A note on decay property of nonlinear Schr\"odinger equations

TL;DR

This paper constructs a special solution to the 3D defocusing cubic nonlinear Schrödinger equation that exhibits extremely slow scattering behavior, highlighting the influence of initial data decay properties on asymptotic dynamics.

Contribution

It demonstrates the existence of solutions with slow scattering in 3D NLS due to delayed backward profiles, emphasizing the role of initial data decay conditions.

Findings

Existence of solutions with arbitrarily slow scattering rate

Slow convergence caused by delayed backward scattering profiles

L¹ initial data prevents slow scattering phenomena

Abstract

In this note, we show the existence of a special solution $u$ to defocusing cubic NLS in $3d$, which lives in $H^{s}$ for all $s>0$, but scatters to a linear solution in a very slow way. We prove for this $u$, for all $\epsilon>0$, one has $\sup_{t>0}t^{\epsilon}\|u(t)-e^{it\Delta}u^{+}\|_{\dot{H}^{1/2}}=\infty$. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of $u(0)$ be in $L^{1}$. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why $L^{1}$ condition of initial data will get rid of this phenomena.