )+O(ε +|d| )]
(2.11)
B
!
√
√
α0 d1 [1 + O(α0 )] − α0 −1 d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
√
.
×
√
√
√
− α0 d2 [i + O(α0 )] − [1/ 3 + O( α0 )]ν α0 d21 + O(ε2 |d| + |d2 |2 + |d1 |3 )
We have now collected all the necessary results from [13]. In the next sections, we set up the rescaling and
transition charts for (2.8).
2.3
The rescaling chart
We set
z=−
α B
+
2
A
(2.12)
and define the rescaling coordinates by
A2 =
A
,
ε
z2 =
z
,
ε
ε2 = ε,
In these coordinates, the Swift–Hohenberg equation
Ar
A
α B
ε∂s A2 = ∂r A2 =
=
− +
+
ε
ε
2
A
α2 =
α
1
1
=
= ,
ε
εr
s
s = εr.
expressed as (2.8) becomes the system
1
1
1
RA = εA2 z2 + RA = ε A2 z2 + 2 RA
ε
ε
ε
and
ε∂s z2 = ∂r z2
=
=
=
=
Substituting
Br
B
1 α2
+
− 2 Ar
ε 2
A
A
2
2
ε
1
B2
B
1 α
+
+ c3 |A|2 + RB − 2 − 2 RA
ε 2
4
A
A
A
2
2
ε
1
α 2 z + α/2
1 α
+
+ c3 |A|2 + RB − z +
−
R
A
ε 2
4
A
2
A2
1 + α22
z2 + α2 /2
1
ε
+ c3 |A2 |2 − α2 z2 − z22 −
R
+
R
A
B .
4
ε 2 A2
ε 3 A2
α2 (A, B, α, ε) = ε2 A2 , ε22 A2 z2 +
, α2 ε2 , ε2
2
7
(2.13)
into the expressions (2.10) for the remainder terms, we obtain after some tedious but straightforward calculations
that
α2 1
2
R
ε
A
,
ε
,
α
ε
,
ε
= |A2 |O |ε2 |2
A
z
+
A
2
2
2
2
2
2
2
2
ε2
2
1
α2 2
,
α
ε
,
ε
= O |ε2 |2 .
R
ε
A
,
ε
A
z
+
2
2
2
B
2
2
2
2
2
ε 3 A2
2
In particular, the remainder terms in the rescaling chart are of order O(|ε2 |2 ), and we arrive at the system
∂s A2 = A2 z2 + O(|ε2 |2 )
(2.14)
1 + α22
+ c3 |A2 |2 − α2 z2 − z22 + O(|ε2 |2 )
4
0
∂s z2
=
∂ s ε2
=
∂s α2
= −α22 .
Thus, the variable ε2 serves as a parameter. Setting ε2 = 0, we obtain the system
∂s A2
=
∂s z2
=
∂s ε2
=
∂s α2
=
A2 z2
1 + α22
+ c3 |A2 |2 − α2 z2 − z22
4
0
(2.15)
−α22 ,
which has two families of equilibria, namely Q+ (ε2 ) = (0, 21 , O(ε22 ), 0) with eigenvalues { 21 , −1, 0, 0} and Q− (ε2 ) =
(0, − 12 , O(ε22 ), 0) with eigenvalues {− 21 , 1, 0, 0}, where the neutral eigendirection points in the direction along these
families. We are interested in the four-dimensional center-stable manifold W cs (Q− ) of the family of equilibria
Q− that contains all solutions of (2.8) of size ε that decay as r → ∞ for ε > 0.
2.4
The transition chart
We transform (2.8) into coordinates (A1 , z1 ) that are obtained from rescaling (A, z) by α = 1/r; recall from
(2.12) that z = −α/2 + B/A. Specifically, we set
A
,
α
In these coordinates, (2.8) becomes
A1 =
∂r A1 =
z1 =
z
1
B
=− +
,
α
2 αA
ε1 =
ε
,
α
α1 = α.
(2.16)
Aαr
1
1
Ar
− 2 = αA1 + αA1 z1 + RA = α A1 + A1 z1 + 2 RA
α
α
α
α
and
∂r z1
=
=
=
=
Br
BAr
Bαr
−
− 2
αA
αA2
α A
α
ε2
B − α2 A + B + RA
− 2 B + 4 A + c3 |A|2 A + RB
B
−
+
αA
αA2
A
2
1
2
z1 + 2
ε
1
1
1
+ αc3 |A1 |2 − α z1 +
+ α z1 +
−
RA + 2 RB
4α
2
2
αA1
α A1
z1 + 12
1 ε21
1
2
2
α −z1 + +
+ c3 |A1 | − 2
RA + 3 RB ,
4
4
α A1
α A1
where RA and RB are now evaluated at (A, B) = (α1 A1 , α12 A1 (z1 + 21 )). We also need the r derivatives of ε1
and α1 , which are given by
εαr
∂r ε1 = − 2 = ε = αε1
α
∂r α1 = −αα1 .
8
The common factor α in these equations suggests to introduce the new independent variable ρ = ln r, or r = eρ .
Using this variable, the system in the transition chart becomes
∂ ρ A1
= A1 [1 + z1 ] +
∂ρ z1
= −z12 +
∂ρ ε1
= ε1
∂ρ α1
= −α1 .
1
RA
α12
(2.17)
z1 + 1
1 + ε21
1
+ c3 |A1 |2 − 2 2 RA + 3 RB
4
α1 A1
α1 A1
It remains to express RA and RB given by


2
X
RA (A, B, α, ε) = O 
|Aj B 3−j | + |α|3 |A| + |α|2 |B| + (|A| + |B|)5 + ε2 |α|(|A| + |B|)
j=0
RB (A, B, α, ε)
=


1
X
O
|Aj B 3−j | + |α|3 |B| + ε2 (ε2 + |α|3 + |A|2 )|A| + (|A| + |B|)5 + ε2 |α||B| .
j=0
in terms of (A1 , z1 ). We will see below that we need the property that the remainder terms in (2.17) vanish at
(A1 , z1 , ε1 ) = (0, − 12 , 0). To establish this property, we introduce the notation z− = z1 + 12 to get
1
RA (α1 A1 , α12 A1 z− , α1 , α1 ε1 )
α12
2
X
1
O
|(α1 A1 )j (α12 z− A1 )3−j | + |α1 |3 |α1 A1 | + |α1 |2 |α12 z− A1 |
α12
j=0
!
=
+(|α1 A1 | + |α12 z− A1 |)5 + |α12 ε21 ||α1 |(|α1 A1 | + |α12 z− A1 |)
1
O |α1 |4 |A1 |
2
α1
= A1 O |α1 |2 ,
=
so that
z−
RA = z− O |α1 |2 ,
2
α A1
and
1
RB (α1 A1 , α12 A1 z− , α1 , α1 ε1 )
3
α1 A1
1
X
1
=
O
|(α1 A1 )j (α12 A1 z− )3−j |
α13 A1
j=0
(2.18)
+ |α1 |3 |α12 A1 z− | + |ε1 α1 |2 (|ε1 α1 |2 + |α1 |3 + |α1 A1 |2 )|α1 A1 |
!
+(|α1 A1 | +
=
=
|α12 A1 z− |)5
+ |ε1 α1 |
2
|α1 ||α12 A1 z− |
1
O |α1 |5 |A1 |5 + |α1 |5 |A1 ||z− |
α13 A1
|α1 |2 O |A1 |4 + |z− | + |ε1 |2 .
+ |ε1 |2 |α1 |5 |A1 | + |ε21 ||α1 |5 |A1 z− |
(2.19)
The system in the transition chart is therefore given by
∂ρ A1
= A1 1 + z1 + O(|α1 |2 )
∂ρ z1
= −z12 +
∂ ρ ε1
= ε1
∂ρ α1
= −α1 .
1+
4
ε21
(2.20)
2
2
4
+ c3 |A1 | + |α1 | O |A1 | + z1 +
9
1 2
+ |ε1 |
2
When c3 < 0, which is the case we are primarily interested in, (2.20) has precisely two equilibria, namely
P± := (0, ± 12 , 0, 0), and we refer to Figure 3 for an illustration of their locations. Since we will need to gain a
detailed understanding of the dynamics near these equilibria, we introduce two sets of coordinates that move
these equilibria to the origin. First, near the equilibrium P+ , we use the new variables (A+ , z+ , ε+ , α+ ) =
(A1 , z1 − 21 , ε1 , α1 ) to get
3
+ z+ + O(|α+ |2 )
(2.21)
∂ρ A+ = A+
2
ε2
2
∂ρ z+ = −z+ − z+
+ + + c3 |A+ |2 + O |α+ |2
4
∂ ρ ε+ = ε+
∂ρ α+
= −α+ .
The linearization of (2.21) about the origin has the eigenvalues { 32 , −1, 1, −1}. Near P− , we choose the variables
(A− , z− , ε− , α− ) = (A1 , z1 + 21 , ε1 , α1 ) and obtain the system
1
∂ρ A− = A−
+ z− + O(|α− |2 )
(2.22)
2
ε2
2
∂ρ z− = z− − z−
+ − + c3 |A− |2 + |α− |2 O |A− |4 + |z− | + |ε− |2
4
∂ρ ε− = ε−
∂ρ α−
= −α− ,
whose linearization about the origin has the eigenvalues { 12 , 1, 1, −1}. Since the final matching analysis will be
carried out in the transition chart near the equilibrium P− , we express the core manifold given in (2.11) in the
coordinates (A− , z− ) and obtain
W−cu (ε)|α=α0 :
A−
=
z−
=
(2.23)
i
−3/2
e
α0 d1 [1 + O(α0 )] − α0 d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
√
√
−d2 [i + O(α0 )] − [1/ 3 + O( α0 )]νd21 + O(ε2 |d| + |d2 |2 + |d1 |3 )
.
α0 d1 [1 + O(α0 )] − d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
i[−π/4+O(α20 )+O(ε2 +|d|2 )]
h
− 21
Finally, we record that the coordinates in the transition and rescaling charts are related by the transformation
A1 =
A2
,
α2
z1 =
z2
,
α2
ε1 =
1
,
α2
α1 = ε2 α2 = εα2 ,
(2.24)
and we can, for instance, transform from one chart to the other in the transverse section ε1 = α2 = 1.
2.5
Singular connecting orbits
We now discuss the dynamics of the system in the transition and rescaling charts. Our goal is to show the
existence of a heteroclinic orbit that connects the equilibrium P+ in the transition chart to the equilibrium Q−
in the rescaling chart in the limit ε = 0; see Figure 3 for an illustration of this orbit and the location and
stability of the equilibria P+ and Q− . First note that the subspace α1 = ε2 = 0 is invariant under the flow of
the equations in the transition and rescaling charts. Indeed, setting α1 = 0 in (2.20), we obtain the system
∂ρ A1
=
A1 (1 + z1 )
∂ρ z1
= −z12 +
∂ρ ε1
=
1+
4
ε1 ,
10
(2.25)
ε21
+ c3 |A1 |2
Figure 3: A cartoon of the heteroclinic orbit that connects P+ in the transition chart to Q− in the rescaling chart in the
invariant subspace α1 = ε2 = 0.
while equation (2.14) at ε2 = 0 becomes
∂s A2
=
∂s z2
=
∂s α2
=
A2 z2
1 + α22
+ c3 |A2 |2 − α2 z2 − z22
4
−α22 .
(2.26)
Note also that the transformation (2.24) between the transition and rescaling charts maps α1 = 0 into ε2 = 0.
We start by analysing (2.26): rewriting this system as an equation for A2 , we obtain the non-autonomous real
Ginzburg–Landau equation
∂s2 A2 = −
∂s A2
A2
A2
+ 2+
+ c3 A32 ,
s
4s
4
A2 ∈ R,
(2.27)
where we restrict A2 to be real-valued.
Lemma 2.3 Assume that c3 < 0 and that Hypothesis (H1) is met; equation (2.27) then has a bounded nontrivial
solution A2 (s) = q(s), and there are constants q0 > 0 and q+ 6= 0 so that

