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Lecture Note Sketches Hermann Riecke - ESAM Home Page

∗ fronts Ψ+ and Ψ− can interact and form a stable pair: wide localized wave train . A = 0 x L A = A c Figure 14: Localized wave trains in convection of water-alcohol mixtures. Space-time plot of the envelope of the left- (top panel) and the right-traveling wave component (bottom panel). In this parameter regime the localized convection waves are not stable but evolve chaotically (‘dispersive chaos’) [4, 5]. 86 x R

4 Fronts and Their Interaction Consider nonlinear PDEs with spatial translation symmetry that have multiple stable spatially homogeneous solutions ψ • ⇒ there must be also solutions that connect the stable states: fronts or kinks • these fronts are heteroclinic in space: they connect two different fixed points for x → ±∞. They are topologically stable: they cannot disappear except at infinity or by collision with ‘anti-fronts’. • This is to be compared to homoclinic solutions which connect to the same fixed point for x → ±∞, i.e. localized ‘humps’ (like the solitons). They are not topologically stable since they can disappear, e.g., due to a sufficiently large perturbation. stable stable unstable Figure 15: Fronts connecting two stable and one unstable spatially homogeneous state. Questions: • Do such fronts travel? What determines their speed? • How do the fronts interact? Can they from stable bound states: localized domains? 87 x

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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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- Page 39 and 40: with O(1) forcing (cf. in Sec.1.2.1
- Page 41 and 42: Thus, if ∆ > 0 both solutions R2
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- Page 45 and 46: which diverge at the upper limit fo
- Page 47 and 48: as in the case a > 0. But: now the
- Page 49 and 50: • if the integration by parts lea
- Page 51 and 52: First consider maximum at the lower
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- Page 55 and 56: • since we kept only the first no
- Page 57 and 58: It looks like a case for Watson’s
- Page 59 and 60: Extend integration to (−∞, +∞
- Page 61 and 62: Bound the integral term ���
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- Page 79 and 80: Notes: Thus: ˙x = 1 2 m ˙x2 + V (
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- Page 95 and 96: Analogously for x > xm: 0 = ǫ∂Tx
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