Abstract
In this chapter we analyze the effect of dispersion on self-focusing pulses.
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Notes
- 1.
In Sect. 2.11 we used a similar argument to calculate the diffraction length \(L_\mathrm{diff}\).
- 2.
We already reached this conclusion in Sect. 36.2.3.
- 3.
i.e., of the two-dimensional plane \((x,y,t \equiv \text{ constant })\).
- 4.
i.e., when \(L_\mathrm{diff}\ll L_\mathrm{disp}\), see Sect. 36.2.2.
- 5.
\(Z_\mathrm{c}(t) = \infty \) if the \(t\) cross-section does not collapse.
- 6.
In Sect. 36.8.5 we will derive this relation from the reduced equations.
- 7.
See Sect. 26.3.2 for why an increase of small dispersion leads to an increase (and not a decrease) of temporal self-focusing.
- 8.
- 9.
In the case of (36.16), the transition is from a 2D critical collapse to a 3D supercritical collapse. In the case of the critical NLS with a small focusing supercritical nonlinearity, however, the transition is from a 2D critical collapse to a 2D supercritical collapse.
- 10.
- 11.
The second splitting turned out to be a numerical artifact (Sect. 36.8.8).
- 12.
See Sect. 31.1.1.
- 13.
- 14.
- 15.
Since \(\zeta (\cdot , t)\) assumes its maximum at \(t_\mathrm{m}\) and since \(\lim _{t \rightarrow \pm \infty } \zeta = 0\), there must be \(t\) cross-sections for which \(\zeta _{tt}>0\).
- 16.
- 17.
- 18.
For example, as \(L \rightarrow 0\), Eqs. (36.21b) and (36.29) reduce to
$$ -L^3 L_{zz} = \beta \sim \tilde{\gamma }_2 (Z_\mathrm{c}'(t))^2 \frac{1}{L^2}. $$It can be verified that this equation admits blowup solutions of the form \(L \sim c \,(Z_\mathrm{c}-z)^{\frac{1}{3}}\) [88]. For these blowup solutions, however, we have that \(\beta \rightarrow \infty \), which implies that the validity of the reduced equations breaks down.
- 19.
Obtaining a quantitative agreement between the NLS (36.1) and the reduced equations turned out to be a nontrivial task. For one thing, the two equations are in agreement only when modulation theory is valid, i.e., after the NLS solution approaches the \(\psi _{R^{(0)}}\) profile, but before dispersion becomes comparable to nonlinearity or diffraction. In addition, because modulation theory is only \(O\!(\beta )\) accurate, there is an \(O\!(\beta )\) error in extracting the reduced-equations variables from the NLS solution. To obtain a quantitative agreement, however, this error should be smaller than the temporal variation of the reduced-equations variables. Moreover, the temporal variation of the reduced-equations variables by itself has to be small, since dispersion should be smaller than nonlinearity and diffraction for modulation theory to be valid.
- 20.
- 21.
These values correspond to the pulse-splitting experiment of Ranka, Schirmer, and Gaeta (Sect. 36.10).
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Fibich, G. (2015). Normal and Anomalous Dispersion. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_36
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