s→0
 q0 s1/2 + O(s3/2 )
−s/2
q(s) =
(2.28)
e
 (q+ + O(e−s/2 )) √
s → ∞.
s
In addition, the linearization of (2.27) about q(s) does not have a nontrivial solution that is bounded uniformly
on R+ . If c3 > 0, then the only bounded solution of (2.27) on R+ is A2 (s) ≡ 0.
Proof. All assertions except for the asymptotic behavior of the solutions q(s) in the limits s → 0 and s → ∞
follow from Hypothesis (H1) upon rescaling or have been proved in [13, Lemma 4]. The asymptotics for s → 0
follow easily from the variation-of-constants formula and a contraction argument for (2.27). To derive the
√
expansion for s → ∞, we can use the variable Â2 = sA2 that transforms (2.27) into the autonomous equation
∂s2 Â2 = Â2 /4 + c3 A32 , and a standard application of the stable-manifold theorem proves the assertion.
Next, we write the solution q(s) via
z2 =
∂s A2
,
A2
α2 =
1
,
s
A1 = sA2 ,
z1 = sz2 =
s∂s A2
,
A2
ε1 = s,
ρ = ln s
in the coordinates of the transition and rescaling charts and conclude that the functions
q 0 (eρ ) ρ
q 0 (s) 1
∗ ∗
∗
(A∗1 , z1∗ , ε∗1 )(ρ) = eρ q(eρ ), eρ
,
e
,
(A
,
z
,
α
)(s)
=
q(s),
,
2 2
2
q(eρ )
q(s) s
11
(2.29)
(2.30)
Figure 4: The anticipated construction of spot B is illustrated: we follow the center-stable manifold of Q− from the
rescaling chart backwards in time along the singular heteroclinic orbit to the equilibrium P+ in the transition chart, then
along the heteroclinic orbit from P+ to P− , where it is finally matched with the core manifold.
satisfy (2.25) and (2.26), respectively. We now show that this solution lies in the unstable manifold of P+ and
the center-stable manifold of Q− , and that these manifolds intersect transversally inside the real subspace of
(2.25) and (2.26).
Lemma 2.4 Assume that c3 < 0, and consider (2.25) and (2.26) in R3 . The solution (2.30) is a connecting
orbit that forms a transverse intersection of the unstable manifold W u (P+ ) of the equilibrium P+ = (0, 12 , 0) of
(2.25) and the center-stable manifold W cs (Q− ) of the equilibrium Q− = (0, − 21 , 0) of (2.26).
Proof. It is easy to check that P+ and Q− are equilibria of (2.25) and (2.26), respectively, and that W u (P+ )
and W cs (Q− ) are both two-dimensional. Using the asymptotic expansions of q(s) from (2.28) and the expression
(2.30) for the resulting solution of (2.25) and (2.26), it is straightforward to verify that this solution indeed lies
in the intersection of W u (P+ ) and W cs (Q− ). Finally, if these two manifolds would not intersect transversally in
R3 , this would yield, upon tracing back our coordinate changes, a nonzero bounded real-valued solution of the
linearization of (2.27) around q(s) in contradiction to Lemma 2.3.
We remark that the real-valued heteroclinic orbit given in (2.30) generates a one-parameter family
(A1 , z1 , ε1 )(ρ) = (eiγ A∗1 , z1∗ , ε∗1 )(ρ),
(A2 , z2 , α2 )(s) = (eiγ A∗2 , z2∗ , α2∗ )(s)
(2.31)
of heteroclinic orbits of (2.25) and (2.26) that are parametrized by γ ∈ R when these equations are posed in
C2 × R. Note that the orbits corresponding to the choices γ = 0 and γ = π are both real.
3
The construction of planar spots in the transition chart
The remainder of the existence proof of the planar spot B solution will take place in the transition chart. The
goal is to follow the center-stable manifold W cs (Q− ) near the heteroclinic orbits constructed in (2.31) for γ = 0, π
backwards in time past the equilibria P+ and P− and then match the resulting manifold with the core manifold
W−cu as illustrated in Figure 4. Before proceeding with this proof, we outline its main features formally in the
3
next section: the arguments given there will also explain the amplitude scaling µ 8 obeyed by the spot B profile
and motivate the ansatz and scaling we will employ in the rigorous analysis.
12
Figure 5: This schematic picture illustrates the various steps involved in tracking the center-stable manifold W cs (Q− ) in
the transition chart from P+ to P− .
3.1
Formal arguments for the amplitude scaling
3
Our goal is to outline a simple argument that explains the scaling of µ 8 for spot B profiles for c3 < 0. In
the following, we will proceed formally: we will neglect all terms that we anticipate to be of higher order and
set every allowable constant to one to simplify the expressions. In particular, we will restrict (A1 , z1 ) to the
real subspace that is invariant in the truncated normal form. In §2.5, we found two singular heteroclinic orbits
between P+ and Q− that exist for α1 = ε2 = 0 and lie in the transverse intersection of W u (P+ ) and W cs (Q− ).
For each small ε > 0, we will follow the center-stable manifold of the equilibrium Q− (ε) backwards along the
singular orbits, then past P+ to P− following the heteroclinic orbit between them, and finally past P− before
matching with the core manifold: the various aspects of this analysis are also indicated in Figure 4. The key will
therefore be the dynamics near P± , where we use the charts introduced in §2.4. Linearizing (2.21) and (2.22) for
P+ and P− about the origin, we obtain respectively the equations
∂ρ A+ =
3
A+ ,
2
∂ρ z+ = −z+ ,
∂ρ ε1 = ε1 ,
∂ρ α1 = −α1
and
1
A− ,
∂ρ z− = z− ,
∂ρ ε1 = ε1 ,
∂ρ α1 = −α1 .
2
In fact, to gain additional insight, we will consider the more general equations
∂ ρ A− =
∂ρ A+ =
5−n
A+ ,
2
∂ρ z+ = −z+ ,
∂ρ ε1 = ε1 ,
∂ρ α1 = −α1
(3.1)
and
3−n
A− ,
∂ρ z− = z− ,
∂ρ ε1 = ε1 ,
∂ρ α1 = −α1
(3.2)
2
that arise in the same fashion when seeking radial spots for x ∈ Rn , so that the planar case considered here
corresponds to taking n = 2. We are interested in ε2 = ε > 0: if we switch from the rescaling to the transition
chart at ε1 = α2 = 1, equation (2.24) implies that α1 (ρ) = εe−ρ . If we match at r = r0 = 1, we need to solve
(3.1) and (3.2) on [ρ∗ , 0] where α1 (ρ∗ ) = εe−ρ∗ = 1/r0 = 1. Thus, we obtain ρ∗ = ln ε and can ignore the
equations for α1 and ε1 from now on.
∂ρ A− =
Figure 5 indicates that the center-stable manifold W cs (Q− (ε)) in the cross-section ε1 = 1 near P+ can be
parametrized as
W cs (Q− (ε))ε1 =1 :
(A+ , z+ ) = (δ, −ã)
with
δ = ±1, |ã| 1,
where δ = ±1 corresponds to setting γ = 0 or γ = π. Solving (3.1) with these initial data gives
5−n
(A+ , z+ )(ρ) = δe 2 ρ , −ãe−ρ
13
where ρ ≤ 0. We switch from P+ to P− at z1 = 0 which corresponds to z+ = − 21 and z− = 21 . Thus, we restrict
ourselves to ã > 0 and obtain z+ (ρ+ ) = − 21 at ρ+ = ln 2ã. Hence,
5−n
1
2
,−
(A+ , z+ )(ρ+ ) = δ(2ã)
2
which corresponds to
5−n 1
2
,
.
(A− , z− )(0) = δã
2
(3.3)
5−n
upon ignoring the extra factor 2 2 in the A+ -component. Note that we now need to solve (3.2) from ρ = 0 to
ρ = ln(ε/ã) since we already expended ρ+ = ln ã time from the total time-of-flight given by ρ∗ = ln ε. Solving
(3.2) with the initial conditions (3.3), and ignoring the factor 21 in the second component, we obtain
5−n 3−n
(A− , z− )(ρ) = δã 2 e 2 ρ , eρ
and evaluating at ρ = ρ− = ln(ε/ã) gives
3−n
ε
.
(A− , z− )(ρ− ) = δãε 2 ,
ã
(3.4)
We can now match with the core manifold given in (2.23) in the same coordinates as
h 1
i
2
2
2
−
−3/2
W−cu |α=α0 : A− = ei[−π/4+O(α0 )+O(ε +|d| )] α0 2 d1 [1 + O(α0 )] − α0 d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
√
√
−d2 [i + O(α0 )] − [1/ 3 + O( α0 )]νd21 + O(ε2 |d| + |d2 |2 + |d1 |3 )
.
z− =
α0 d1 [1 + O(α0 )] − d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
Ignoring all remainder terms and subsequently setting α0 = 1 gives
−id2 − d21
cu
W− |α0 =1 : (A− , z− ) = d1 − id2 ,
.
d1 − id2
Setting d2 = 0 finally gives
W−cu |α0 =1 :
(A− , z− ) = (d1 , −d1 ) .
(3.5)
Matching (3.4) and (3.5) requires solving the system
δãε
3−n
2
= d1 ,
ε
= −d1
ã
for (ã, d1 ). Since ã, ε > 0, we need δ = −1 and obtain
d1 = −ε
5−n
4
= −µ
5−n
8
< 0,
ã = ε
n−1
4
.
(3.6)
In particular, we see that the amplitude near the core is necessarily negative. In addition, we find that the spot
1
3
amplitude scales as d1 = −µ 8 for n = 2 and as d1 = −µ 4 for n = 3 as claimed. It is interesting to note that
