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Lecture Note Sketches Hermann Riecke - ESAM Home Page

a) R 2 R 2 K 2 b) Ω Figure 9: Bifurcation diagrams arising from cuts at fixed Ω (a) and at fixed K (b) in the phase diagram shown in Fig.8. 2 Asymptotic Evaluation of Integrals 2 Motivation: Many functions and solutions of differential equations are given or can be written in terms of definite integrals that depend on additional parameters. E.g. modified Bessel function K0(x) = e −x � ∞ 0 � t 2 + 2t � −1/2 e −xt dt To get analytical insight into the behavior of such a complicated solution it is often useful to extract the behavior of that solution in a limiting case like large argument. Often one could analyse the differential equation leading to that special function in that limiting case. But then not all boundary conditions are exploited and one looses information (prefactors). If one has an integral representation of that function one can analyze that in the limit, which retains the full boundary information. E.g. y ′ = xy + 1 y(0) = 0 For large x one can get the approximate solution via y ′ = xy d 1 ln y = x ⇒ y = y0e 2 dx x2 To obtain the prefactor y0 one needs to use the initial condition. However, the approximate solution is only valid for x → ∞ and not for x = 0. 2 This chapter follows quite closely Bender & Orszag. 42

Here we can get an integral expression for the solution using an integrating factor Using the initial condition y(0) = 0 we get d � e dx −1 2x2 � y = e −1 2x2 y(x) = e 1 2 x2 � x 0 e −1 2t2 dt From this integral one can get the asymptotic behavior of y(x) for x → ∞ including the prefactor y0. 2.1 Elementary Examples Consider I(x) = For large x one can easily approximate � 2 0 I(x) → � 1 � 2 3 cos t + t x � 2 0 � 1 4 cos [0] dt = 2 Not alway one can interchange the limit with the integral! If f(x, t) is asymptotic to f0(t) for x → x0, f(x, t) ∼ f0(t) x → x0 uniformly in the integration interval [a, b] then � dt � b � b I(x) = f(x, t)dt ∼ f0(t)dt if the integral on the r.h.s. is finite and nonzero. a In fact, if f(x, t) is given in terms of a series that is asymptotic to f(x, t) uniformly in [a, b] f(x, t) ∼ ∞� fn(t) (x − x0) αn n=0 then the integration can be performed term by term **Note**s: � b (f(x, t) dt ∼ a a x → x0, α > 0 ∞� (x − x0) αn � b fn(t) dt x → x0 n=0 43 a

- 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: Thus, if ∆ > 0 both solutions R2
- 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
- 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
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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