- Text
- Solution,
- Integral,
- Equation,
- Forcing,
- Amplitude,
- Term,
- Integration,
- Solutions,
- Consider,
- Maximum,
- Lecture,
- Note,
- Sketches,
- Hermann,
- Riecke,
- Home,
- People.esam.northwestern.edu

Lecture Note Sketches Hermann Riecke - ESAM Home Page

Then �� ∞ 〈ψ1, iLψ2〉 = ℜ �� = ℜ ⎛ � ⎜ = ℜ ⎜ ⎝ �� = ℜ −∞ ψ ∗ 1 ψ ∗ � 1iLψ2dΘ � = −ω0ψ2 + 1 2 λ2 0 ∂2 Θ ψ2 + � 2|ψ0| 2 ψ2 + ψ 2 0 ψ∗ 2 � � � dΘ −ω0ψ ∗ 1ψ2 + 1 2 λ20 ∂2 Θψ1 ψ2 + 2|ψ 2 0 |ψ∗ 1 ψ2 + ψ 2 0ψ∗ 1 ψ∗ 2 � �� � (ψ∗2 0 ψ1 ψ2) ∗ (iLψ1) ∗ � ψ2 dΘ = 〈iLψ1, ψ2〉 = ⎞ ⎟ dΘ⎟ ⎠ Thus, with this scalar product the left eigenvectors are identical to the right eigenvectors. **Note**: • of course Liψ0 = 0 as well, but iψ0 is a left-eigenvector of iL but not of L • (iL) 2 iΘψ0 = 0 and (iL) 2 (Θ∂Θψ0 + ψ0) = 0 as expected of generalized eigenvectors Focus here on a simple, purely dissipative perturbation (µ, α, γ ∈ R), P(Ψ) = µΨ + α|Ψ| 2 Ψ + γ|ψΨ| 4 Ψ (37) Then we need only the eigenvector Ψφ associated with the phase invariance φ → φ + ∆φ Ψφ0 = iψ0 Thus using iL, i.e. after multiplying O(ǫ)-equation by i, � 0 = 〈iψ0, iLψ1〉 = ℜ � = ψ ∗ � 0 −∂Tψ0 + µψ0 + αψ 3 0 + γψ5 � 0 dΘ Use ψ0 = λ 1 cosh Θeiφ0 and � 1 cosh 2 dΘ = 2 Θ to get **Note**s: −iψ ∗ 0 i � −∂Tψ0 − iω1ψ0 − iλ0λ1∂ 2 Θψ0 + e −iφ P � dΘ = � 1 cosh 4 4 dΘ = Θ 3 d 2 λ = µλ + dT 3 αλ3 + 8 15 γλ5 � 1 cosh 6 16 dΘ = Θ 15 • the dissipative perturbations P lead to a slow evolution of the amplitude of the perturbed soliton close to the soliton family of solutions: slow manifold 84 (38)

• with increasing amplitude the perturbed soliton becomes narrower • non-trivial fixed points – α < 0: supercritical pitch-fork bifurcation λ 2 = 3 µ + h.o.t. if µ > 0. 2 α Within the amplitude equation (38) the fixed point is stable. However: do not expect this localized soliton-like solution to be stable within the full NLS since Ψ = 0 is unstable for µ > 0: perturbations will grow far away from the soliton – α > 0: subcritical pitch-fork bifurcation λ 2 1,2 α 15 = −5 ± 8 γ 16γ � 4 9 α2 − 32 15 µγ two soliton-like solutions created in saddle-node bifurcation at α 2 = 24 5 µγ within (38) the one with larger amplitude is stable, the other unstable. Background state Ψ = 0 is linearly stable for µ < 0. • full solution consists of four coupled evolution equations for λ, q, x0, φ0: **Note**s: – would have to check that for the perturbation (37) the equations for q, x0, and φ0 have stable fixed points with q = 0, x0 = const. and φ0 = const. – a general perturbation can make soliton travel, q �= 0, d dT x0 �= 0. • experiments in convection of water-alcohol mixtures: onset of convection via a subcritical Hopf bifurcation • quintic complex Ginzburg-Landau equation – for strong dispersion, i.e. large α and β, the complex Ginzburg-Landau equation can be considered as a perturbed NLS: expect localized solutions in the form of perturbed solitons [7]. – for weak dispersion perturbation approach via interacting fronts [6]: subcritical bifurcation (cf. Sec.4). ∗ bistability between conductive state (Ψ = 0) and convective state (Ψ = Ψ0 �= 0) ∗ front solutions Ψ±(x, t) → 0 for x → ∓∞ and Ψ±(x, t) → Ψ0 for x → ±∞ 85

- 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 and 36: • a nonlinear saturating term nee
- Page 37 and 38: • 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 ���
- 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 Note: ∂φ0 � i 1 2 λ20 ψ0
- Page 87 and 88: 4 Fronts and Their Interaction Cons
- Page 89 and 90: a) • this equation can be read as
- Page 91 and 92: • the coefficient of ψ is chosen
- 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