Abstract
In this chapter we consider NLS solutions that collapse on a \(d\)-dimensional sphere.
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Notes
- 1.
The variable \(\rho \) measures the distance from the ring peak in units of \(L(z)\). Therefore, \(\rho \) is negative for \(\,0 \le r < r_\mathrm{max}\) and positive for \(\,r_\mathrm{max}<r<\infty \).
- 2.
For clarity, in this chapter we denote the solutions of the one-dimensional NLS by \(\phi \).
- 3.
We drop the \(^{(0)}\) superscript, because in one dimension there is a unique solitary wave.
- 4.
The dimension \(d\) does not have to be an integer. See [74, Sect. 6.2] for a numerical study of standing-ring blowup solutions of the quintic NLS with \(d=3/2\).
- 5.
See Sect. 14.2.1.
- 6.
- 7.
In Chap. 26 we use the NGO method to explain why strongly nonlinear super-Gaussian initial conditions evolve into a ring profile.
- 8.
This filamentation pattern is typical for elliptic input beams, see Lemma 25.1.
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Fibich, G. (2015). Singular Standing-Ring Solutions \(\big (\psi _F\big )\) . In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_22
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DOI: https://doi.org/10.1007/978-3-319-12748-4_22
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