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## i.e. Thus 4 (X − Y ) =

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 Notes: (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

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