Abstract
In this chapter we present the explicit singular peak-type solution \(\psi _Q^\mathrm{explicit}\) of the supercritical NLS.
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Notes
- 1.
- 2.
See Definition 11.1.
- 3.
If \(c_1 = 0\), then \(Q \sim c_2 Q_2\). Therefore, the integral is finite, and the proof is immediate.
- 4.
The reason why the condition of a finite Hamiltonian is “more important” than that of a finite power will become clear in Sect. 21.1.
- 5.
This phase represents focusing with \(F(z) = -{L}/{L_z}\), see Sect. 2.12.2.
- 6.
See, e.g., Sect. 17.6.2.
- 7.
For example, the unique value of \(a\) that corresponds to an admissible solution of the cubic three-dimensional NLS is
$$ a_Q(\sigma =1,d=3) \approx 0.917. $$ - 8.
Using the Matlab code in Fig. 12.1.
- 9.
When \(d=1\) we can set \(\gamma =1\), since \(\gamma _0 = -\frac{1}{\sigma }\) and \(\gamma _1 +2 = 3+\frac{1}{\sigma }\) (Lemma 12.7).
- 10.
See Lemma 12.8.
- 11.
Thus, for any \(J\), \(Q_{K=0,J}\) converges to the trivial solution \(R^{(-1)}(\rho ) \equiv 0\), \(Q_{K=1,J}\) converges to the ground state \(R^{(0)}(\rho )\), etc.
- 12.
This is different from the critical case, where all the explicit peak-type blowup solutions \(\psi _{R^{(n)}}^\mathrm{explicit}\) are unstable (Sect. 10.6). It is, however, analogous to the explicit ring-type blowup solutions \(\psi _G^\mathrm{explicit}\) of the critical NLS, for which the single-ring solution is stable but the multi-rings solutions are unstable (Sect. 11.4.1). In that case, the single-ring solution is the “least non-monotone”.
- 13.
Recall that for \(H^1\) NLS solutions, blowup of the \(H^1\) norm implies blowup of the \(L^p\) norms for \( 2 \sigma +2 \le p \le \infty \) (Corollary 5.8).
- 14.
The value of \(c_2\) cannot be equal to zero, see Lemma 12.12.
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Fibich, G. (2015). The Explicit Supercritical Singular Peak-Type Solution \(\psi _Q^\mathrm{{explicit}}\) . In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_12
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DOI: https://doi.org/10.1007/978-3-319-12748-4_12
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