1
the exponent 5−n
8 equals 2 for n = 1 which is indeed the observed amplitude of the two pulse solutions that are
known to emerge in the normal-form equation in one space dimension.
We now proceed with the rigorous arguments by analysing the dynamics near P+ , in between P+ and P− , and
finally near P− before matching with the core manifold: see Figure 5 for an illustration. Our analysis will exploit
the scaling (3.6) for the variable ã that parametrizes the stable manifold of Q− .
3.2
The dynamics near P+
Our goal is to track the center-unstable manifold W cs (Q− ) for ε > 0 in backwards time as it passes near the
equilibrium P+ . We consider the equation in the transition chart using the variable z+ := z1 − 12 in which P+
14
corresponds to the origin. The resulting system (2.21) is given by
3
2
+ z+ + O(|α+ | )
∂ρ A+ = A+
2
ε2
2
∂ρ z+ = −z+ − z+
+ + + c3 |A+ |2 + O(|α+ |2 )
4
∂ρ ε+ = ε+
∂ρ α+
(3.7)
= −α+ .
It is convenient to simplify this system by flattening out the unstable manifold and transforming away some of
the higher-order terms.
Lemma 3.1 There is a smooth change of coordinates of the form
z̃+ = z+ + h+ (A+ , ε+ , α+ ),
2
h+ (A+ , ε+ , α+ ) = O(|A+ |2 + ε2+ + α+
)
(3.8)
that transforms (3.7) near the origin into
3
+ O(|A+ | + |z̃+ | + |ε+ | + |α+ |)
2
= −z̃+ [1 + O(|A+ | + |z̃+ | + |ε+ | + |α+ |)]
∂ρ A+
∂ρ z̃+
= A+
∂ ρ ε+
= ε+
∂ρ α+
= −α+ .
(3.9)
Proof. First, we claim that we can find a smooth change of coordinates of the form
ẑ+ = z+ + h0 (α+ )
(3.10)
2
) so that the right-hand side of the equation for ẑ+ vanishes at (A+ , ẑ+ , ε+ ) = 0. Since
with h0 (α+ ) = O(α+
n
with n ≥ 2 in the equation for z+ is resonant, this claim follows, for instance, from
none of the monomials α+
the Poincaré–Dulac theorem [2, pp 181–184] together with [6, Theorem of Equivalence]. Thus, with this choice
of h0 , the full system (3.7) becomes
3
∂ρ A+ = A+
+ O(|A+ | + |ẑ+ | + |ε+ | + |α+ |)
(3.11)
2
ε2
2
2
∂ρ ẑ+ = −ẑ+ − ẑ+
+ + + c3 |A+ |2 + α+
[A+ g1 + ẑ+ g2 + ε+ g3 ]
4
∂ρ ε+ = ε+
∂ρ α+
=
−α+ ,
where gj = gj (A+ , ẑ+ , ε+ , α+ ) with j = 1, 2, 3 are smooth functions. Next, we wish to find a change of coordinates
that flattens the unstable manifold W u (0) of (3.11). Since the subspace {α+ = 0} is invariant for (3.11), it follows
that the unstable manifold W u (0) is of the form (ẑ+ , α+ ) = (h1 (A+ , ε+ ), 0) with h1 = O(|A+ |2 + |ε+ |2 ). Thus,
introducing the new variable z̃+ = ẑ+ − h1 (A+ , ε+ ) leads to W u (0) = {(z̃+ , α+ ) = 0}, and the resulting equation
for z̃+ must be of the form
∂ρ z̃+ = −z̃+ [1 + O(|A+ | + |z̃+ | + |ε+ | + |α+ |)]
since (z̃+ , α+ ) = 0 is now invariant, while the α+ -axis continues to be invariant.
We are interested in tracking the center-stable manifold W cs (Q− ) near P+ in backwards time. To find an
expression for this manifold near P+ , we interpret the transversality of W cs (Q− ) and W u (P+ ) stated in Lemma 2.4
in the coordinates constructed in Lemma 3.1.
15
Lemma 3.2 For each sufficiently small δ0 > 0, there are constants a0 , ε0 > 0 such that the following is true.
Define the section Σ0 := {ε+ = δ0 }, then
ε
W cs (Q− ) ∩ Σ0 = (A+ , z̃+ , α+ ) = −eiγ [η(δ0 ) + O(ã)] + O(ε2 ), −ã + O(ε2 ),
: ã ∈ (−a0 , a0 ), ε < ε0 ,
δ0
3/2
where η0 (δ0 ) = q0 δ0 (1 + O(δ0 )) is smooth, q0 > 0 is the constant defined in (2.28), and γ ∈ R is arbitrary.
Proof. First, note that the stable direction ∂ρ α+ = −α+ decouples and that α+ = ε/ε+ . Using (2.28), it is easy
to see that the heteroclinic orbit (2.30) has the claimed expansion in terms of δ0 . For ε = 0, the transversality
stated in Lemma 2.4 together with the S 1 -symmetry of the normal form implies that we can parametrize W cs (Q− )
as claimed by ã ∈ R and γ ∈ R. Including the parameter ε yields the additional O(ε2 ) terms, where we used
that the remainder terms in the rescaling chart (2.14) are of order O(ε2 ).
We start with ρ = 0 for initial data in Σ0 and need to track solutions until ρ = ρ∗ , where ρ∗ < 0 is such that
α+ (ρ∗ ) = α0 = 1/r0 . Since α+ (0) = ε/δ0 in Σ0 , we find from (3.9) that
ρ∗ = ln
ε
,
α0 δ0
(3.12)
and we consequently solve (3.9) only for ρ∗ ≤ ρ ≤ 0. We now choose a second constant δ1 > 0 and track an
appropriate part of the center-stable manifold W cs (Q− ) in backwards time under the evolution of (3.9) from Σ0
1
to Re z̃− = −δ1 . We will exploit that our formal analysis led to (3.6), which predicts that ã = O(ε 4 ).
Lemma 3.3 For each fixed choice of 0 < δ0 , δ1 , κ 1, there is an ε0 > 0 such that solutions of (3.9) associated
with initial data of the form
!
1
4
1
aε
ε
(A+ , z̃+ , ε+ , α+ )(0) = −eiγ η0 (δ0 ) + O(ε 4 −κ ), −
+ O(ε2 ), δ0 ,
(3.13)
δ0
δ0
in W cs (Q− ) ∩ Σ0 with a ∈ [εκ , ε−κ ] and ε ∈ (0, ε0 ) land after the time
1
ρ1 = ln
aε 4
≥ ρ∗
δ0 δ1
(3.14)
at the point
1
aε 4
δ0 δ1
! 32
1
eiγ η0 (δ0 )(1 + O(δ0 + δ1 + ε 4 −κ ))
A+ (ρ1 )
= −
z̃+ (ρ1 )
= −δ1 (1 + O(δ0 + δ1 + ε 4 −κ ))
ε+ (ρ1 )
=
aε 4
δ1
α+ (ρ1 )
=
δ1 ε 4
.
a
1
1
3
Proof. We begin by solving (3.9) given by
∂ρ A+
∂ρ z̃+
3
+ O(|A+ | + |z̃+ | + |ε+ | + |α+ |)
2
= −z̃+ [1 + O(|A+ | + |z̃+ | + |ε+ | + |α+ |)]
= A+
∂ ρ ε+
= ε+
∂ρ α+
= −α+
16
(3.15)
with initial conditions
(A+ , ε+ )(0) = (A0 , δ0 ),
(z̃+ , α+ )(ρ̃) = (B1 , α1 )
for arbitrary but small A0 , B1 ∈ C and α1 > 0 on the interval [ρ̃, 0] for arbitrary ρ̃ −1. We obtain immediately
that
ε+ (ρ) = δ0 eρ ,
α+ (ρ) = α1 eρ̃−ρ ,
and it remains to solve
∂ρ A+
∂ρ z̃+
3
ρ
ρ̃−ρ
= A+
+ O(|A+ | + |z̃+ | + δ0 e + α1 e
) ,
2
= −z̃+ 1 + O(|A+ | + |z̃+ | + δ0 eρ + α1 eρ̃−ρ ) ,
A+ (0) = A0
(3.16)
z̃+ (ρ̃) = B1
on [ρ̃, 0]. Using a standard contraction mapping argument in exponentially weighted spaces that exploits the
special structure of the nonlinearity, we find that (3.16) has a unique solution and that this solution depends
smoothly on (A0 , B1 , α1 , ρ̃) and is given by
= A0 e3ρ/2 (1 + O(|A0 | + |B1 | + α1 + δ0 ))
A+ (ρ)
(3.17)
= B1 eρ̃−ρ (1 + O(|A0 | + |B1 | + α1 + δ0 ))
z̃+ (ρ)
uniformly in ρ̃ ≤ ρ ≤ 0 and |A0 |, |B1 |, α1 1. Inspecting the initial conditions (3.13) for which we want to solve,
and substituting ρ̃ = ρ1 with ρ1 as in (3.14), we obtain
3
iγ
A0 = −e η0 (δ0 ) + O(ε
1
4 −κ
ε
δ1 ε 4
α1 = e−ρ1 =
.
δ0
a
),
Similarly, the initial condition for z̃1 (0) becomes
1
1
1
B1 aε 4
aε 4
!
B1 e (1 + O(|A0 | + |B1 | + α1 + δ0 )) =
(1 + O(|B1 | + δ0 + ε 4 −κ )) = −
+ O(ε2 ),
δ0 δ1
δ0
ρ1
1
which has the unique solution B1 = −δ1 (1 + O(δ0 + δ1 + ε 4 −κ )). Substituting these expressions into (3.16) and
(3.17) and evaluating at ρ = ρ̃ = ρ1 gives (3.15) as claimed.
Inverting the coordinate transformation (3.8) and reverting to the original transition-chart variables with z1 =
z− + 21 , we obtain
A01
z10
ε01
3
3
:= A1 (ρ1 ) = −a 2 ε 8 η1 eiγ
1
1
− δ1 (1 + O(δ0 + δ1 + ε 4 −κ ))
:= z1 (ρ1 ) =
2
1
aε 4
:= ε1 (ρ1 ) =
δ1
(3.18)
3
α10
δ1 ε 4
:= α1 (ρ1 ) =
a
with
3
−3
1
1
η1 := η0 (δ0 )(δ0 δ1 )− 2 (1 + O(δ0 + δ1 + ε 4 −κ )) = q0 δ1 2 (1 + O(δ0 + δ1 + ε 4 −κ )) > 0.
(3.19)
Next, we transport this manifold to a neighborhood of the equilibrium P− .
3.3
The dynamics between P+ and P−
Next, we fix a small constant δ2 > 0 and integrate the transition-chart system
∂ρ A1 = A1 1 + z1 + O |α1 |2
1 ε2
∂ρ z1 = −z12 + + 1 + c3 |A1 |2 + O |α1 |2
4
4
∂ρ ε1 = ε1
∂ρ α1
=
−α1
17
(3.20)
with initial conditions given by (3.18) backwards in time until z1 is approximately equal to − 12 + δ2 . More
precisely, we set
δ1 δ2
ρ2 = ln
(1 − δ1 )(1 − δ2 )
and integrate (3.20) from ρ = 0 to ρ = ρ2 . We initially set (A1 , ε1 , α1 ) = 0 so that (3.20) with the initial
condition (3.18) for z1 becomes the complex differential equation
1
∂ρ z1 = −z12 + ,
4
z1 (0) = z10 =
1
1
− δ1 (1 + O(δ0 + δ1 + ε 4 −κ )),
2
whose solution z1∗ (ρ) evaluated at ρ = ρ2 is given by
1
1
z1∗ (ρ2 ) = − + δ2 (1 + O(δ0 + δ1 + ε 4 −κ )).
2
Next, we expand the time-ρ2 map of (3.20) with initial condition (A01 , z10 , ε01 , α10 ) at ρ = 0 around (0, z10 , 0, 0) and
obtain




