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Lecture Note Sketches Hermann Riecke - ESAM Home Page

Relevant integrals: Thus **Note**s: • Interaction � e L 2 0 � L e 2 0 � L e 2 0 � L e 2 0 dL/dt s (1 + s2 1 ) 3ds = 4 � 1 1 − (1 + eL ) 2 � s2 1 √ 3ds = 1 + s2 2 L + ln 2 − 1 + O(e−L ) s 4 √ 1 + s 2 s 6 √ 1 + s 2 1 4 3ds = L + ln 2 − 2 3 + O(e−L ) 1 23 3ds = L + ln 2 − 2 15 + O(e−L ) ∂TL = −16 e−2L ǫ + 2√ 3c1 (42) Figure 20: Dependence of growth rate of domain on domain size L. – attactive ⇒ fixed point L = L0 for c > c0, i.e. without interaction the fronts would be drifting apart – decays with distance – ⇒ localized state is unstable: for L > L0 the attraction is insufficient and the fronts drift apart • This localized state corresponds to a critical droplet in a first-order phase transition – ψ = 0 corresponds to the gas phase, say, and ψ = ψ0 to the liquid phase 96 L

– L = 0 corresponds to a pure gas phase, L → ∞ to a pure liquid phase. – the localized state separates these two stable phases ⇒ if there is only one such localized state it has to be unstable. • the interaction between the fronts is exponential and monotonic • in a more general system the interaction could be non-monotonic, e.g., dL dt = a cosκL e−αL + bc1 then there are multiple localized states, alternating stable and unstable dL/dt Figure 21: Oscillatory interaction between fronts would allow multiple localized states, stable and unstable. • for oscillatory interaction fronts can ‘lock’ into each other at multiple positions, arrays of fronts can be spatially chaotic [1]. 97 L

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Lecture Note Sketches Perturbation

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4 Fronts and Their Interaction 87 4

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References [1] P. Coullet, C. Elphi

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1.1.1 The Mathieu Equation Consider

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• case δ1 = + 1 2 � � 1 ü2

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• we can assume from the start th

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1.1.2 Floquet Theory In the discuss

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• it is also convenient to introd

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Goal: determine α(δ, ǫ). For a r

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Forcing Strength ε 1 0.5 0 -0.5 n=

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10 5 −10 0 0 −5 t 50 100 150 20

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O(ǫ): with Note: y0(ψ, T) = R(T)

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i.e. Now the nonlinear problem: Fix

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• α < 0: Notes: - in-phase solut

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Since L is singular we expect that

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• using the other left 0-eigenvec

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Thus, m = n + 1 or αmn = 0 Alterna

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• a nonlinear saturating term nee

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• Non-resonant forcing does not i

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with O(1) forcing (cf. in Sec.1.2.1

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Thus, if ∆ > 0 both solutions R2

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Here we can get an integral express

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