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3.2 Soliton Solutions of the NLS For s = +1 the NLS has exact localized solution of the form Inserting into NLS yields ∂tψ − i 2 ∂2 xψ − si|ψ|2 � ψ = λ Thus we need s = +1 and **Note**: 1 ψ(x, t) = λ cosh ρx eiωt 1 i iω − cosh ρx 2 ρ2cosh2 ρx − 2 cosh 3 ρx − siλ2 1 cosh 3 � e ρx iωt = i = λ cosh 3 � cosh ρx 2 � ρx ω − 1 2 ρ2 � + ρ 2 − sλ 2 � e iωt ρ = λ ω = 1 2 λ2 1 ⇒ ψ(x, t) = λ cosh λ (x − x0) ei1 2 λ2 (t−t0) • the parameter λ is arbitrary: there is a one-parameter family of solutions with different amplitudes and associated different frequencies and width • the soliton solution is not an attractor: small perturbations do not relax and the solution does not come back to the unperturbed solution • in fact, (36) corresponds already to a three-parameter family due to translation symmetry in space and time, x0 and t0 are arbitrary. NLS also allows transformations into moving frames of reference (boosts): u = x − ct Consider ˜ψ(x, t) = ψ(t, x − ct)e iqx+iωt where ψ(t, x) is a solution. For ˜ ψ to be a solution, as well, q and ω have to satisfy certain conditions. Insert ˜ ψ into NLS ∂t ˜ ψ − i 2 ∂2 x ˜ ψ − si| ˜ ψ| ˜ � ψ = ∂tψ − c∂uψ + iωψ − i � 2 ∂uψ + 2iq∂uψ − q 2 2 ψ � − si|ψ| 2 � ψ e iqx+iωt = � � = ∂uψ (−c + q) + iψ ω + 1 2 q2 �� e iqx+iωt using that ψ(t, u) satisfies NLS. Require **Note**: c = q ω = − 1 2 q2 80 (36)

• the boost velocity c or the background wavenumber q is a free parameter generating a continuous family of solutions Thus: The focussing Nonlinear Schrödinger equation has a four-parameter family of solutions of solitons 1 ψ(x, t) = λ cosh (λ(x − qt − x0)) eiqx+i12 (λ2−q2 )t+iφ0 After a perturbation (change) in any of the four parameter q, λ, x0, φ the solution does not relax back to the unperturbed solution but gets shifted along the corresponding family of solutions. **Note**: • Surprising feature of solitons: during collisions solutions become quite complicated, but after the collisions the solitons emerge unperturbed except for a shift in position x0 and the phase φo. In particular, the other two parameters, λ and q, are unchanged, although there is also no ‘restoring force’ to push them back to the values before the collision. • general solution can be described in terms of a nonlinear superposition of many interacting solitons and periodic waves (captured by inverse scattering theory). 3.3 Perturbed Solitons Consider soliton-like solutions of the (focussing) NLS with small dissipative perturbations ∂tΨ − i 2 ∂2 xΨ − i|Ψ| 2 Ψ = ǫP(Ψ, ∂xΨ, . . .) For small perturbations expect slow evolution along the family of solutions: T = ǫt λ = λ(T) q = q(T) x0 = x0(T) φ0 = φ0(T) For simplicity: focus on perturbations for which soliton remains stationary: c = 0 = q Slow changes in the frequency ω = 1 2 (λ2 − q 2 ): need to deal with the phase Ansatz for the expansion where ω(T) = d dt φ Ψ = ψ(Θ, T) e iφ = (ψ0(Θ, T) + ǫψ1(Θ, T) + . . .)e iφ 1 ψ0(Θ) = λ(T) cosh(Θ − Θ0) eiφ0 , Θ = λ(T)x Θ0, φ0 = const. 81

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Lecture Note Sketches Perturbation

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4 Fronts and Their Interaction 87 4

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References [1] P. Coullet, C. Elphi

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1.1.1 The Mathieu Equation Consider

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• case δ1 = + 1 2 � � 1 ü2

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• we can assume from the start th

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1.1.2 Floquet Theory In the discuss

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• it is also convenient to introd

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Goal: determine α(δ, ǫ). For a r

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Forcing Strength ε 1 0.5 0 -0.5 n=

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10 5 −10 0 0 −5 t 50 100 150 20

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O(ǫ): with Note: y0(ψ, T) = R(T)

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i.e. Now the nonlinear problem: Fix

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• α < 0: Notes: - in-phase solut

- Page 29 and 30: Since L is singular we expect that
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- Page 47 and 48: as in the case a > 0. But: now the
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- Page 51 and 52: First consider maximum at the lower
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- Page 57 and 58: It looks like a case for Watson’s
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