η2 A01 (1 + O(|A01 | + |ε01 | + |α10 |))
A1 (ρ2 )


 z (ρ )   z1∗ (ρ2 ) + O(|A01 | + |ε01 | + |α10 |) 
 1 2  

1
=

,
aδ2 ε 4 (1 + O(δ1 + δ2 ))

 ε1 (ρ2 )  
3


ε4
α1 (ρ2 )
(1 + O(δ1 + δ2 ))
aδ2
where the constant η2 that appears in (3.21) is given by η2 = a1 (ρ2 ), and a1 is the solution to the linear equation
∂ρ a1 = (1 + z1∗ (ρ))a1 ,
a1 (0) = 1.
This equation can be solved explicitly, and we obtain
3
1
η2 = δ12 δ22 (1 + O(δ1 + δ2 )).
Substituting the initial conditions (3.18), we arrive at


3
3
−a 2 ε 8 η3 eiγ


A1 (ρ2 )
 1

1
− + δ2 (1 + O(δ0 + δ1 + ε 4 −κ ))
 z (ρ )  


 1 2   2
,
1

=

4
aδ2 ε (1 + O(δ1 + δ2 ))
 ε1 (ρ2 )  

3


ε4
α1 (ρ2 )
(1 + O(δ1 + δ2 ))
aδ2
(3.21)
where η3 is given by
1
1
(3.19)
1
η3 := η1 η2 (1 + O(ε 4 −κ )) = q0 δ22 (1 + O(δ0 + δ1 + δ2 + ε 4 −κ )).
3.4
(3.22)
The dynamics near P−
It remains to solve the system (3.20) with initial conditions given by (3.21) for the remaining time
1
ρ3 = ρ∗ − ρ1 − ρ2 = ln
3
ε
aε 4
δ1 δ2
ε 4 (1 − δ1 )(1 − δ2 )
− ln
− ln
= ln
α0 δ0
δ0 δ1
(1 − δ1 )(1 − δ2 )
aα0 δ2
(3.23)
near the equilibrium P− . Using the variable z− = z1 + 21 , we therefore need to solve the system (2.22) given by
1
∂ρ A− = A−
+ z− + O(|α− |2 )
(3.24)
2
ε2
2
∂ρ z− = z− − z−
+ − + c3 |A− |2 + |α− |2 O(|A− |4 + |z− | + |ε− |2 )
4
∂ ρ ε− = ε−
∂ρ α−
=
−α−
18
with initial conditions
A− (0)
z− (0)
3
3
= −a 2 ε 8 η3 eiγ
= δ2 (1 + O(δ0 + δ1 + ε
(3.25)
1
4 −κ
))
1
4
ε− (0)
= aδ2 ε (1 + O(δ1 + δ2 ))
α− (0)
=
3
ε4
(1 + O(δ1 + δ2 ))
aδ2
from ρ = 0 to ρ = ρ3 .
Lemma 3.4 For all fixed sufficiently small constants α0 , δj , κ > 0 with j = 0, 1, 2, there is an ε0 > 0 such that
the solution of (3.24) with initial condition (3.25), evaluated at ρ = ρ3 with ρ3 from (3.23), is given by
3
1
aε 4 q0 eiγ 1 + O(α0 + δ0 + δ1 + δ2 + ε 4 −κ )
√
α0
3 4
1
ε
z− (ρ3 ) =
1 + O(α0 + δ0 + δ1 + δ2 + ε 4 −κ )
aα0
ε
= εr0
ε− (ρ3 ) =
α0
1
α− (ρ3 ) = α0 =
r0
A− (ρ3 )
=
−
(3.26)
uniformly in a ∈ (εκ , ε−κ ) and ε ∈ (0, ε0 ), where q0 > 0 is the constant given in (2.28).
Proof. Our choice of ρ∗ in (3.12) was made to ensure that α− (ρ3 ) = α0 = 1/r0 , and the statement for ε−
follows from its definition in (2.16). In particular, we have
ε− (ρ) = ε− (0)eρ ,
for 0 ≤ ρ ≤ ρ3 . Next, we write
1
ε− (0) = O(ε 4 −κ ),
A− (ρ) = Ã− (ρ)eρ/2 ,
α− (ρ) = α0 eρ3 −ρ
z− (ρ) = z̃− (ρ)eρ
(3.27)
and obtain the system
∂ρ Ã−
=
Ã− z̃− eρ + e2(ρ3 −ρ) O(α02 )
∂ρ z̃−
=
1
2
−eρ z̃−
+ ε− (0)2 eρ + c3 |Ã− |2 + α02 e2(ρ3 −ρ) O(|Ã− |4 + |z̃− | + |ε− (0)|2 ),
4
(3.28)
which we consider with the initial conditions (3.25), which become
3
3
Ã− (0) = −a 2 ε 8 η3 eiγ =: Ã0− ,
1
0
z̃− (0) = δ2 (1 + O(δ0 + δ1 + ε 4 −κ )) =: z̃−
.
(3.29)
We write (3.28)–(3.29) as the fixed-point equation
Z ρ
h
i
Ã− (ρ) = Ã0− +
Ã− (y) z̃− (y)ey + e2(ρ3 −y) O(α02 ) dy
(3.30)
0
Z ρ
1
0
z̃− (ρ) = z̃−
+
−ey z̃− (y)2 + ε− (0)2 ey + c3 |Ã− (y)|2 + α02 e2(ρ3 −y) O(|Ã− (y)|4 + |z̃− (y)| + |ε− (0)|2 ) dy
4
0
3
0
on [ρ3 , 0]. Using that Ã0− = O(ε 8 (1−κ) ), z̃−
= O(δ2 ), and |ρ3 | ≤ | ln ε|, we can apply the contraction mapping
principle to show that (3.30) has a unique solution (Ã− , z̃− ) in an appropriate small ball centered at the origin
in C 0 ([ρ3 , 0], C2 ). Furthermore, there is a uniform constant C with
1
1
0
0
kÃ− k ≤ C|Ã0− |,
kz̃− k ≤ C |z̃−
| + ε 4 −κ + |ρ3 | |Ã0− |2 ≤ C |z̃−
| + ε 4 −κ .
19
Using these estimates together with (3.29) and (3.30) in (3.27), we obtain
3
A− (ρ3 )
(3.23)
3
3
−a 2 ε 8 eρ3 /2 η3 eiγ (1 + O(α0 + δ2 )) = −
=
3
(3.22)
−
=
aε 4 q0 eiγ
√
α0
aε 4 η3 eiγ
1
(α0 δ2 ) 2
1
1 + O(α0 + δ0 + δ1 + δ2 + ε 4 −κ )
and
1
z− (ρ3 ) = δ2 1 + O(α0 + δ0 + δ1 + ε 4 −κ ) eρ3
(3.23)
=
(1 + O(α0 + δ2 ))
3
1
ε4 1 + O(α0 + δ0 + δ1 + δ2 + ε 4 −κ ) ,
aα0
which completes the proof.
3.5
Matching core and far field
It remains to find nontrivial intersections of the center-stable manifold W cs (Q− ) and the core manifold W−cu (ε)
at α = α0 . To simplify the following expressions, we will write
1
∆ := O(α0 + δ0 + δ1 + δ2 + ε 4 −κ )
(3.31)
with a slight abuse of notation that should not cause confusion as the Laplacian will not be used in this section.
At α = α0 , we then have the expression (3.26)
3
aε 4 q0 eiγ
(1 + ∆),
A− = − √
α0
3
ε4
z− =
(1 + ∆)
aα0
with a ∈ (εκ , ε−κ ) for the center-stable manifold W cs (Q− ) and the expansion (2.23)
i
h 1
2
2
2
−
−3/2
A− = ei[−π/4+O(α0 )+O(ε +|d| )] α0 2 d1 [1 + O(α0 )] − α0 d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
√
√
−d2 [i + O(α0 )] − [1/ 3 + O( α0 )]νd21 + O(ε2 |d| + |d2 |2 + |d1 |3 )
z− =
α0 d1 [1 + O(α0 )] − d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
with d = d(d1 , d2 ) ∈ R2 for the core manifold W−cu (ε) in the (A− , z− ) coordinates. Setting these expressions
equal to each other gives the system
3
−
aε 4 q0 eiγ
(1 + ∆)
√
α0
2
+|d|2 )]
h
−1
−3/2
d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
ei[−π/4+O(α0 )+O(ε
=
√
√
−d2 [i + O(α0 )] − [1/ 3 + O( α0 )]νd21 + O(ε2 |d| + |d2 |2 + |d1 |3 )
α0 d1 [1 + O(α0 )] − d2 [i + O(α0 )] + O(ε2 |d| + |d|2 )
3
ε4
(1 + ∆)
aα0
2
=
α0 2 d1 [1 + O(α0 )] − α0
i
that we need to solve. Next, we let γ = γ̃ − π4 + O(α02 ) + O(|ε|2 + |d|2 ), and use the scaling (d1 , d2 ) = (ε 4 d˜1 , ε 2 d˜2 )
to obtain
3
0
0
3
3
˜
= aq0 eiγ̃ (1 + ∆) + d˜1 (1 + O(α0 )) + O(ε 4 |d|)
(3.32)
3
˜ + a d˜2 (i + O(α0 )) + √ν + O(√α0 ) d˜2 + O(ε 43 |d|)
˜ .
= (1 + ∆) d˜1 (1 + O(α0 )) + O(ε 4 |d|)
1
3
Initially setting ε = 0, we arrive at the system
= aq0 (cos γ̃ + i sin γ̃) + d˜1 (1 + ∆)
ν
0 = d˜1 (1 + ∆) + a d˜2 (i + ∆) + √ + ∆ d˜21 .
3
0
20
(3.33)
Figure 6: The left panel contains profiles of spot B solutions that are plotted as functions of the rescaled spatial variable
√
s = µr for a range of values of the parameter µ. Note that the envelope of the largest profile is not monotone, though
the value of µ is quite large. The left panel contains the spot B profile for µ = 0.0059, plotted again as a function of the
rescaled variable s: the region of non-monotonicity has shifted to the second and third maxima.
Now, we formally set ∆ = 0 and separate (3.33) into real and imaginary parts: solving the resulting system is
then equivalent to finding zeros of the mapping


aq0 cos γ̃ + d˜1
 aq sin γ̃ 
0


F (d˜1 , γ̃, a, d˜2 ) =  ˜
.
 d1 + a √ν3 d˜21 
ad˜2
(3.34)
It is readily seen that the vector
 s

s√
√
3q
3
0
(d˜∗1 , γ̃ ∗ , a∗ , d˜∗2 ) = −
, 0,
, 0
ν
q0 ν
is a root of F with Jacobian

1
0

DF (d˜∗1 , γ̃ ∗ , a∗ , d˜∗2 ) = 
−1
0
0
a∗ q0
0
0
q0
0
q0
0

0
0

.
0
a∗
Since q0 > 0, the Jacobian is invertible, and we can therefore solve (3.33) uniquely for all sufficiently small ∆,
that is, for α0 , δ0 , δ1 , δ2 small enough, and subsequently (3.32) for all 0 < ε 1. Reversing the scaling for d, we
find that
s√
3
1
3q0 d1 = −µ 8
1 + O(α0 + δ0 + δ1 + δ2 + µ 8 )
ν
d2
=
3
1
µ 4 O(α0 + δ0 + δ1 + δ2 + µ 8 ),
which completes the proof of Theorem 1.
21
3.6
Shape and monotonicity of planar spot B envelopes
We now summarize the asymptotic expressions for the planar spot B solutions we constructed in the preceding
sections. We found a spot B with the asymptotics

! 12
√


3
3q
√
0


µ 8 J0 (r) + O( µ)
0 ≤ r ≤ r0
at the core
−


ν




3


—
r0 ≤ r ≤ O(µ− 8 )
near P−
(3.35)
u(r) =
9
3
−
16 )

