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Lecture Note Sketches Hermann Riecke - ESAM Home Page

F ∗ : m − n = −1 ⇒ k − l = 4 k = 4 l = 0 ⇒ F ∗ A 4 Balance saturation and forcing A 3 ∼ FA 2 ⇒ F ∼ A With this scaling terms of order O(F 2 A) would be of the same order, i.e. m + n = 2, m = 2 n = 0 ⇒ k − l = −5 ⇒ F 2 A ∗5 m = 1 n = 1 ⇒ k − l = 1 ⇒ |F | 2 A m = 0 n = 2 ⇒ k − l = 7 ⇒ F ∗2 A 7 To leading order symmetry and scaling results in an equation of the form **Note**s: A ′ = � µ + β|F | 2� A − γ|A| 2 A + δFA ∗2 • Through the term β|F | 2 the forcing modifies the linear coefficient of the equation – depending on the sign of βr the forcing can enhance or reduce the damping – through βi|F | 2 the frequency of small-amplitude oscillations are modified by the forcing • The forcing will lead to qualitatively new phenomena only through the term involving A ∗2 because only it breaks the symmetry A → Ae iϕ for ϕ �= 2π 3 . • For consistent scaling we need again µ = O(A 2 ) 1.3.4 Non-resonant Forcing Consider ω = αω0 with α not an integer. Selection Rule lowest-order term k − l + α (m − n) = 1 ⇒ m = n k = l + 1 |F | 2 A ⇒ the forcing appears only through |F | 2 , i.e. the phase of the forcing does not play a role and there is no resonance between the oscillator and the forcing Balance saturation and forcing resulting in the amplitude equation **Note**s: A 3 ∼ F 2 A ⇒ F ∼ A A ′ = � µ + |F | 2� A − γ|A| 2 A 36 (16)

• Non-resonant forcing does not introduce new terms in the equation of the unforced oscillator (at any order), it only modifies its coefficients. None of the terms are phasesensitive. – in principle, all coefficients depend on |F | 2 – the strongest effect of the forcing is on the bifurcation parameter µ because it is small – only for the linear term is the shift of the coefficient of the same order as the coefficient itself and therefore relevant at leading order • In the resonant cases - as those discussed before - all the coefficients depend also on |F | 2 , but again most effects are of higher order. More relevant are, however, the new terms that are phase-sensitive. • Resonant forcing with higher resonances (m : 1 with m ≥ 4) do not lead to additional terms in the lowest order amplitude equation (see homework) – but they introduce new higher-order terms that are phase-sensitive – to capture aspects of their impact on the system in a leading-order amplitude equation one may have to consider singular limits like |γ| ≪ 1, i.e. consider higher singular points. 1.4 Forced Duffing Oscillator II Consider now explicitly the Duffing oscillator with forcing near 3ω0 with initial conditions y(0) = δ, ˙y(0) = 0. ¨y + ǫβ ˙y + ω 2 0y + ǫαy 3 = f cosωt To get an overview of possible resonance consider first general ω and O(1)-forcing O(1): O(ǫ): Insert y0 into f1 using y0(t) = δ cosω0t + ¨y0 + ω 2 0 y0 = f cosωt f ω2 (cosωt − cosω0t) 0 − ω2 ¨y1 + ω 2 0y1 = −β ˙y0 − y 3 0 ≡ f1 K ≡ f ω 2 0 − ω 2 f1 = β {ω0 sin ω0t + K [ω sin ωt − ω0 sin ω0t]} − {K cosωt + (1 − K) cosω0t} 3 = βω0 (1 − K)sin ω0t + βω sin ωt − − � K 3 cos 3 ωt + 3K 2 (1 − K) cos 2 ωt cosω0t + 3K (1 − K) 2 cos ωt cos 2 ω0t + (1 − K) 3 cos 3 ω0t � 37

- Page 1 and 2: Lecture Note Sketches Perturbation
- Page 3 and 4: 4 Fronts and Their Interaction 87 4
- Page 5 and 6: References [1] P. Coullet, C. Elphi
- Page 7 and 8: 1.1.1 The Mathieu Equation Consider
- Page 9 and 10: • case δ1 = + 1 2 � � 1 ü2
- Page 11 and 12: • we can assume from the start th
- Page 13 and 14: 1.1.2 Floquet Theory In the discuss
- Page 15 and 16: • it is also convenient to introd
- Page 17 and 18: Goal: determine α(δ, ǫ). For a r
- Page 19 and 20: Forcing Strength ε 1 0.5 0 -0.5 n=
- Page 21 and 22: 10 5 −10 0 0 −5 t 50 100 150 20
- Page 23 and 24: O(ǫ): with Note: y0(ψ, T) = R(T)
- Page 25 and 26: i.e. Now the nonlinear problem: Fix
- Page 27 and 28: • α < 0: Notes: - in-phase solut
- Page 29 and 30: Since L is singular we expect that
- Page 31 and 32: • using the other left 0-eigenvec
- Page 33 and 34: Thus, m = n + 1 or αmn = 0 Alterna
- Page 35: • a nonlinear saturating term nee
- 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 ���
- Page 63 and 64: π i to ensure decay of the exponen
- 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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4 Fronts and Their Interaction Cons

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a) • this equation can be read as

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• the coefficient of ψ is chosen

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Using that ψL,R satisfy the O(ǫ 0

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Analogously for x > xm: 0 = ǫ∂Tx

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- L = 0 corresponds to a pure gas p