It turns out that only the edge-splitting occurs, which seems a natural consequence from .... models, and hence seem to a natural framework (see [16] and [6]). Let us consider the ...... Since Mo > 0 (see S7 and Lemma 2.1), the interaction ..... Our claim is that 0(t, x) plus freezing term satisfies the equation with small error for 0 ...
Japan J. Indust. App!. Math., 18 (2001), 181-205
2n-Splitting or Edge Splitting? -
— A Manner of Splitting in Dissipative Systems — To the memory of Masaya Yamaguti Shin-ichiro Eit, Yasumasa
NISHIURA$*
and Kei-ichi VEDA
t Graduate School of Integrated Science, Yokohama City University, Yokohama 236-0027, Japan #Laboratory of Nonlinear Studies and Computations, Research Institute for Electronic Science, Hokkaido University, Kita -ku, Sapporo 060-0812, Japan Received August 30, 2000 Revised October 13, 2000 Since early 90's, much attention has been paid to dynamic dissipative patterns in laboratories, especially, self-replicating pattern (SRP) is one of the most exotic phenomena. Employing model system such as the Gray-Scott model, it is confirmed also by numerics that SRP can be obtained via destabilization of standing or traveling spots. SRP is a typical example of transient dynamics, and hence it is not a priori clear that what kind of mathematical framework is appropriate to describe the dynamics. A framework in this direction is proposed by Nishiura-Ueyama [16], i.e., hierarchy structure of saddle-node points, which gives a basis for rigorous analysis. One of the interesting observation is that when there occurs self-replication, then only spots (or pulses) located at the boundary (or edge) are able to split. Internal ones do not duplicate at all. For 1D-case, this means that the number of newly born pulses increases like 2k after k-th splitting, not 2''1 -splitting where all pulses split simultaneously. The main objective in this article is two-fold: One is to construct a local invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other is to study the manner of splitting by analysing the resulting ODE, and answer the question "2'-splitting or edge-splitting?" starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natural consequence from a physical point of view, because the pulses at edge are easier to access fresh chemical resources than internal ones. Key words: self-replicating pattern, reaction diffusion system, pulse solution, Turing pattern, wave splitting, saddle-node bifurcation
One of my favorites is transient for us, but eternal in nature. Masaya Yamaguti 1. Introduction Since the early 90's, a variety of chemical patterns have been observed in chemical laboratories ([1], [12], [13]), and among them, self-replicating patterns (SRP) is one of the spectacular examples. Self-replicating patterns have been also observed numerically in reaction diffusion systems, typically for the Gray-Scott model ([19], [22], [14], [21], [20]). Theoretical investigations about the existence and stability of spots and pulses have been done by many authors, for instance, * This research was supported in part by Grant-in-Aid for Scientific Research 09440071 and 09874039, Ministry of Education, Science and Culture, Japan.
182
S. EI, Y. NISHIURA
and K. UEDA
[23], [3], [4], [15] and [25]. More recently mildly strong interaction between two pulses, namely its distance is not very far away, was analysed by [5]. A crucial observation by [16] is that a hierarchy structure of saddle-node points is responsible for the occurrence of SRP, which, to the authors' best knowledge, the first clear geometrical characterization for SRP. Note that this approach is also powerful to predict the onset and termination of spatio-temporal chaos (see [17]). The main issue here is about the manner of splitting in 1D space. It is clearly visible in Fig. 1.1 that only pulses located at both ends are able to split, and hence the number of newly born pulses increases like 2k after k -th splitting, while internal ones remain the same as before. Intuitively it may happen that all the pulses start to split simultaneously, i.e., 2'-splitting. Hence a natural question is "2'-splitting or edge-splitting?". In order to answer this, what we shall discuss in this article is two-fold: One is to construct a locally invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other is to study the manner of splitting by analysing the resulting ODE. First we find the existence of critical distance 2, for splitting, that is, a pulse can split only when the distance to neighbouring pulses exceeds .£,. Second there appears an ordering among distances of pulses when time becomes large, which is a key to answer " 2?-
Fig. 1.1. Self-replicating patterns for the Gray-Scott model for D u = 2 x 10 -5 D v = 1 x 10 -5 , F = 0.04, L = 3.0, (a) k = 0.0606, (b) k = 0.06075. The second case (b) is closer to the location of the saddle-node point, so it takes more time to split than (a). In both cases the pulses located at edge are able to split while the internal ones do not. ,
2'1 -Splitting or Edge-Splitting?
