Abstract
In this chapter we present an asymptotic theory that approximates any perturbed critical NLS equation by reduced equations that are independent of the transverse variable \(\mathbf x\).
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Notes
- 1.
Here we only consider peak-type blowup solutions, since ring-type blowup solutions are azimuthally unstable (Sect. 19.3).
- 2.
Continuations of singular solutions are discussed in Chap. 38.
- 3.
- 4.
Otherwise, the leading-order dynamics is affected by all three terms in (31.4), i.e., the solution “jumps” from stage 1 to stage 5.
- 5.
This is the case, e.g., with linear damping (see Sect. 33.1.5).
- 6.
This is the case, e.g., with the NLS with small normal dispersion after the pulse splitting (see Sect. 36.8.8 and in particular Fig. 36.4). In some cases, however, the leading order-dynamics during the fifth stage is only determined by two terms, one of which is the perturbation. For example, when \(F\,{=}\,|\psi |^{\frac{4}{d}+2} \psi \) and \(0<\epsilon \ll 1\), as \(\psi _\mathrm{coll}\) collapses, the supercritical nonlinearity eventually dominates over the critical nonlinearity, and so the leading-order dynamics is determined by the supercritical nonlinearity and diffraction (Sect. 32.3.2). Similarly, when \(F = -\Delta ^{2} \psi \) and \(0<\epsilon \ll 1\), as \(\psi _\mathrm{coll}\) collapses, fourth-order diffraction eventually dominates over diffraction, and so the leading-order dynamics is determined by fourth-order diffraction and nonlinearity.
- 7.
See (17.65).
- 8.
In the case of multiple perturbations, Condition 1 reads \( \epsilon _k F_k \ll \Delta \psi , |\psi |^{\frac{4}{d}} \psi \) for \(k=1, \ldots , K\).
- 9.
Unlike the derivation in the unperturbed case, we allow \(V^{\epsilon }\) to depend on \(\varvec{\xi }\) (rather than only on \(\rho \)). This will enable us to consider anisotropic perturbations such as vectorial effects [76], propagation in fiber arrays [93], and propagation of ultrashort pulses in planar waveguides with fourth-order dispersion [81].
- 10.
i.e. that \(V_0^{\epsilon }\) is quasi steady.
- 11.
This approach was applied in [88] to the time-dispersive NLS (31.19).
- 12.
This is not always the case, see, e.g., relation (34.17).
- 13.
i.e., those satisfying \(f_1' \ll f_2\), see Conclusion 31.5.
- 14.
See Sect. 31.1.2.
- 15.
i.e., neglecting the coupling between \(\psi _\mathrm{coll}\) and \(\psi _\mathrm{{outer}}\).
- 16.
By (31.12a) and (31.50), \( [f_1] = L^{2+\frac{d}{2}} [F]\) and \([f_1] = L^{-2}\), respectively. Therefore, \([F] = L^{-4-\frac{d}{2}}\). In addition, \(\left[ \, |\psi |^{\frac{4}{d}} \psi \right] \sim \left[ \, |\psi _{R^{(0)}}|^{\frac{4}{d}} \psi _{R^{(0)}}\right] = L^{-2-\frac{d}{2}}\), see (31.35a). Therefore, (31.61) follows.
- 17.
- 18.
Here we make the approximation \(\nu ^\epsilon (\beta ) \sim \nu (\beta )\), i.e., we assume that nonadiabatic effects are, to leading order, unaffected by the perturbation. See [164] for a detailed analysis of nonadiabatic effects.
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Fibich, G. (2015). Modulation Theory. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_31
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DOI: https://doi.org/10.1007/978-3-319-12748-4_31
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