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## Then �� ∞ 〈ψ1,

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 Notes: −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: Notes: – 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

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