183
splitting or edge-splitting?" Combining these observations, we are able to prove that edge-splitting really occurs under natural assumptions. This seems a natural consequence from a physical point of view, because the edge-pulses are easy to access fresh chemical resources than internal ones. The discussion in Section 4 is focused on the principal part of the flow on the invariant manifold. The proof that takes care of the remainder terms is more involved and lengthy, so it will be reported elsewhere. For definiteness we employ the following Gray-Scott model ([9]), however all the arguments are valid under more general setting mentioned in the next section.
au at
= D„V 2 u—uv 2 +F(1—u) (1.1)
b
y = D„V 2 v + uv 2 — (F + k)v, át where u and v are concentrations of the chemical materials U and V, respectively, D and D v the diffusion coefficients, F the in-flow rate of U from outside, and F + k is the removal rate of v from the reaction field. Note that (u, v) = (1, 0) is a stable homogeneous state independent of (k, F) even in the PDE-sense. du
^t = —uv 2 + F(1 — u) dv
2
(1.2)
—=uv = uv — (F + k)v. The main feature of the associated ODE (1.2) is that it has a Bogdanov-Takens (BT)-point at (k, F) = (1/16,1/16). BT-point is a singularity of codim 2 where saddle-node and Hopf bifurcations merge in 2-dimensional parameter space (k, F), and most of the interesting PDE-dynamics also appear near these two bifurcation lines (see [19], [14], [16], [17]).
2. Pulse Interaction (Weak Interaction) by PDE Approach In this section, we describe the dynamics of widely-spaced 1-pulses near saddlenode point by PDE approach in a general setting including the Gray-Scott model. All the assumptions listed below can be checked • at least numerically for several models, and hence seem to a natural framework (see [16] and [6]). Let us consider the equation of the form u t = £(u; k), t > 0, x E R',
(2.1)
where u E R' 2 , G(u; k) = Du,,,, + F(u; k) and k is a bifurcation parameter such as k in (1.1). Suppose (2.1) has a bifurcation structure with respect to k as follows: Si) 0 = (0, ... , 0) E RT' is always a stable equilibrium of (2.1). That is, G(0; k) 0 and all the spectrum of the linearized matrix F'(0; k) lie in the left hand side of the imaginary axis for any k. For (1.1), (1,0) should be shifted to (0,0).
S. EI, Y. NISHIURA and K. UEDA
184
S2)
There exists k = k, such that for k < k, 0 is globally stable in (2.1), and for k > k,, there exist two branches of stationary solutions {Ps (x; k)} and {P' (x; k)} of (2.1) except 0. Both are symmetric and P 8 (x; k) is stable, Pu(x; k) is unstable (see Fig. 2.1).
PS
y^'
'
á
Cik kc
Fig. 2.1. P' and P' merge at the saddle-node point at k = k.. Splitting occurs slightly left to k = k. (small negative e).
S3)
Let X = {L 2 (R I )} with the norm • and let the linearized operators of (2.1) in X with respect to stationary solutions P'(x; k) and Pu(x; k) respectively be L'(k) = L'(P'(x; k)) and L'(k) = £'(P'(x; k)'). For k > k, close to k,, L 3 (k) has two critical simple eigenvalues 0 and .) 9 (k) < 0. Other spectra of L 8 (k) are in the left hand side uniformly apart from imaginary axis, say El(L 8 (k)) C {z E C; Re(z) < —po} for a p 0 > 0. The associated eigenfunction corresponding to 0 is Pte, which comes from translation invariance. Let ^'(x; k) be an associated eigenfunction corresponding to as(k) (see Fig. 2.2). We suppose similarly for the operator L'(k). That is, the spectrum of Lu(k) consists of {0}, {A (k)} which is positive, and other spectra L1(Lu(k)) C {z E C; Re(z) 0 such that -p
P(x)
e - ` I ' la, .(x) -^ e 'Hb, 0*(x) -> ±e "l es la *, Z * (x) e -011 xI b * -
-
as x -- ±oo.
REMARK 2.1. The bifurcation structure Si) - S4) and the symmetry of eigenfunctions S5) are numerically checked for (1.1). The profile of t; is shown in Fig. 2.2(b), which accelerates the splitting of solutions. 2.1. Weak interaction of pulses Let us consider the interaction of N + 1 pulses near the saddle-node point. Let N
P(x;
h)
=
P(x-x), j=0
N
^(x;h,r) = j=0
N
S(x; h, r) =
{P(x - xj) + rj e(x - x j )} = P(x; h) + ^(x, h, r), j=0
where xo = 0 and xj = x(h) = >z =1 h i (j > 1) for h = (hl, h 2i ... , hN) ER N and r = (ro, r 1 ,. . . , rN) ERN+1. We define the translation operator S by S (l)u = u(x - 1) and define a set .M(h*, r*) = {5(l)S(•; h, r); 1 E R', min h > h*, Ir j < r*}, a quantity ö(h) = sup I G(P(x; h))1, xER1
2' -Splitting or Edge-Splitting?
