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Lecture Note Sketches Hermann Riecke - ESAM Home Page

Figure 16: Localized wave trains in convection of water-alcohol mixtures. Top view of annular convection cell. In this regime the localized waves are spatially more extended, resembling bound states of fronts. Two slowly drifting, stable localized wave trains are seen [3]. 4.1 Single Fronts Connecting Stable States Consider a simple nonlinear diffusion equation ∂tψ = ∂ 2 x ψ + f(ψ) ≡ ∂2 x ψ − ∂ψV (ψ; λ) where λ is a control parameter of the system. This equation can be written in variational form ∂tψ = − δV{ψ} δψ with V{ψ} = � 1 2 (∂xψ) 2 + V (ψ; λ)dx Assume V (λ; ψ) has two minima at ψ = ψ1,2, corresponding to stable, spatially homogeneous solutions. Look for ‘wave solution’, i.e. a steadily propagating front solution 6 which satisfies **Note**s: ψ = ψ(x − vt) ∂ 2 x ψ + v∂xψ = +∂ψV (ψ) ≡ −∂ψ ˆ V (ψ) with ˆ V (ψ; λ) = −V (ψ; λ). 6 for clarity could introduce ζ = x − vt. 88

a) • this equation can be read as describing the position ψ of a particle in the potential ˆV (ψ) and experiencing friction with coefficient v. • we are interested in solutions that start at ψ1 and end at ψ2 ψ(x) → ψ1 for x → −∞ ψ(x) → ψ2 for x → +∞ • since in terms of ˆ V (ψ) the‘positions’ ψ1,2 are actually maxima, the ‘friction’ v must be tuned exactly such that the particle, starting at one maximum, stops at the other maximum. The velocity is uniquely determined. • depending on the relative heights of the maxima the ‘friction’ may be negative. V v>0 ψ b) V v

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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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- Page 79 and 80: Notes: Thus: ˙x = 1 2 m ˙x2 + V (
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