Abstract
In this chapter we study solutions that undergo a quasi self-similar collapse with the \(\psi _{R^{(0)}}\) profile.
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Notes
- 1.
- 2.
The explicit solution \(\psi _{R^{(0)}}^\mathrm{explicit}\) also collapses with the \(\psi _{R^{(0)}}\) profile. This solution, however, is “truly” self-similar, and is unstable.
- 3.
I.e., by letting \(\psi _{R^{(0)}}\) have the same amplitude at \(r=0\) as \(\psi \).
- 4.
- 5.
In light of Lemma 5.5, the question whether the collapsing core becomes radial is only of interest if \(\psi _0\) is not radial.
- 6.
In Sect. 3.6.3 we saw that noise has a destabilizing effect on infinite-power, plane wave solutions.
- 7.
As discussed in Sect. 3.2, the self-focusing process accelerates because of a nonlinear feedback mechanism. Thus, as the beam collapses, the intensity at its center increases. Hence so does the index of refraction. This, in turn, increases the attraction towards the beam center, thereby accelerating the self-focusing process.
- 8.
i.e., the value of \(Z_\mathrm{c}\) for which \(r^2\) of the regression curve (14.13) is closest to 1.
- 9.
Indeed, even if the blowup rate is a square root, the fitted numerical value of \(p\) will never be exactly \(1/2\).
- 10.
When \(G(0) \approx {R^{(0)}}(0)\), the difference between \(G\) and \({R^{(0)}}\) is evident only for \(\rho = O(\alpha ^{-2}) \gg 1\). This region, however, is way outside the collapsing-core domain where \(\psi \) converges to a self-similar profile.
- 11.
See Sect. 14.2 on how to determine the blowup rate numerically.
- 12.
The Townes profile in Fig. 14.15 has a triangular shape, because the \(y\)-axis is on a logarithmic scale.
- 13.
More recent simulations suggest that there may exist radially-stable solutions that collapse with a ring-type blowup profile which is different from \(\psi _{R^{(0)}}\) (Sect. 14.6.8).
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i.e., under the adiabatic approximation (Sect. 18.4).
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- 19.
In Chap. 11 we saw that the critical NLS admits the explicit ring-type solutions \(\psi _G^\mathrm{explicit}\) that collapse with a self-similar profile which is different from \(\psi _{R^{(0)}}\), at a square root blowup rate. These solutions, however, are not in \(H^1\).
- 20.
When the dimension \(d\) is an integer, these ring-type solutions are unstable under nonradial perturbations.
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Fibich, G. (2015). The Peak-Type Blowup Profile \(\psi _{R^{(0)}}\) . In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_14
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DOI: https://doi.org/10.1007/978-3-319-12748-4_14
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