187
and functions H(h) = (£(P(x + x^; h)), 0* )L2, î(h) = (I (P(x + xj; h)), e* ) L2 for j = 0,1, . .. , N, where min h = min{h l , . .. , hN }. Note that 8(h) = O( e-a by S6). Let 4 1 = A,(h, r, E) = 6(h) + Iri 2 + Iej. Then, we have
min
h)
THEOREM 2.1. There exist positive constants h*, r*, e*, Co and a neighborhood U of M(h*, r*) such that if the initial data u(0) E U, then there exist functions l(t) E R 1 , h(t) E R N and r(t) E R N+1 such that II u(t) — S (l(t))S(h(t), r(t)) II „ < Codl (h(t), r(t), E)
(2.3)
holds as long as minh(t) > h*, I r i (t) I < r* and I EI < E*, where u(t) is a solution of (2.2). h and r are estimated by (2.4)
1, h, r = O(A 1 ). Here, we give the explicit forms of the equations for h and r. Let
Mo = 2a (Da, a*) , Mo = —2a (Da, b*) , M1 = 2(F/F(P)
*)r2, Ma= — (g(P)
* )L2.
Suppose S7) All constants Mo , Mo , Ml and M2 are positive. REMARK 2.2. The sign of constant Mo determines whether interaction between pulses is attractive or repulsive. For the Gray-Scott model, it is known that they are positive (see [6]), which implies repulsion. Fix ß such that 3 a h* and
ri(t)I < r* :
i = —Ho(h) + O(Ao),
(2.5)
h^ = HJ -1 (h) — HH (h) + O( A _1 + A^ ),
(2.6)
r^ = M1 r — eM2 — H^ (h) + 0(A^)
(2.7)
S. EI, Y. NISHIURA and K. UEDA
188
hold. Hj and Hj are represented by Hj (h) =
Mo
(e —aha+i — e
—
aha)
(1 + O(e-7 min h)) (.Í = 1,... ,N —1),
Hj ( h) = Mo (e + ' + e'3) (1 + O( e-7 min h)) (j = 1, ... , N — 1), Ho h = Moe —aha (1 + O( e —ryminh )) (
)
HN(h) = — Moe —ah N (1+O(e-7 min h)) Ho (h) = Mo e
1
(1 + O( e-7 min h)\J,
HN(h) = Moe — a h N (1+O( e-7min h)\ for a positive constant y THEOREM 2.3. There exist CI > 0, C2 > 0 such that rj (t) is monotone increasing satisfying
rj > C2(rj + IEI +gj(h))
as long as I rj I > Cl./Sj + I EI while rj satisfies rj = (Mi +O(4 ))r — EM2 — H^(h) + (bj + IEl)O( e—minh + Y Imo! + IrI)
(2.8)
as long as I r j 1 < Cl Sj + I ei 3. Proof of Theorems 2.1, 2.2 and 2.3 In what follows, C denotes a generic positive constant independent of h for sufficiently large min h. We take h* in Theorem 2.1 to be sufficiently large so as to match the following lemmas. 3.1. Proof of Theorem 2.1 We shall give the outline of the proof, since it can be done in a parallel way to that of Theorem 2.1 in [6]. Let X = {L 2 (R I )}n and L(h) = L'(P(x; h)), and let L*(h) be the adjoint operator of L(h). Then the following two propositions can be obtained in a similar manner to [6]. PROPOSITION 3.1. There exist positive constants C and h* such that for h with min h > h*, the operator L(h) has 2(N + 1) semi-simple eigenvalues {Aj (h)}j = o,...,2N +I with jAj(h)I < C6(h). Multiple eigenvalues are repeated as many times as their multiplicities indicate. Other spectra of L(h) are in the lefthand side of z = —p o for a positive constant po . Let E(h) be the eigenspace corresponding to eigenvalues {Aj(h)}j=o,...,2N+1. The adjoint operator L*(h) has also 2(N + 1) semi-simple eigenvalues
2' -Splitting or Edge-Splitting?
