Abstract
In this chapter we analyze the explicit blowup solutions \(\psi _G^\mathrm{explicit}\).
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Notes
- 1.
- 2.
This is different from \(\psi _{R^{(n)}}^\mathrm{explicit}\), which is unstable under radial perturbations for any \(n\), and not just for the excited states.
- 3.
In Sect. 17.3 the profile of the collapsing core is denoted by \(V_0\), in order to emphasize that it is peak-type and not ring-type.
- 4.
In Lemma we show that all self-similar radial solutions of the critical NLS are of the form (11.5).
- 5.
Denoted therein by \(V_0\).
- 6.
In Corollary 11.4 we will see that if \(0<G_0 \ll 1\) then \(r_{\max } \gg 1\). The condition \(r_{\max }\le \gamma ^{-\frac{1}{2}}\) ensures that \(r _{\max }\) is in the ring region and not in the tail region.
- 7.
i.e., when \(\alpha =\alpha ^{(1)}(G_0)\), see Sect. 11.2.5.
- 8.
i.e., calculating \(c_G\) as a function of \(\alpha \), and finding the minimum points \(\alpha ^{(n)}\).
- 9.
See e.g., the top curve in Fig. 11.6.
- 10.
The conclusion that \(G = O(1)\) in the ring region also follows from (11.31).
- 11.
In the absence of noise, the radius of the (rescaled) \(G\) profile is \(\rho _{\max }\approx 19\) (Fig. 11.11b). Once noise is added, however, the solution approaches a different \(G\) profile with \(\rho _{\max } \approx 20\) (Fig. 11.13c). This is because the noise used in this simulation increases the input power. Therefore, the solution collapses with a self-similar ring profile that has more power, hence a larger radius (Sect. 11.2.5).
- 12.
The fact that the perturbed solution in Fig. 11.13 collapses with a different single-ring profile (Footnote 11), does not imply that \(\psi _G^\mathrm{explicit}\) is unstable, because perturbations of a stable singular solution are allowed to lead to a continuous change of the blowup profile (Sect. 10.5).
- 13.
See Sect. 9.7 for the advantage of using a deterministic azimuthal perturbation.
- 14.
Here we use the assumption that \(\delta _k\) is real.
- 15.
This is typical for a modulational instability (Sect. 3.6.3).
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Fibich, G. (2015). The Explicit Critical Singular Ring-Type Solution \(\psi _G^\mathrm{explicit}\) . In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_11
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DOI: https://doi.org/10.1007/978-3-319-12748-4_11
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