Global existence and scattering for quadratic NLS with potential in 3D
Tristan L\'eger
TL;DR
This paper proves global existence and scattering for a quadratic nonlinear Schrödinger equation in three dimensions with a small, time-dependent potential and localized initial data, using space-time resonance and wave operator techniques.
Contribution
It establishes the first global well-posedness and scattering results for this class of quadratic NLS with potential in 3D.
Findings
Solutions exist globally in time
Solutions scatter to linear solutions asymptotically
Method combines space-time resonance and wave operator analysis
Abstract
In this article we study the asymptotic behavior of a quadratic NLS equation with small, time-dependent potential and small spatially localized initial data. We prove global existence and scattering of solutions. The two main ingredients of the proof are the space-time resonance method and the boundedness of wave operators for the linear Schr\"{o}dinger equation with potential.
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Taxonomy
TopicsAdvanced Mathematical Physics Problems · Stability and Controllability of Differential Equations · Mathematical Analysis and Transform Methods