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Lecture Note Sketches Hermann Riecke - ESAM Home Page

Since ψ0(x) breaks the continuous translation symmetry L is singular and the eigenvector associated with the 0 eigenvalue is the translation mode ∂xψ0. **Note**: • if ψ0 did not break the translation symmetry, ∂xψ0 would vanish and not represent an eigenvector and there would be no 0 eigenvalue associated with the translation symmetry. L is self-adjoint ⇒ ∂xψ0 is also its left 0-eigenvector. Project (39) on ∂xψ0 Thus **Note**s: v1 � ∞ −∞ 0 = v1 � +∞ −∞ ∂xψ0 � −v1∂xψ0 − λ1∂λ∂ψ ˆ � � � V (λ; ψ) � dx λ=0,ψ=ψ0 � ∞ (∂xψ0) 2 dx = −λ1∂λ −∞ � +∞ −∞ ∂ψV (0, ψ0) ∂xψ0 dx � �� � ∂xV (0;ψ0(x)) (∂xψ0) 2 dx = −λ1 ∂λ [V (λ; ψ0(x))]| λ=0 | +∞ −∞ ≈ − V (λ1; ψ0)| +∞ −∞ • the lhs of the equation can be read as the amount of work performed by the friction � +∞ � β −∞ dx � dx dt dt dt � �� � dx • the rhs of the equation can be read as the difference in potential energy between initial and final state • the perturbation method does not rely on the existence of a potential ⇒ works also when there are multiple coupled components ψj(x, t) satisfying nonlinear PDEs that cannot be derived from a potential. 4.2 Interaction between Fronts Consider fronts of the nonlinear diffusion equation **Note**s: ∂tψ = ∂ 2 x ψ − ψ + cψ3 − ψ 5 • the coefficients of ∂ 2 x ψ, ψ, and of ψ5 can be chosen to have magnitude 1 by rescaling of space, time and ψ. 90

• the coefficient of ψ is chosen negative: ψ = 0 is linearly stable • the coefficient of ψ 5 is chosen negative: saturation at large values of ψ Homogeneous stationary states: linearly stable linearly unstable V Figure 18: Potential with minima at ψ0 and ±ψ0. ψ = 0 or ψ 2 0 = c + √ c 2 − 4 2 ψ 2 u = c − √ c 2 − 4 2 Consider fronts that connect ψ = 0 with ψ = ψ0 ψ ψ R L L x x ψ L m Figure 19: Front positions. Goal: evolution equations for xL and xR, which describe the interaction between the two fronts. We need separation of time scales • individual fronts should move slowly • interaction between the fronts weak: for xR − xL large the fronts deform each other only weakly 91 x R ψ ψ R

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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

- Page 39 and 40: with O(1) forcing (cf. in Sec.1.2.1
- Page 41 and 42: Thus, if ∆ > 0 both solutions R2
- Page 43 and 44: Here we can get an integral express
- 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
- Page 53 and 54: Is this series for I(x; ǫ) asympto
- 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 65 and 66: using � ∞ 0 e−u ln u du = γ
- Page 67 and 68: using the series expansion from the
- Page 69 and 70: Example: Behavior near the saddle p
- Page 71 and 72: Limiting behavior of the contours v
- Page 73 and 74: Write xρ = (X + iY ) ρ ≡ Φ + i
- Page 75 and 76: 3 Nonlinear Schrödinger Equation C
- Page 77 and 78: 3.1 Some Properties of the NLS Cons
- Page 79 and 80: Notes: Thus: ˙x = 1 2 m ˙x2 + V (
- Page 81 and 82: • the boost velocity c or the bac
- Page 83 and 84: and Note: ∂φ0 � i 1 2 λ20 ψ0
- Page 85 and 86: • with increasing amplitude the p
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- Page 93 and 94: Using that ψL,R satisfy the O(ǫ 0
- Page 95 and 96: Analogously for x > xm: 0 = ǫ∂Tx
- Page 97 and 98: - L = 0 corresponds to a pure gas p