Abstract
In this chapter we consider the variance identity and its applications.
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Notes
- 1.
This second proof will become useful when we discuss the variance identity on a bounded domain (Sect. 16.8.1).
- 2.
By equivalent we mean that both conservation laws follow from the invariance of the action integral under dilations (see [220, 247, 249] for derivation of NLS conservation laws via Noether Theorem).
- 3.
See Conclusion 2.12.
- 4.
The condition \(H(\psi _0)<0\) is not sharp, i.e., collapse can also occur for real initial conditions with a positive Hamiltonian (Sect. 7.4). Hence, nonlinearity can be stronger than diffraction even when \(H(\psi _0)>0\).
- 5.
In the critical case, it can be rigorously shown that if \(0< -F\le Z_\mathrm{c}\), where \(Z_\mathrm{c}\) is the blowup point of \(\psi \), then \(\tilde{\psi }\) does not blowup (Sect. 8.4.6). In the supercritical case, it can be informally shown that collapse is arrested when \(-F\) is sufficiently small (Sect. 8.4.2).
- 6.
The converse statement is not true, i.e., the condition \(H\big (\tilde{\psi }_0\big )> 0\) does not necessarily imply that nonlinearity is weaker than the combined effects of diffraction and the lens.
- 7.
We already reached this conclusion when we analyzed the NLS with the aberrationless approximation method (Sect. 3.5).
- 8.
- 9.
This is the case, e.g., when \(\psi _0\) is real or radially-symmetric.
- 10.
Knowing the amount of power that collapses into the singularity is important in applications where one wants to maximize (or control) the amount of power at the “target”.
- 11.
- 12.
These results can be generalized to focused initial conditions by using Lemma 7.11.
- 13.
See Sect. 8.4.3 for an alternative proof, which is based on the lens transformation.
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Fibich, G. (2015). Variance Identity. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_7
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DOI: https://doi.org/10.1007/978-3-319-12748-4_7
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