8)
r
≈
O(µ
O(µ
between
P
and
P
−
+




3
1


—
O(µ− 8 ) ≤ r ≤ δ0 µ− 2
near P+




1
 √ √
µq( µr) cos [r(1 + O(µ))] + O(µ)
r ≥ δ0 µ− 2
in the rescaling chart,
where ”—” indicates that the solution changes its asymptotics in this regime. We recall from Lemma 2.3 that
q(s) satisfies q(0) = q(∞) = 0.
Equation (3.35) shows immediately that the profile of spot B is not monotone; this follows also directly from
the fact that z1 = Ar /(rA) changes sign in the transition chart. To verify this prediction, we continued spot B
numerically towards small µ. The main difficulty is that the profile stretches out as µ tends to zero: this is
√
reflected in the rescaling chart where s = µr gives the correct scaling of the far-field profile. Thus, in order
to avoid having to enlarge the domain on which we compute our profiles as we continue spot B solutions, we
√
employ this scaling. Setting s = µr and 0 = d/ds, the stationary radial planar Swift–Hohenberg equation (1.1),
written as a first-order system, becomes
u0
=
u1
u01
=
u2
u02
=
u03
=
u3
2u
2 u1
1
1 u1
3
2
3
+
u
−
−
u
−
,
−(1
+
µ)u
+
νu
−
u
−
2
2
µ2
µ s
s2 s
s
which we solve in auto-07p on a fixed interval (0, L) with L 1 together with Neumann boundary conditions
u1 (0) = 0,
u3 (0) = 0,
u1 (L) = 0,
u3 (L) = 0
using the methods discussed in [14]. The results of our computations, with L = 75 and ν = 1.6, are shown
in Figure 6: as predicted, the profiles become non-monotone for sufficiently small µ, which provides additional
validation of the theory presented in this paper.
4
Spots in three space dimensions
In this section, we discuss localized spot solutions for the Swift–Hohenberg equation
ut = −(1 + ∆)2 u − µu + νu2 − u3 ,
x ∈ R3
(4.1)
posed in three space dimensions in the region 0 < µ 1. More precisely, we seek solutions u(x, t) = u(|x|) of
(4.1) that are bounded as |x| → 0 and satisfy lim|x|→∞ u(|x|) = 0. We have the following two existence results
1
for such solutions. The first result pertains to spot B solutions whose amplitude scales with O(µ 4 ).
p
Theorem 3 Fix ν > ν∗ := 27/38; then there is a µ0 > 0 such that equation (4.1) has a stationary localized
1
radial solution uB (|x|) of amplitude O(µ 4 ) for each µ ∈ (0, µ0 ). Furthermore, there is a constant q0 > 0 such
22
that uB (|x|) has the expansion
r

1
q0 sin r
√


+ O( µ)
−4µ 4


νπ r




—


B
1
u (r) =
O(µ 2 )






—



 √ √
µq( µr) cos [r(1 + O(µ))] + O(µ)
0 ≤ r ≤ r0
at the core
− 14
r0 ≤ r ≤ O(µ
− 14
r ≈ O(µ
− 14
O(µ
)
near P−
)
between P− and P+
− 21
)≤r≤µ
near P+
− 12
r ≥ δ0 µ
in the rescaling chart,
where ”—” indicates that the solution changes its asymptotics in this regime, and q(x) denotes the ground state
of the 3D nonlinear Schrödinger equation (see Lemma 4.3 below).
√
There is a second family of spots, which exists for each ν > 0 and whose amplitude scales with O( µ).
Theorem 4 Fix ν > 0; then there is a µ0 > 0 such that equation (4.1) has a stationary localized radial solution
1
uA (|x|) of amplitude O(µ 2 ) for each µ ∈ (0, µ0 ). More precisely, there is a constant d > 0 such that uA (|x|) has
the expansion
1 sin r
uA (r) = dµ 2
+ O(µ)
r
uniformly on bounded intervals [0, r0 ] as µ → 0.
In the remainder of this section, we present the proofs of these theorems. The general strategy is analogous
to that presented earlier for planar spots, and we will therefore keep the details to a minimum and focus on
highlighting the differences between the planar and the three-dimensional cases.
4.1
The core manifold
Radial steady states of (4.1) satisfy the non-autonomous differential equation
2
2
∂r2 + ∂r + 1 u = −µu + νu2 − u3 ,
r
r > 0,
and we seek solutions u(r) that are bounded as r → 0 and satisfy limr→∞ u(r) = 0. We rewrite this equation as
2
∂r2 + ∂r + 1 u1 = u2
(4.2)
r
2
∂r2 + ∂r + 1 u2 = −µu1 + νu21 − u31
r
where u1 := u. Setting (u3 , u4 ) := (∂r u1 , ∂r u2 ), we obtain the first-order system

Ur = AU + F(U, µ),
0
0

A=
−1
0
0
0
1
−1
1
0
− 2r
0

0
1 

,
0 
− 2r


0


0


F(U, µ) = 
,


0
2
3
−µu1 + νu1 − u1
(4.3)
where U = (u1 , u2 , u3 , u4 )t .
We begin by characterizing all solutions of (4.3) that are bounded as r → 0. The arguments below closely
follow [13, §2] where an identical analysis was done for the planar Swift–Hohenberg equation. The linear system
23
Ur = AU admits the linearly independent solutions
V1 (r)
=
V2 (r)
=
t
sin r
cos r sin r
, 0,
− 2 ,0
r
r
r
2 cos r 2 sin r
sin r
2 sin r cos r sin r
− cos r,
,
− 2 + sin r,
−
r
r
r
r
r
r2
t
cos r
sin r cos r
=
, 0, −
− 2 ,0
r
r
r
t
2 cos r
2 sin r 2 cos r
,
=
sin r,
, cos r, −
−
r
r
r2
V3 (r)
V4 (r)
t
where V1 (r) and V2 (r) remain bounded as r → 0, while V3 (r) and V4 (r) blow up at the origin. In contrast, V1 (r)
and V3 (r) converge to zero as r → ∞, while V2 (r) and V4 (r) remain bounded but do not decay as r → ∞.
Lemma 4.1 For each fixed r0 > 0, there is a constant δ0 > 0 such that the set W−cu (µ) of solutions U (r) of
(4.3) for which sup0≤r≤r0 |U (r)| < δ0 is, for each |µ| < δ0 , a smooth two-dimensional manifold. Furthermore,
each U ∈ W−cu (µ) can be written uniquely as
U (r0 )
= d1 V1 (r0 ) + d2 V2 (r0 ) + V3 (r0 )O(|µ||d| + |d|2 )
π
1
+V4 (r0 ) − + O
νd21 + O(|µ||d| + |d1 |3 + |d2 |2 ) ,
8
r0
(4.4)
where d = (d1 , d2 ) ∈ R2 is small, and the right-hand side in (4.4) depends smoothly on (d, µ).
Proof. Using the independent variable ρ = ln r transforms (4.3) into the system

0
0

Ur = 
0
0
0
0
0
0
0
0
−2
0

0
0

 U + O(eρ )G(U, µ),
0
−2
where G is smooth. A standard center-unstable manifold construction in the limit ρ → −∞ shows that W−cu (µ)
exists and is a two-dimensional smooth manifold for each fixed r. It remains to verify the claimed expansion.
The adjoint equation Wr = −At W has the four independent solutions
W1 (r)
=
W2 (r)
=
W3 (r)
=
W4 (r)
=
t
r2
r2
cos r + r sin r, − cos r, r cos r,
sin r
2
2
t
1
r
0, (cos r + r sin r), 0, cos r
2
2
t
1
r
r cos r − sin r, (r cos r + (−1 + r2 ) sin r), −r sin r, (r cos r − sin r)
2
2
t
1
r
0, (r cos r − sin r), 0, − sin r ,
2
2
which satisfy hVi (r), Wj (r)i = δij for i, j = 1 . . . , 4 and all r > 0. It is easy to check that the core manifold is
formed of solutions of the fixed-point equation
U (r)
=
2
X
dj Vj (r) +
j=1
=
2
X
j=1
2
X
Z
r0
j=1
dj Vj (r) +
2
X
j=1
r
Vj (r)
Z
hWj (s), F(U (s), µ)ids +
4
X
r0
Wj,4 (s), F4 (U (s), µ)ds +
24
0
j=3
r
Vj (r)
r
Z
Vj (r)
4
X
j=3
Z
Vj (r)
0
hWj (s), F(U (s), µ)ids
r
Wj,4 (s), F4 (U (s), µ)ds
on C 0 ([0, r0 ], R4 ). Using this equation, the expansion (4.4) can now be derived, and the quadratic coefficient in
d1 in front of the V4 (r) term is given by
Z
0
r0
ν
W4,4 (s)νV1,1 (s) ds = −
2
2
using the asymptotic expansion Si(r) :=
4.2
r0
Z
0
Rr
0
hπ
i
s=r0
sin3 s
ν
ds = − [3 Si(s) − Si(3s)] s=0 = −ν
+ O(r0−1 )
s
8
8
sin s
s ds
=
π
2
+ O(1/r) as r → ∞.
The far-field equation
Now that we understand small bounded solutions of (4.3) on intervals [0, r0 ] for each fixed r0 1, we focus on
the dynamics for large r in the limit r → ∞. In this limit, it is convenient to make (4.3) autonomous by adding
the variable α = 1/r, which satisfies the differential equation αr = −α2 . This leads to the equation

  
u3
u1

  

u2  
u4

 
d 
u3  = 
,
(4.5)
−u1 + u2 − 2αu3




dr   
2
3
u4  −u2 − 2αu4 − µu1 + νu1 − κu1 
−α2
α
which we study for 0 < α 1. Introducing the coordinates
 
 
0
1
 2i 
0
 
 
U = Ã   + B̃   + c.c.,
1
i
−2
0
Ã
B̃
!
1
=
4
2u1 − i(2u3 + u4 )
−u4 − iu2
!
(4.6)
turns (4.5) into the system
Ãr
=
(i − α)à + αà + B̃ + O((|µ| + |Ã| + |B̃|)(|Ã| + |B̃|))
B̃r
=
(i − α)B̃ − αB̃ + O((|µ| + |Ã| + |B̃|)(|Ã| + |B̃|))
αr
=
(4.7)
−α2 .
The following lemma shows that the right-hand side can be further simplified.
Lemma 4.2 Write µ = ε2 , then there is a change of coordinates
!
!
A
Ã
−iφ(r)
=e
[1 + T (α)]
+ O (ε2 + |Ã| + |B̃|)(|Ã| + |B̃|)
B
B̃
(4.8)
such that (4.7) becomes
Ar
=
Br
=
αr
=
−αA + B + RA (A, B, α, ε)
ε2
−αB + A + c3 |A|2 A + RB (A, B, α, ε)
4
−α2 ,
2
(4.9)
where c3 := 34 − 19ν
18 . The transformation (4.8) is polynomial in (Ã, B̃, α) and smooth in ε. The function
T (α) = O(α) is linear and upper triangular for each α, while φ(r) satisfies
φr = 1 + O(ε2 + |α|2 + |A|2 ),
25
φ(0) = 0.
The remainder terms are of the form


2
X
j
3−j
3
2
5
2
RA (A, B, α, ε) = O 
|A B | + |α| |A| + |α| |B| + (|A| + |B|) + ε |α|(|A| + |B|)
(4.10)
j=0
RB (A, B, α, ε)
=


