Upper and lower $L^2$-decay bounds for a class of derivative nonlinear Schr\"odinger equations

TL;DR

This paper establishes the optimal decay rate in $L^2$ for small solutions to a class of derivative nonlinear Schrödinger equations with weak dissipation, showing solutions decay like $( ext{log } t)^{-1/4}$ as time progresses.

Contribution

It provides the first precise decay bounds and demonstrates the optimality of the decay rate for this class of equations.

Findings

Solutions decay like $( ext{log } t)^{-1/4}$ in $L^2$ as $t o o + abla$

The decay rate is proven to be optimal via matching lower bounds

The results apply to cubic derivative nonlinear Schrödinger equations with weak dissipation

Abstract

We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as $t\to +\infty$. Furthermore, we find that this $L^2$-decay rate is optimal by giving a lower estimate of the same order.