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We study the focusing NLS equation in $$R\mathbb{R}^N$$ in the mass-supercritical and energy-subcritical (or intercritical ) regime, with $H^1$ data at the mass-energy threshold $$\mathcal{ME}(u_0)=\mathcal{ME}(Q)$$, where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the $H^1$-critical case, in dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, $$Q^\pm$$, besides the standing wave $$e^{it}Q$$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blow-up occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixed-point argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations.
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Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on
Abstract We consider the $$\mathbb {T}^{4}$$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U - V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U - V estimates then seamlessly conclude the $$H^{1}$$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $$H^{1}$$ uniqueness for the $$ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.
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- PAR ID:
- 10327808
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 10
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2021.16