1
X
O
|Aj B 3−j | + |α|3 |B| + ε2 (ε2 + |α|3 + |A|2 )|A| + (|A| + |B|)5 + ε2 |α||B| .
j=0
Proof. The planar case was handled previously in [18, Lemma 3.10 and Corollary 3.14] and [13, Lemma 2].
The only difference between these cases is the treatment of the terms that are linear in (A, B), and we therefore
focus on the justification of the linear part of the normal form (4.9). Proceeding as in [18, Lemma 3.10], it is
easy to check that the transformation
B = B̃ −
1
B̃,
1 + 2ir
A = Ã +
1
i + 2r
à +
B
1 + 2ir
2r(i − 2r)
transform the linear part of (4.7) into
Ar = (i − α)(1 + O(α2 ))A + (1 + O(α2 ))B,
Br = (i − α)(1 + O(α2 ))B.
The remainder of the proof proceeds as in the references cited above by using normal-form transformations applied
to (A, B, α) and removing the purely imaginary linear terms through an appropriate phase transformation.
4.3
The rescaling and transition charts
Next, we set z = −α + B/A and introduce the coordinates
A
,
α
A
A2 = ,
ε
A1 =
z
B
= −1 +
,
α
αA
z
α
B
z2 = = − +
,
ε
ε
εA
z1 =
ε1 =
ε
,
α
ε2 = ε,
α1 = α,
r = eρ
α
,
ε
s = εr.
α2 =
The coordinates (A2 , z2 , ε2 , α2 ) are referred to as the rescaling chart, while the variables (A1 , z1 , ε1 , α1 ) correspond
to the transition chart. Note that these coordinates are related via
A1 =
A2
,
α2
z1 =
z2
,
α2
ε1 =
1
,
α2
α1 = ε2 α2 = εα2 ,
(4.11)
and we can transform from one to the other chart in the transverse section ε1 = α2 = 1.
In the rescaling chart, (4.9) becomes
∂ s A2
∂s z2
∂s ε2
∂s α2
= A2 z2 + O(|ε2 |2 )
1
− z22 − 2α2 z2 + c3 |A2 |2 + O(|ε2 |2 )
=
4
= 0
(4.12)
= −α22 ,
where the estimates for the remainder terms can be derived from (4.10) as in the planar case. This system admits
the one-parameter family Q− = (O(|ε2 |2 ), − 12 + O(|ε2 |2 ), ε2 , 0) of equilibria.
In the transition chart, we obtain
∂ρ A1 = A1 1 + z1 + O(|α1 |2 )
∂ρ z1
=
−z1 (1 + z1 ) +
∂ρ ε1
=
ε1
∂ρ α1
=
−α1 ,
ε21
4
(4.13)
+ c3 |A1 |2 + O |α1 |2 (|1 + z1 | + |ε1 |2 + |A1 |4 )
26
where the estimates for the remainder terms are again analogous to those for the planar case. This system has
the equilibria P+ = (0, 0, 0, 0) and P− = (0, −1, 0, 0). It is convenient to use the the coordinates
A− = A1 ,
z− = 1 + z1 ,
ε− = ε1 ,
α− = α1
(4.14)
near P− in which (4.13) becomes
∂ ρ A−
4.4
= A− z− + O(|α− |2 )
∂ρ z−
= z− (1 − z− ) +
∂ρ ε−
= ε−
∂ρ α−
= −α− .
ε2−
4
(4.15)
+ c3 |A− |2 + O |α− |2 (|z− | + |ε− |2 + |A− |4 )
Singular connecting orbits
We return to the equations in the transition and rescaling chart, and set α1 = 0 and ε2 = 0. In the transition
chart, we then obtain
∂ρ A1
= A1 (1 + z1 )
∂ρ z1
= −z1 (1 + z1 ) +
∂ρ ε1
= ε1 ,
(4.16)
ε21
4
+ c3 |A1 |2
while (4.13) in the rescaling chart becomes
∂s A2
∂s z2
∂s α2
= A2 z2
1
− z22 − 2α2 z2 + c3 |A2 |2
=
4
= −α22 .
(4.17)
Written as a second-order equation for A2 , equation (4.17) becomes
A2
2
+ c3 A32 ,
∂s2 A2 + ∂s A2 =
s
4
where we restrict A2 to be real-valued.
A2 ∈ R,
(4.18)
Lemma 4.3 ([11] or [5, Lemma 2.1]) If c3 < 0, then (4.18) has a unique positive bounded nontrivial solution
A2 (s) = q(s) for s ∈ (0, ∞), and there are constants q0 > 0 and q+ 6= 0 so that

 q0 + O(s)
s→0
−s/2
q(s) =
(4.19)
 (q+ + O(e−s/2 )) e
s → ∞.
s
Moreover, the linearization of (4.18) about q(s) does not have a nontrivial uniformly bounded solution on R+ .
Next, we write the solution q(s) via
z2 =
∂s A2
,
A2
α2 =
1
,
s
A1 = sA2 ,
z1 = sz2 =
s∂s A2
,
A2
ε1 = s,
ρ = ln s
in the transition and rescaling charts and conclude that the functions
q 0 (s)
q 0 (s) 1
(A∗1 , z1∗ , ε∗1 )(ln s) = s q(s),
,1 ,
(A∗2 , z2∗ , α2∗ )(s) = q(s),
,
q(s)
q(s) s
(4.20)
satisfy (4.16) and (4.17), respectively. It is now easy to obtain the following lemma, whose proof we omit.
Lemma 4.4 Assume that c3 < 0, and consider (4.16) and (4.17) in R3 . The solution (4.20) is a connecting
orbit that forms a transverse intersection of the unstable manifold W u (P+ ) of the equilibrium P+ = (0, 0, 0) of
(4.16) and the center-stable manifold W cs (Q− ) of the equilibrium Q− = (0, − 12 , 0) of (4.17).
27
4.5
The dynamics near P+
Recall that equation (4.13) in the transition chart is given by
∂ρ A1
=
A1 1 + z1 + O(|α1 |2 )
∂ρ z1
=
−z1 (1 + z1 ) +
∂ρ ε1
=
ε1
∂ρ α1
=
−α1 .
ε21
4
(4.21)
+ c3 |A1 |2 + O |α1 |2 (|1 + z1 | + |ε1 |2 + |A1 |4 )
As in the planar case, there is a smooth coordinate transformation z̃1 = z1 + O(|A1 |2 + ε21 + α12 ) that brings
(4.21) near the origin into the form
∂ρ A1
= A1 [1 + O(|A1 | + |z̃1 | + |ε1 | + |α1 |)]
∂ρ z̃1
= −z̃1 [1 + O(|A1 | + |z̃1 | + |ε1 | + |α1 |)]
∂ρ ε1
= ε1
∂ρ α1
= −α1 .
(4.22)
We are interested in tracking the center-stable manifold W cs (Q− ) near P+ in backwards time. To find an
expression for this manifold near P+ , we interpret the transversality of W cs (Q− ) and W u (P+ ) stated in Lemma 4.4
in the coordinates (A1 , z̃1 , ε1 , α1 ). The following lemma can be proved as in the planar case.
Lemma 4.5 For each sufficiently small δ0 > 0, there are constants a0 , ε0 > 0 such that the following is true.
Define the section Σ0 := {ε1 = δ0 }, then
ε
: ã ∈ (−a0 , a0 ), ε < ε0 ,
W cs (Q− ) ∩ Σ0 = (A1 , z̃1 , α1 ) = −eiγ [η(δ0 ) + O(ã)] + O(ε2 ), −ã + O(ε2 ),
δ0
where η0 (δ0 ) = q0 δ0 (1 + O(δ0 )) is smooth, q0 > 0 is the constant defined in (4.19), and γ ∈ R is arbitrary.
We start with ρ = 0 for initial data in Σ0 and need to track solutions until ρ = ρ∗ , where ρ∗ < 0 is such that
α1 (ρ∗ ) = α0 = 1/r0 . Since α1 (0) = ε/δ0 in Σ0 , we find from (4.22) that
ρ∗ = ln
ε
,
α0 δ0
(4.23)
and we consequently solve (4.22) only for ρ∗ ≤ ρ ≤ 0. We now choose a second constant δ1 > 0 and track an
appropriate part of the center-stable manifold W cs (Q− ) in backwards time under the evolution of (4.22) from
Σ0 to Re z̃1 = −δ1 . Let I denote any open interval that contains the point 2(q0 νπ)−1 .
√
Lemma 4.6 Fix any open interval I that contains the point 2/ q0 νπ. For each fixed choice of 0 < δ0 , δ1 , there
is an ε0 > 0 such that solutions of (4.22) associated with initial data of the form
!
1
1
aε 2
ε
iγ
2
(A1 , z̃1 , ε1 , α1 )(0) = −e η0 (δ0 ) + O(ε 2 ), −
+ O(ε ), δ0 ,
(4.24)
δ0
δ0
in W cs (Q− ) ∩ Σ0 with a ∈ I and ε ∈ (0, ε0 ) land after time
1
ρ1 = ln
aε 2
≥ ρ∗
δ0 δ1
28
(4.25)
at the point
1
1
aε 2 iγ
e η0 (δ0 )(1 + O(δ0 + δ1 + ε 2 ))
δ0 δ1
A1 (ρ1 )
= −
z̃1 (ρ1 )
= −δ1 (1 + O(δ0 + δ1 + ε 2 ))
ε1 (ρ1 )
=
aε 2
δ1
α1 (ρ1 )
=
δ1 ε 2
.
a
(4.26)
1
1
1
Proof. We begin by solving (4.22) given by
∂ρ A1
= A1 [1 + O(|A1 | + |z̃1 | + |ε1 | + |α1 |)]
∂ρ z̃1
= −z̃1 [1 + O(|A1 | + |z̃1 | + |ε1 | + |α1 |)]
∂ρ ε1
= ε1
∂ρ α1
= −α1
with initial conditions
(A1 , ε1 )(0) = (A0 , δ0 ),
(z̃1 , α1 )(ρ̃) = (B1 , β1 )
for arbitrary but small A0 , B1 ∈ C and β1 > 0 on the interval [ρ̃, 0] for arbitrary ρ̃ −1. We obtain immediately
that
ε1 (ρ) = δ0 eρ ,
α1 (ρ) = β1 eρ̃−ρ ,
and it remains to solve
∂ρ A1
∂ρ z̃1
= A1 1 + O(|A1 | + |z̃1 | + δ0 eρ + β1 eρ̃−ρ ) ,
= −z̃1 1 + O(|A1 | + |z̃1 | + δ0 eρ + β1 eρ̃−ρ ) ,
A1 (0) = A0
(4.27)
z̃1 (ρ̃) = B1
on [ρ̃, 0]. Using a standard contraction mapping argument in exponentially weighted spaces that exploits the
special structure of the nonlinearity, we find that (4.27) has a unique solution and that this solution depends
smoothly on (A0 , B1 , β1 , ρ̃) and is given by
A1 (ρ)
z̃1 (ρ)
=
=
A0 eρ (1 + O(|A0 | + |B1 | + β1 + δ0 ))
B1 e
ρ̃−ρ
(4.28)
(1 + O(|A0 | + |B1 | + β1 + δ0 ))
uniformly in ρ̃ ≤ ρ ≤ 0 and |A0 |, |B1 |, β1 1. Inspecting the initial conditions (4.24) for which we want to solve,
and substituting ρ̃ = ρ1 with ρ1 as in (4.25), we obtain
1
ε
δ1 ε 2
β1 = e−ρ1 =
.
δ0
a
1
2
iγ
A0 = −e η0 (δ0 ) + O(ε ),
Similarly, the initial condition for z̃1 (0) becomes
1
B1 eρ1 (1 + O(|A0 | + |B1 | + β1 + δ0 )) =
1
1
B1 aε 2
aε 2
!
(1 + O(|B1 | + δ0 + ε 2 )) = −
+ O(ε2 ),
δ0 δ1
δ0
1
which has the unique solution B1 = −δ1 (1 + O(δ0 + δ1 + ε 2 )). Substituting these expressions into (4.27) and
(4.28) and evaluating at ρ = ρ̃ = ρ1 gives (4.26) as claimed.
Reverting back to the variable z1 , we obtain
A01
z10
1
:= A1 (ρ1 ) = −aε 2 η1 eiγ
1
:= z1 (ρ1 ) = −δ1 1 + O(δ0 + δ1 + ε 2 )
1
ε01
aε 2
:= ε1 (ρ1 ) =
δ1
α10
:= α1 (ρ1 ) =
1
δ1 ε 2
,
a
29
(4.29)
where
η1 :=
1
1
q0
η0 (δ0 )
(1 + O(δ0 + δ1 + ε 2 )) = (1 + O(δ0 + δ1 + ε 2 )).
δ0 δ1
δ1
Next, we transport this manifold to a neighborhood of the equilibrium P− .
4.6
The dynamics between P+ and P−
We fix a small constant δ2 > 0, set
ρ2 = ln
and integrate the transition-chart system
∂ρ A1
δ1 δ2
,
(1 − δ1 )(1 − δ2 )
A1 (1 + z1 + O(|α1 |2 ))
=
∂ρ z1
=
−z1 (1 + z1 ) +
∂ρ ε1
=
ε1
∂ρ α1
=
−α1
ε21
4
(4.30)
+ c3 |A1 |2 + O |α1 |2 (|1 + z1 | + |ε1 |2 + |A1 |4 )
with initial conditions given by (4.29) backwards in time from ρ = 0 to ρ = ρ2 . We initially set (A1 , ε1 , α1 ) = 0
so that (4.30) with the initial condition (4.29) for z1 becomes the complex differential equation
∂ρ z1 = −z1 (1 + z1 ),
1
z1 (0) = z10 = −δ1 (1 + O(δ0 + δ1 + ε 2 )),
whose solution z1∗ (ρ) evaluated at ρ = ρ2 is given by
1
z1∗ (ρ2 ) = −1 + δ2 (1 + O(δ0 + δ1 + ε 2 )).
Next, we expand the time-ρ2 map of (4.30) with initial condition (A01 , z10 , ε01 , α10 ) at ρ = 0 around (0, z10 , 0, 0) and
obtain




