nonlinear dynamics of one-dimensional fronts - ISC - Cnr

NONLINEAR DYNAMICS OF
ONE-DIMENSIONAL FRONTS
Paolo Politi
Institute for Complex Systems
National Research Council (CNR)
Florence (Italy)

• Examples of experimental fronts
Outline of the lecture
• Representative Partial Differential Equations and nonlinear behaviors
• Linear stability analysis and relevant time/length scales
• Multiple scales analysis (the anharmonic oscillator)
• The amplitude and the phase equations

Meandering during Cu-MBE growth
[T. Maroutian et al (CEA-Saclay). L = 640˚A]
Fire propagation in paper
[Zhang et al (Physica A, 1992). L = 8cm]
Electromigration step-bunching on Si(111)
[K. Sudoh (Osaka University)]
Domain wall dynamics in Co film
[Lemerle et al (PRL, 1998). L = 90µm]

Segregation of granular materials
[SW Morris (Toronto University)]
Fronts, but not only fronts
Rayleigh-Benard convection
[S. Ciliberto et al (INO-Florence)]

Unstable growth of a high-symmetry surface
Deposition, diffusion and nucleation,
step-edge barriers
∂th = F − ∂xJ
J = A(∂xh) + K∂xxxh
u(x, t) = ∂xh
∂tu(x, t) = −∂xx[(δu − u 3 ) + uxx] = ∂xx
F =
Cahn-Hilliard equation
dx
1
2 u2 x +
−δ u2
2
u4
+
4
δF
δu
dF
dt
= −
dx
∂x δF
δu
2
≤ 0

Meandering of a single step
Deposition, diffusion without nucleation,
step-edge barriers and desorption
∂u
∂t = −δ∂2 u
∂x2 − ∂4u + u∂u
∂x4 ∂x
Kuramoto-Sivashinsky equation
F does not exist

Linear stability analysis
Cahn − Hilliard ut = −∂xx[(δu − u 3 ) + uxx]
Kuramoto − Sivashinsky ut = −δuxx − uxxxx + uux
ω(q) = δq 2 − q 4
u(x, t) = Aq(t) exp(iqx)
ω
0.3
0
Aq(t) = Aq(0) exp(ωt)
q 2
-0.3
0 0.5 1
q
dAq
dt = (δq2 − q 4 )Aq
Modes with
arbitrarily large λ
are unstable

If translational invariance is broken . . .
Swift − Hohenberg ut = δu − (1 + ∂xx) 2 u − u 3
ω(q) = δ−(1−q 2 ) 2
ω
0.4
0.2
0
-0.2
-0.4
q 1
q c
0 0.5 1 1.5 2
q
q 2 Only a band
of finite q
is unstable

t → ω0t ω0 =
¨x + x = −ɛx 3
Naive perturbation theory
m¨x + kx = −k3x 3 Anharmonic (quartic) oscillator
k
m
ɛ = k3
k
V (x) = x2
2
+ ɛx4
4
x = x0 + ɛx1 + . . . x0 = A cos(t + ϕ)
¨x1 + x1 = −x 3 0 = −A3 cos 3 (t + ϕ) = −A 3 1 3 cos 3(t + ϕ) + 3 4
x1(t) = − A3
16
sin(t + ϕ)(6t + sin 2(t + ϕ))
⇑
secular
term
cos(t + ϕ)
⇑
resonant
forcing

¨x + x = −ɛx 3
x = x0 + ɛx1 + . . .
t0 = t, t1 = ɛt, . . .
∂ 2 x0
∂t 2 0
d
dt
Multiple scales method
= ∂
∂t0
+ ɛ ∂
∂t1
+ . . .
+ x0 = 0 x0 = A0(t1) cos(t0 + ϕ0(t1))
∂ 2 x1
∂t 2 0
+ x1 = −2 ∂2 x0
∂t0∂t1
= 2 dA0
dt1
− x 3 0
dϕ0
sin(t0 + ϕ0) + 2A0
dt1
d2 ∂2
dt2 =
∂t2 0
+ 2ɛ ∂2
∂t0∂t1
− 3
4 A3 0 cos(t0 + ϕ0)
− A3 0
4 cos(3(t0 + ϕ0))

Resonant forcing can be eliminated
dϕ0
2A0
dt1
x0(t0, t1) = A0 cos
dA0
dt1
= 0 ⇒ A0 ≡ constant
− 3
4 A3 0 = 0 ⇒ ϕ0(t1) = 3
8 A2 0 t1
x0(t) = A0 cos
t0 + 3
8 A20 t1
1 + 3
8 A20 ɛ
t
We can continue with x1(t) and higher order terms. . .
t0 = t, t1 = ɛt

ω(q) = δ − (q − 1) 2
≡ ɛ 2 − (q − 1) 2
ω
0.4
0.2
0
-0.2
-0.4
The amplitude equation
q 1
q c
q 2
0 0.5 1 1.5 2
q
q = 1 + k ω(k) = ɛ 2 − k 2 k = ɛQ, ω = ɛ 2 Ω Ω = 1 − Q 2
e Ωt e iqx = e Ωɛ2 t e i(1+ɛQ)x = e ΩT e iQX e ix
If T = ɛ 2 t and X = ɛx
∂A
∂T = A + ∂2A + nonlinear terms
∂X 2
u(x, t) = A(X, T )e ix + c.c.

The Ginzburg-Landau equation
∂A
∂T = A + ∂2 A
∂X 2 − |A|2 A A → Ae iα
Steady states: As(X) = A0e iQX 1 − A 2 0 − Q2 = 0
As(X) =
1 − Q 2 e iQX
What about the stability of these solutions? (Secondary instability)

∂A
∂T = A + ∂2 A
∂X 2 − |A|2 A As(X) = A0e iQX
A(X, T ) = A0(1 + r)e i(QX+φ)
∂r
∂T = −2A20 r + ∂2r ∂φ
− 2Q
∂X 2 ∂X
∂φ ∂r
= 2Q
∂T ∂X + ∂2φ ∂X 2
Adiabatic elimination procedure . . .

∂r
∂T = −2A20 r + g(T )
If g(T ) is slowly varying on the scale T ∗ = 1
2A 2 0
∂r
∂T = −2A2 0 r +
∂ 2 r
∂X
∂φ
∂T = D ∂2 φ
∂X 2
Q 2 < 1 3
Q 2 > 1 3
2 − 2Q ∂φ
∂X
,
∂r
∂T
= 0
1
= 0 ⇒ r ≈ −
A2 Q
0
∂φ
∂X
D = 1 − 2Q2
A 2 0
D > 0 Stability
= 1 − 3Q2
A 2 0
D < 0 Eckhaus Instability

Swift-Hohenberg
equation
Cahn-Hilliard type
equation
ω
ω
0.3
0
0.4
0.2
0
-0.2
-0.4
q 1
q c
q 2
0 0.5 1 1.5 2
q
-0.3
0 0.5 1
q
creation elimination
of rolls of rolls

∂φ
∂T = D ∂2φ ∂X 2
The phase diffusion coefficient
D =
1 − 3Q2
A 2 0
A 2 0
(1 − 3Q 2 ) = d
dQ (Q − Q3 ) = d
dQ (QA2 0 )
In many cases, D = [positive D0] ×
= 1 − Q2
d I(q)
• The phase is stable/unstable if I is an increasing/decreasing function
• The relation D ∼ λ2
t
gives the coarsening exponent
dq

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