- 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

i.e. Thus 4 (X − Y ) = 2X ⇔ X = 2Y ⇔ arg (x) = arctan 1 2 • for arg (x) < arctan 1 2 • for arg (x) > arctan 1 2 **Note**s: (and arg (x) < π) I(x) ∼ 1 � π 2 x e−2x I(x) ∼ i − 4 68x e−4x(1+i) • analogously, for arg (x) < − arctan 1 the endpoint at t = +1 is dominant and one obtains 2 I(x) = − i + 4 68x e−4x(1−i) • at the Stokes lines the dominant and subdominant terms interchange their roles • with varying arg x the contours of steepest descent can also suddenly switch and change or omit a saddle point −2 U −1 −2 0 0.0 −0.5 −1.0 V −1.5 −2.0 −1 U 2 1 0 0 V −1 −2 −2.0 −1.5 1 U −1.0 2 −0.5 −2 0.5 0.0 0.0 −0.5 −1.0 V −1.5 −2.0 U −1 −2 0 0.0 −0.5 1 −1.0 V U −1 1 0 V −1 −2 0 1 Figure 13: Dependence of contour of steepest descent on ϕ = arg x. ϕ = 0, 1, 2.01, 2.05, 2.5. While for ϕ < arctan 4 ≈ 1.32 the descent is towards more negative U it is towards more positive U for ϕ > arctan4. At ϕ = π − arctan 2 the two contours reconnect and the contour of steepest descent is asymptotic to that emerging from t = +1 and no connecting contour crossing the saddle point arises any more. This reconnection does not affect the leading-order behavior of the integral, though, since the saddle point contributes only a subdominant term for ϕ > arctan 1 2 . 74 −1.5 −2.0 2

3 Nonlinear Schrödinger Equation Consider oscillations in a nonlinear conservative system, i.e. a system without dissipation. Classic example: pendulum of length L without damping: ∂ 2 t ψ = −ω2 0 sin ψ with ω2 0 = g L More generally, the right-hand side could be any function f(ψ) (with f(0) = 0) Consider nonlinear oscillations in a continuum (many coupled pendula) **Note**: ∂ 2 t ψ − c2 ∂ 2 x ψ + ω2 0 sin ψ = 0 (31) • this nonlinear equation is called the sine-Gordon equation in analogy to the linear Klein-Gordon equation ∂ 2 t ψ − c 2 ∂ 2 xψ + ω 2 0ψ = 0 The Klein-Gordon equation allows simple traveling waves ψ = Ae iqx−iωt + A ∗ e −iqx+iωt with ω 2 = ω 2 0 + c2 q 2 Weakly nonlinear regime: expect traveling waves with slightly different frequency and slightly different wave form. Aim: weakly nonlinear theory for such waves that allows also spatially slow modulations of the waves, e.g. spatially varying wavenumbers or wave packets **Note**: • expect similarities between these traveling waves and the oscillations and waves arising from a Hopf bifurcation To derive a weakly nonlinear description one can use a multiple-scale analysis. We will not do that here (is standard topic in the class Methods of Nonlinear Analysis). But guess the resulting equation using symmetry arguments: Ansatz for right-traveling wave with T = ǫ 2 t, X = ǫx. ψ = ǫA(X, T, . . .)e iqx−iωt + ǫA ∗ (X, , T, . . .)e −iqx+iωt + ǫ 2 ψ2 + ǫ 3 ψ3 + . . . (32) For a wave packet one would have A → 0 for X → ±∞. The sine-Gordon equation (31) is equivariant under reflections in space: x → −x reflections in time: t → −t 75

- 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: Write xρ = (X + iY ) ρ ≡ Φ + i
- 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
- 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