Unconditional Uniqueness Results for the Nonlinear Schrödinger Equation

Abstract

We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation. We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that solutions to the cubic NLS give rise to solutions of the Gross-Pitaevskii hierarchy, which is an infinite-dimensional system of linear equations. By using the uniqueness analysis of the Gross-Pitaevskii hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on rectangular tori, which cover the full scaling-subcritical regime in high dimensions. In fact, we prove a more general result which is conditional on the domain. In addition, we observe that well-posedness of the cubic NLS in Fourier-Lebesgue spaces implies unconditional uniqueness.

arXiv commentversion 2: minor corrections

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Cited by in corpus (8)

• The rigorous derivation of the ${\mathbb{T}}^2$ focusing cubic NLS from 3D
• Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space
• Unconditional uniqueness for the Benjamin-Ono equation
• The unconditional uniqueness for the energy-supercritical NLS
• The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$
• Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel
• Well/Ill-posedness of the Boltzmann Equation with Soft Potential
• On the mean-field and semiclassical limit from quantum $N$-body dynamics