η2 A01 (1 + O(|A01 | + |ε01 | + |α10 |))
A1 (ρ2 )


 z (ρ )   z1∗ (ρ2 ) + O(|A01 | + |ε01 | + |α10 |) 
 1 2  

1
(4.31)
=

,
aδ2 ε 2 (1 + O(δ1 + δ2 ))

 ε1 (ρ2 )  
1


ε2
α1 (ρ2 )
(1 + O(δ1 + δ2 ))
aδ2
where the constant η2 is given by η2 = a1 (ρ2 ), and a1 is the solution to the linear equation
∂ρ a1 = (1 + z1∗ (ρ))a1 ,
a1 (0) = 1.
This equation can be solved explicitly, and we obtain
η2 = δ1 (1 + O(δ2 )).
Substituting the initial conditions (4.29), we arrive at


1


−aε 2 η3 eiγ
A1 (ρ2 )
1


 z (ρ )  −1 + δ2 (1 + O(δ0 + δ1 + ε 2 ))

 1 2  
1
,

=
aδ2 ε 2 (1 + O(δ1 + δ2 ))

 ε1 (ρ2 )  
1


ε2
α1 (ρ2 )
(1 + O(δ1 + δ2 ))
aδ2
where η3 is given by
1
η3 := η1 η2 (1 + O(ε 2 )) =
1
1
q0
(1 + O(δ0 + δ1 + ε 2 ))δ1 (1 + O(δ2 )) = q0 (1 + O(δ0 + δ1 + δ2 + ε 2 )).
δ1
30
(4.32)
Using the notation
1
∆ := O(α0 + δ0 + δ1 + δ2 + ε 2 )
(4.33)
and transforming (4.32) into the coordinates (4.14) near P+ , we obtain


1


−q0 aε 2 eiγ (1 + ∆)
A− (0)


δ2 (1 + ∆)
 z (0)  

 −  

1
 =  aδ2 ε 2 (1 + ∆)  ,


 ε− (0)  
1


ε2
α− (0)
(1 + ∆)
aδ2
(4.34)
where we have also reset time back to zero.
4.7
The dynamics near P−
It remains to solve equation (4.15), given by
∂ ρ A−
= A− (z− + O(|α− |2 ))
∂ρ z−
= z− (1 − z− ) +
∂ρ ε−
= ε−
∂ρ α−
= −α− ,
ε2−
4
(4.35)
+ c3 |A− |2 + O |α− |2 (|z− | + |ε− |2 + |A− |4 )
with the initial data (4.34) for the remaining time
1
ρ3 = ρ∗ − ρ1 − ρ2 = ln
ε 2 (1 − δ1 )(1 − δ2 )
.
aα0 δ2
(4.36)
We have the following result.
Lemma 4.7 For all fixed sufficiently small constants α0 , δj > 0 with j = 0, 1, 2, there is an ε0 > 0 such that the
solution of (4.35) with initial condition (4.34), evaluated at ρ = ρ3 with ρ3 from (4.36), is given by
A− (ρ3 )
=
z− (ρ3 )
=
ε− (ρ3 )
=
α− (ρ3 )
=
1
−q0 aε 2 eiγ (1 + ∆)
(4.37)
1
2
ε
(1 + ∆)
aα0
ε
= εr0
α0
1
α0 =
r0
uniformly in a ∈ I and ε ∈ (0, ε0 ), where q0 > 0 is the constant given in (4.19).
Proof. Our choice of ρ∗ in (4.23) was made to ensure that α− (ρ3 ) = α0 = 1/r0 , and the statement for ε−
follows from its definition. In particular, we have
ε− (ρ) = ε− (0)eρ ,
1
ε− (0) = O(ε 2 ),
α− (ρ) = α0 eρ3 −ρ
for 0 ≤ ρ ≤ ρ3 . Next, we can flatten the center manifold of (4.35) by a transformation of the form ẑ− =
z− + O(|A− |2 ) to get the system
∂ρ A− = A− (ẑ− + O(|A− |2 + |α− |2 )),
∂ρ ẑ− = ẑ− (1 + O(ẑ− + |A− | + |α− | + |ε− |)) + O(ε2− ).
We write
ẑ− (ρ) = z̃− (ρ)eρ
A− (ρ) = Ã− (ρ),
31
(4.38)
and obtain the system
∂ρ Ã−
∂ρ z̃−
= Ã− z̃− eρ + O(|Ã− |2 ) + e2(ρ3 −ρ) O(α02 )
1
= z̃− O eρ z̃− + |Ã− | + α0 eρ3 −ρ + ε 2 eρ + O(ε)eρ ,
(4.39)
which we consider with the initial conditions (4.34), which become
1
Ã− (0) = −q0 aε 2 eiγ (1 + ∆) =: Ã0− ,
0
z̃− (0) = δ2 (1 + ∆) =: z̃−
.
We write (4.39)–(4.40) as the fixed-point equation
Z ρ
h
i
Ã− (ρ) = Ã0− +
Ã− (y) z̃− (y)ey + O(|Ã− (y)|2 ) + e2(ρ3 −y) O(α02 ) dy
Z 0ρ 1
0
z̃− (y)O ey z̃− (y) + |Ã− (y)| + α0 eρ3 −y + ε 2 ey + O(ε)ey dy
z̃− (ρ) = z̃− +
(4.40)
(4.41)
0
1
0
on [ρ3 , 0]. Using that Ã0− = O(ε 2 ), z̃−
= O(δ2 ), and |ρ3 | ≤ | ln ε|, we can apply the contraction mapping
principle to show that (4.41) has a unique solution (Ã− , z̃− ) in an appropriate small ball centered at the origin
in C 0 ([ρ3 , 0], C2 ). Furthermore, there is a uniform constant C with
kÃ− k ≤ C|Ã0− |,
0
kz̃− k ≤ |z̃−
|(1 + ∆) + Cε.
Using these estimates together with (4.40) and (4.41) in (4.38) and reverting back to the z− -variable using the
inverse transformation z− = ẑ− + O(|A− |2 ), we obtain
1
A− (ρ3 ) = −q0 aε 2 eiγ (1 + ∆),
and
1
(4.36)
z− (ρ3 ) = δ2 (1 + ∆)eρ3 + O(ε) =
ε2
(1 + ∆)
aα0
which completes the proof.
4.8
The core manifold expressed in the transition chart
To match the center-stable manifold W cs (Q− ) near the equilibrium P− with the core manifold W−cu (ε), we express
the latter in the coordinates (A− , z− , ε− , α− ) that we introduced in (4.14).
Lemma 4.8 For each fixed constant 0 < α0 1, there is a constant δ3 > 0 such that W−cu (ε) is given by
2
2
2
π
1
d2
W−cu (ε)α=α :
A− = ei[− 2 +O(α0 +ε +|d| )] d1 (1 + O(α0 )) −
(i + O(α0 )) + O ε2 |d| + |d|2
0
2
α0
π
2
d2 (i + O(α0 )) + 8 + O(α0 ) νd1 + O ε2 |d| + |d2 |2 + |d1 |3
z− = −
(4.42)
d1 α0 (1 + O(α0 )) − d2 (i + O(α0 )) + O (ε2 |d| + |d|2 )
uniformly in |ε| + |d| < δ3 .
Proof. First, we express
!
Ã1
=
B̃1
!
Ã3
=
B̃3
the solutions Vj (r0 ) = Vj (1/α0 ) in the (Ã, B̃)-coordinates (4.6) and obtain
!
!
!
−1
1
+
O(α
)
Ã
−i
+
O(α
)
π
π
α0 i(α−1
1
0
2
0
e 0 −2)
,
= ei(α0 − 2 )
,
2
2
0
B̃2
−iα0 + O(α02 )
!
!
!
−1
i
+
O(α
)
Ã
1
+
O(α
)
π
π
1
α0 i(α−1
0
4
0
−
)
i(α
−
)
e 0 2
,
= e 0 2
,
2
2
0
B̃4
α0 + O(α02 )
32
where (Ãj , B̃j ) corresponds to Vj (α0−1 ). Using equation (4.4) gives
"
!
!
2
2
α
d
(1
+
O(α
))
−
d
(i
+
O(α
))
+
α
(i
+
O(α
))O(ε
|d|
+
|d|
)
Ã
π
1 i(α−1
0
1
0
2
0
0
0
)
−
e 0 2
=
2
−α0 d2 (i + O(α0 ))
B̃
!
#
π
1 + O(α0 )
2
2
3
2
−
+ O(α0 ) νd1
+ O(ε |d| + |d1 | + |d2 | ) ,
8
α0 + O(α02 )
and equation (4.8) then yields
!
2
2
2
π
A
1
= ei[− 2 +O(α0 +ε +|d| )]
2
B
!
α0 d1 (1 + O(α0 )) − d2 (i + O(α0 )) + O(ε2 |d| + |d|2 )
.
−α0 d2 (i + O(α0 )) − α0 [ π8 + O(α0 )]νd21 + O(ε2 |d| + |d2 |2 + |d1 |3 )
The transformation
A− =
A
,
α0
z− =
B
α0 A
finally gives
A−
z−
1 i[− π2 +O(α20 +ε2 +|d|2 )]
d2
e
d1 (1 + O(α0 )) −
(i + O(α0 )) + O(ε2 |d| + |d|2 )
2
α0
2
π
d2 (i + O(α0 )) + 8 + O(α0 ) νd1 + O(ε2 |d| + |d2 |2 + |d1 |3 )
= −
d1 α0 (1 + O(α0 )) − d2 (i + O(α0 )) + O(ε2 |d| + |d|2 )
=
as claimed.
4.9
Matching core and far field
It remains to find nontrivial intersections of the center-stable manifold W cs (Q− ) and the core manifold W−cu (ε)
at α = α0 . Recall the expression (4.37)
1
1
A− = −q0 aε 2 eiγ (1 + ∆),
z− =
ε2
(1 + ∆)
aα0
with
1
∆ := O(α0 + δ0 + δ1 + δ2 + ε 2 )
for the center-stable manifold W cs (Q− ) and the expansion (4.42)
1 i[− π2 +O(α20 +ε2 +|d|2 )]
d2
2
2
A− =
e
d1 (1 + O(α0 )) −
(i + O(α0 )) + O(ε |d| + |d| )
2
α0
π
2
d2 (i + O(α0 )) + 8 + O(α0 ) νd1 + O ε2 |d| + |d2 |2 + |d1 |3
z− = −
d1 α0 (1 + O(α0 )) − d2 (i + O(α0 )) + O (ε2 |d| + |d|2 )
with d = d(d1 , d2 ) ∈ R2 for the core manifold W−cu (ε). Setting these expressions equal to each other gives the
system
1
1 i[− π2 +O(α20 +ε2 +|d|2 )]
d2
2
2
iγ
−q0 aε 2 e (1 + ∆) =
e
d1 (1 + O(α0 )) −
(i + O(α0 )) + O(ε |d| + |d| )
2
α0
1
d2 (i + O(α0 )) + π8 + O(α0 ) νd21 + O ε2 |d| + |d2 |2 + |d1 |3
ε2
(1 + ∆) = −
aα0
d1 α0 (1 + O(α0 )) − d2 (i + O(α0 )) + O (ε2 |d| + |d|2 )
that we need to solve. We set γ = γ̃ −
0
0
π
2
1
+ O(α02 + ε2 + |d|2 ) and use the scaling (d1 , d2 ) = (ε 2 d˜1 , εd˜2 ) to obtain
1
˜
(4.43)
2q0 aeiγ̃ (1 + ∆) + d˜1 (1 + O(α0 )) + O(ε 2 |d|)
νπ
1
1
2
˜
˜
˜
˜
˜
2
2
= (1 + ∆) d1 (1 + O(α0 )) + O(ε |d|) + a d2 (i + O(α0 )) +
+ O(α0 ) d1 + O(ε |d|) .
8
=
33
Initially setting ε = 0, we arrive at the system
0
0
2q0 a(cos γ̃ + i sin γ̃) + d˜1 (1 + ∆)
νπ
= d˜1 (1 + ∆) + a d˜2 (i + ∆) +
+ ∆ d˜21 .
8
=
(4.44)
We formally set ∆ = 0 and separate (4.44) into real and imaginary parts: solving the resulting system is then
equivalent to finding zeros of the mapping