189
{.^(h)}^=o,... ' 2N+1 with ^A^(h)l h^. _ Fix p l > 0 and define H(h, p i ) = {h = (h l , ... , hv) E R N ; h < h < h + pl }, M = M(h, p l ) = {S (l)S(•, h, r); l E R l , h E H(h, p i ), Iri h* for sufficiently large h* such that for any h E H(h, p i ) the map 17(h) satisfies IIH(h)II, J1H-1(h)Ii, áh H(h) < Cl , J H(h)i,
l
IIn (h)II^, á -1
H(h)
II
0 and C2 > 0 be constants such that if 11w Mw h* and I ri h*} min h>h*
such that if (h(0), r(0), w(0)) E U, then the solution of (3.11), (3.12) and (3.14) is attracted exponentially and remains in d l neighborhood of the set as long as minh > h* and Iri < r*, that is, IIw(t)II < C4 1 (h(t), r(t), e). Since the solution u of (2.2) is given by u = S (l){S(h, r) + H(h)w} by using the solution (h, r, w) of (3.11), (3.12) and (3.14), this reads
lIu(t) - S (l){S(h(t), r(t))}II < lI17(h)wli < Cz i (h(t), r(t), e) and the proof is complete.
n
3.2. Proof of Theorem 2.2 We will give the estimates of o, = o,(h, r, e) (x) constructed in the previous subsection. as IxI - oo, Let X(x) > 0 be a symmetric function satisfying X(x) -> where Q is a constant fixed as 3a < ß < a. Fixing arbitrary 0 < jt < N, we put xt = xjt, lt = 1 + xt, Pt(.; h) = "( - x t )P( . ; h), fi t ('; h) = ,F ( - x t )^ t ( - ; h), fit(•; h, r) = S (-xt)^t(•; h, r) and St(•; h, r) = S (-xt)$(•; h, r). Let vt = E (-xt)17(h)o,. Then, we have LEMMA 3.1.
v t (x)I
Cmin{e' (t +IEI
+Ir^t
12
+dilr^tl)Vi(x), 4 iV2(x)}
holds for some functions V1 , V2 E H'(R') Proof. Since vt satisfies vt = £(St + vt) + Eg(St + v t ) + lt
áx {St + vt} a(h r) (h, T),
(3.15)
we have by the transformation v = Xvt vt = Xr-(S t + X -I V) + eXg(S t + X
aast -IV) + l } X ax { S t + X -I V} X a(h r) ' r) (3.16)
2'-Splitting or Edge-Splitting?
193
We shall estimate each term of (3.16). First, we see
x 1 (S t + X -l v) = x£(S t ) + x£'(S t )x -l v + xo(Ix -l vl 2 ) { x (Pt) + xL'(P t )^ t + X0(I^ t 1 2 )}
+ {x1'(P t )x -l v +_xo(I^ t l . Ix -l vl)} + O(Iv_t l - Ivl) = x£'(P t )x -l v + O(b^t + Irlbj t + I ri t 1 2 + Irl 2 b^t + In • II + dill) = XC(P t )X 'v + O(b^t + I
(3.17)
rit 12 + dolel),
where d o = d o (h, r, e) = ö+ Irl +lel. Similarly we have
exg(S t + x - lv) = exg(P) + O(IEI(Int I + b,t
+Iel)),(3.18)
Xax{St +X`ii} = xPP+r,tX +O(b t +
(3.19)
;
(3.20)
X a-(h T) (h,r) = rjtX+O(Ihlbit + Ht).
Let Z = Z(h) = XG'(Pt)X -1 . Since ß < a, Z has similar properties to L with respect to the spectrum. Namely, similar results to Propos it ions 3.1 and 3.2 hold by taking appropriate constant po and functions {^^ (h)(•)}, {e (h)(•)} and {4(h)(.)},
{^^ (h)(.)} (j = 0, ... , N) satisfying properties (3.1) '- Also define projections Q(h) and R(h) similarly. Since R(h)X- {St + X -1 v} = O(8 t + I rit l + Ilvll,,), R(h)Xa^h,l(h,r) = O ((Ir^tI + lhl+Irl)bjt) = O(d l b,r) and lt = O(Al) hold, (3.16) becomes by using (3.18) - (3.20)
vt = Lv + G
(3.21)
with G = G(h,r,v) = O(bj t + II + ICI • Ir^tI + A,Ir^tl + Ire +I 2 + dollvll") O(b^t+l6I+dl lni t I +Irr t 1 2 +_dollvII ). Then, similarly as in the previous subsection, we can prove Ilvll,,
202
S. Ei, Y. NISHIURA and K. UEDA
Mlr oz - rM2 - Moe -af`
(4.28)
C
= Mlro - (1 - 1 MZE
(4.29)
Let to be the time when e -11h1 (and/or e - a h N) reaches e - a e °. If ro (and/or rN) is already positive, the proof is done. If not, let tl be the time when, say ro = i> 0 holds, and p = (1 - 1/c)M2 > 0, then it follows from (4.29) that ti - to
