Abstract
In this chapter we derive the loglog law and the adiabatic laws for solutions of the critical NLS that collapse with the \(\psi _{R^{(0)}}\) profile.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Chap. 30 for what can happen when one tries to compute a singular NLS solution with a standard numerical solver.
- 2.
Since the ratio (18.28) may appear to approach a constant \(\approx 0.6\) in Fig. 18.1b, one might argue that \(L(z)\) did reach the loglog regime. This is wrong, however, because there is no free multiplicative constant in the loglog law. Thus, the ratio (18.28) has to approach \(1\) in the loglog regime. The ratio (18.28) appears to approach a constant simply because both \(L(z)\) and \(L_\mathrm{loglog}(z)\) are ‘almost’ a square root.
- 3.
For example, when we solved the linear Helmholtz equation in Sect. 2.1 using geometrical optics, we first solved the fast-scale Eikonal equation for the phase \(S\), and only then used its solution to solve the slow-scale transport equation for the amplitude \(A\). Similarly, when we solved the weakly-nonlinear Helmholtz equation (1.34) in Sect. 1.7, we first assumed that the fast-scale dynamics is given by the carrier oscillations \(e^{ i k_0 z}\), and only then used this assumption to derive the slow-scale equation for the amplitude \(\psi \).
- 4.
- 5.
Therefore, \(\beta _0>0\) is the asymptotic analog of the necessary condition \(P(0) > P_\mathrm{cr}\).
- 6.
For NLS solutions, the condition \(P>P_\mathrm{cr}\) is not sufficient for collapse (Corollary 13.8 and Chap. 24). The reduced equations, however, are derived under the assumption that the self-similar profile of the collapsing core is close to \({R^{(0)}}\). In that case, the condition \(P>P_\mathrm{cr}\) is asymptotically equivalent to the condition \(H<0\), see Corollary 17.5, and is therefore sufficient for collapse.
- 7.
- 8.
This observation is consistent with the results that nonlinearity and diffraction are balanced when \(P = P_\mathrm{cr}\), and that \(\beta \) is proportional to the excess power above \(P_\mathrm{cr}\).
- 9.
Indeed, since \(\beta = -L^{-3}L_{zz}\), then \(\beta =0 \Longrightarrow L_{zz}=0\), i.e., there is no acceleration or deceleration.
- 10.
This is a consequence of the dual borderline properties of the \(\psi _{R^{(0)}}\) profile (Sect. 17.2).
- 11.
See Conclusion 18.5.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Fibich, G. (2015). Loglog Law and Adiabatic Collapse. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-12748-4_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12747-7
Online ISBN: 978-3-319-12748-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)