2q0 a cos γ̃ + d˜1
 2q a sin γ̃ 


0
F (d˜1 , γ̃, a, d˜2 ) =  ˜
.
νπ
2
˜
 d1 + 8 ad1 
ad˜2
(4.45)
It is readily seen that the vector
(d˜∗1 , γ̃ ∗ , a∗ , d˜∗2 ) =
√
4 q0
2
− √ , 0, √
,0
q0 νπ
νπ
is a root of F with Jacobian

1
0

DF (d˜∗1 , γ̃ ∗ , a∗ , d˜∗2 ) = 
−1
0
0
2q0 a∗
0
0
2q0
0
2q0
0

0
0

.
0
a∗
Since q0 > 0, the Jacobian is invertible, and we can therefore solve (4.44) uniquely for all sufficiently small ∆,
that is, for α0 , δ0 , δ1 , δ2 small enough, and subsequently (4.43) for all 0 < ε 1. Reversing the scaling for d, we
find that
√ q0
1 4
1
4
√
d1 = −µ
1 + O(α0 + δ0 + δ1 + δ2 + µ 4 )
νπ
d2
4.10
1
1
= µ 2 O(α0 + δ0 + δ1 + δ2 + µ 4 ).
Existence proof for spot A in three dimensions
In this section, we prove Theorem 4. Setting α1 = 0 and ε2 = 0 in the equations in the transition and the
rescaling chart, we obtain the systems
∂ρ A1 = A1 (1 + z1 ),
∂ρ z1 = −z1 (1 + z1 ) +
ε21
+ c3 |A1 |2 ,
4
∂ρ ε1 = ε1
(4.46)
and
1
− z22 − 2α2 z2 + c3 |A2 |2 ,
∂s α2 = −α22 ,
4
respectively. These systems admit the explicit solution
1 1 1
(A2 , z2 (s), α2 (s)) =
0, − − ,
2 s s
1
s ρ=ln s
1
(A1 , z1 (s), ε1 (s)) =
0, sz2 (s),
= 0, −1 − , s
=
0, −1 − eρ , eρ ,
α2 (s)
2
2
∂s A2 = A2 z2 ,
∂s z2 =
(4.47)
(4.48)
which satisfies z2 (s) → − 21 as s → ∞ and z1 (s) → −1 as s → 0 and therefore lies in the intersection of W u (P− )
and W cs (Q− ). Using the coordinates (A− , z− , ε− , α− ) from (4.14), we obtain z− (s) = 1 + z1 (s) = −s/2. Next,
for each small δ0 > 0, we consider the intersection of the center-stable manifold W cs (Q− ) with the section Σ0
given by ε− = δ0 . Linearizing (4.46) and (4.47) about the solution (4.48), we see that the tangent space of
34
W cu (Q− ) at this solution in Σ0 is spanned by (A− , z− , ε− , α− ) = (1, 0, 0, 0). Hence, for each small δ0 > 0, there
exist constants a0 , ε0 > 0 such that
δ0
ε
W cs (Q− ) ∩ Σ0 = (A− , z− , α− ) = ãeiγ + O(ε2 ), − + O(ã2 + ε2 ),
: |ã| < a0 , ε < ε0 .
2
δ0
We set ã = εa and obtain the initial data
A− (0) = aεeiγ + O(ε2 ),
z− (0) = −
δ0
+ O(ε2 ),
2
ε− (0) = δ0 ,
α− (0) =
ε
δ0
for which we need to solve equation (4.35),
∂ ρ A−
= A− (z− + O(|α− |2 ))
∂ρ z−
= z− (1 − z− ) +
∂ρ ε−
= ε−
∂ρ α−
= −α− ,
ε2−
+ c3 |A− |2 + O |α− |2 (|z− | + |ε− |2 + |A− |4 )
4
until time
ρ0 = ln
ε
.
α0 δ0
Exploiting that ε− appears to at least quadratic order in the z− -equation, we can now proceed exactly as in the
proof of Lemma 4.7 and find that
A− (ρ0 ) = aεeiγ (1 + ∆),
z− (ρ0 ) = −
ε
(1 + ∆),
2α0
α(ρ0 ) = α0
(4.49)
with
1
∆ := O(α0 + δ0 + ε 2 ).
Matching with the core manifold (4.42) gives the system
1 i[− π2 +O(α20 +ε2 +|d|2 )]
d2
iγ
2
2
e
aεe (1 + ∆) =
d1 (1 + O(α0 )) −
(i + O(α0 )) + O(ε |d| + |d| )
2
α0
2
π
d2 (i + O(α0 )) + 8 + O(α0 ) νd1 + O ε2 |d| + |d2 |2 + |d1 |3
ε
−
(1 + ∆) = −
2α0
d1 α0 (1 + O(α0 )) − d2 (i + O(α0 )) + O (ε2 |d| + |d|2 )
that we need to solve. Substituting d1 = εd˜1 , d2 = ε2 d˜2 , and γ = − π2 + O(α02 + ε2 + |d|2 ) + γ̃ gives
2aeiγ̃ (1 + ∆)
1˜
d1 (1 + ∆) + O(ε)
2
d˜1 (1 + O(α0 )) + O(ε)
hπ
i
= d˜2 (i + O(α0 )) +
+ O(α0 ) ν d˜21 + O(ε),
8
=
which can be solved as before to get
d1 =
4ε
,
πν + O(α0 + δ0 )
d2 = O(ε2 ).
This completes the proof of Theorem 4.
5
Discussion
Using geometric blow-up techniques, we have shown that planar spot B profiles can be constructed by gluing
the Bessel function J0 and a real Ginzburg–Landau pulse together: the Bessel function describes the profile
near the core, while the pulse reflects the far-field envelope of the planar spot. One outcome of our analysis
3
1
is an understanding of the unexpected scaling of the spot amplitude which behaves like µ 8 instead of the µ 2
35
scaling expected from the Ginzburg–Landau equation in the far field. It is worthwhile to emphasize that our
analysis does not rely on the anticipated scaling; instead, it emerges naturally during the analysis as a function
of the eigenvalues at the equilibria involved in its construction. We believe that this approach, and in particular
the formal analysis outlined in §3.1, might also be helpful in other bifurcation settings to extract all possible
amplitude scalings of localized patterns.
We carried out our formal analysis in §3.1 for radial spots of the Swift–Hohenberg equation posed on Rn . The
5−n
results given in (3.6) indicated that the amplitude of spot B profiles in Rn would scale like µ 8 . Note that the
exponent (5−n)/8 is linear in n and equals 12 for n = 1 and 41 for n = 3. In particular, this agrees with the known
√
fact that there are two localized pulses on the 1D Swift–Hohenberg equation whose amplitudes scale with µ.
Thus, we conjecture that, for each 1 ≤ n ≤ 3 (with n treated as a continuous variable, which is meaningful for
√
radial solutions), there are two spots for each µ such that spot A has amplitude µ, while spot B has amplitude
5−n
µ 8 . The supporting data points we have are the proofs for spots for n = 1, 2, 3, and it would be interesting to
treat the general case using the methods of this paper.
Finally, we mention two other phenomena that could be approached using the methods outlined here: both of
these are work in progress. First, it was observed in [13] that spot A solutions of the planar Swift–Hohenberg
equation undergo a fold bifurcation at a positive value of µ at which they turn back towards decreasing values
of µ. In the two-dimensional parameter space (µ, ν), these folds occur along a curve that emerges from the
origin (0, 0). For small values of ν, the spots actually enter the region µ < 0, where they become nonlocalized
stationary target patterns. While the fold curve was proved to exist in [13], the transition to target patterns was
not analysed there.
The second topic relates to oscillons, which are among the most fascinating patterns that have been observed
in experiments: these patterns are localized standing time-periodic structures that are still the subject of much
investigation. In one space dimension, a detailed bifurcation analysis near forced Hopf bifurcations was recently
given in [4]. An analysis of the emergence of planar oscillons in various bifurcation scenarios, including forced
Hopf bifurcations, using the geometric methods outlined here is currently in progress.
Acknowledgments.
A
Sandstede was partially supported by the NSF through grant DMS-0907904.
Numerical verification of Hypothesis (H1)
We are interested in the existence and transversality of positive bounded localized solutions A(s) of (1.3),
Arr +
A
Ar
− 2 = A − A3 ,
r
4r
(A.1)
√
on [0, ∞). Setting A(r) = w(r)/ r, equation (A.1) becomes
wrr = w −
w3
.
r
(A.2)
Localized solutions of either equation will decay exponentially to zero as r → ∞, and boundedness as r → 0
p
implies that A(r) is proportional to (r) as r → ∞ or, equivalently, that w(r) satisfies the boundary conditions
w(0) = 0,
wr (0) =: a
(A.3)
for some a > 0. Transversality means that the linearization of (A.1) or (A.2) does not have a bounded solution on
[0, ∞): an equivalent condition is that the derivative wa (r) of the bounded solution w(r) of (A.2) with boundary
conditions (A.3) with respect to the shooting parameter a diverges as r → ∞. To check the existence of a
transverse localized bounded solution w(r) of (A.2)-(A.3), we posed this problem on a finite interval [0, L] with
36
-6
15
(i)
-8
13
11
1.2
9
0.8
7
0.4
log(|w(L)|)
- 14
- 16
8
1.6
(iii)
w(r)
log(|wa (L)|)
- 10
- 12
2
(ii)
10
12
14
16
L
18
5
8
10
12
14
16
L
18
0
0
4
8
12
r
16
Figure 7: Numerical evidence for the validity of our hypothesis.
L 1 and added asymptotic boundary conditions at r = L that guarantee that w(r) lies in the stable subspace
of the equilibrium w = 0 at r = ∞. We implemented this formulation in auto07p and computed bounded
localized solutions using shooting in a. The profile of the resulting solution w(r) is shown in Figure 7(iii). We
also calculated the derivative wa (r) and plot in panels (i)-(ii) of Figure 7 the values log |w(L)| and log |wa (L)| as
functions of increasing L: as visible in these plots, w(L) goes to zero exponentially as expected, thus indicating
existence, while wa (L) grows exponentially in norm, thus indicating transversality.
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38