Modified Scattering for Nonlocal Nonlinear Schrödinger Equations

TL;DR

The paper analyzes the long-time behavior of small data solutions to the Hartree equation $i\partial_t u+\Delta u=(|\cdot|^{-1}*|u|^2)u$ and the Schrödinger-Bopp-Podolsky equation $i\partial_t u+\Delta u=(\mathcal K*|u|^2)u-|u|^{2/d}u$ in $d\in\{2,3\}$. It employs the testing by wavepackets framework to derive a leading amplitude ODE for $\gamma(t,v)$ with a leading nonlinearity and controlled remainders, yielding a modified scattering expansion with a logarithmic phase corrections $e^{-\frac{i}{2}\log t (|\cdot|^{-1} * |\mathscr W|^2)}$ and $e^{-\frac{i}{2}\log t (\mathcal K * |\mathscr Q|^2)}$. The main outcomes are global well-posedness in $H^{0,\beta}$ with $\beta=\frac{d}{2}+\frac{1}{10}$, sharp $L^\infty$ decay $\|u\|_{L^\infty} \lesssim \varepsilon |t|^{-d/2}$, and explicit asymptotic expansions for $u$ in terms of the profiles $\mathscr W$ and $\mathscr Q$. The work sharpens prior results by reducing regularity requirements and extends modified scattering to nonlocal interactions, providing a template for scattering-critical NLS with minimal regularity.

Abstract

We prove a modified scattering and sharp $L^\infty$ decay result for both the Hartree and Schrödinger-Bopp-Podolsky equations in dimensions $2$ and $3$ using the testing by wavepackets approach due to Ifrim and Tataru. We show that modified scattering and sharp pointwise decay occur for these equations at a regularity much lower than previous results due to Hayashi-Naumkin and Kato-Pusateri, and as a corollary also show that the results on power-type scattering-critical NLS due to Hayashi-Naumkin can be proven under minimal regularity assumptions.

Theorems & Definitions (21)

• Theorem 1
• Remark
• Lemma 1: Real Interpolation of $L^p$ spaces
• remark 1
• Lemma 2
• proof
• remark 2: Convention for convolutions
• Theorem 2
• proof
• Definition 1