Abstract
In this chapter we discuss the loss of phase of NLS solutions beyond the singularity.
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Notes
- 1.
Loss of information at singularities occurs also in e.g., shock waves and black holes.
- 2.
A small perturbation leads to a small relative change in the accumulated phase. Since, however, the amount of accumulated phase is “almost infinity”, a small relative change can correspond to an \(O\)(1) absolute change. Indeed, assume that a perturbation of size \(\epsilon \) arrests collapse at \(z_\mathrm{arrest}=z_\mathrm{arrest}(\epsilon )\). The on-axis accumulated phase at \(z_\mathrm{arrest}\) is \(\zeta (z_\mathrm{arrest}) = \int _0^{z_\mathrm{arrest}} L^{-2}\). Since \(\lim _{\epsilon \rightarrow 0} \zeta (z_\mathrm{arrest}) = \int _0^{Z_\mathrm{c}} L^{-2} = \infty \), small changes in \(\epsilon \) can lead to \(O\)(1) changes in \(\zeta (z_\mathrm{arrest})\).
- 3.
We already saw the high sensitivity of the post-collapse phase in Fig. 38.1b.
- 4.
- 5.
This dynamics is shown in Fig. 38.1a for the un-tilted case \(({c} = 0)\).
- 6.
- 7.
As we already saw in Sect. 27.5.
- 8.
Recall that out of phase beams repel each other (Sect. 3.2.2).
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Fibich, G. (2015). Loss of Phase and Chaotic Interactions. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_39
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DOI: https://doi.org/10.1007/978-3-319-12748-4_39
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