On Schr\"odinger systems with cubic dissipative nonlinearities of derivative type

TL;DR

This paper investigates cubic derivative nonlinear Schrödinger systems in one dimension, identifying conditions under which solutions exhibit enhanced logarithmic decay over time compared to free solutions.

Contribution

It provides structural conditions on the nonlinearity that lead to improved decay rates for small data solutions in Schrödinger systems with derivative nonlinearities.

Findings

Solutions gain logarithmic decay under certain resonance conditions

Structural conditions on nonlinearity are identified for decay enhancement

Results apply to systems with specific mass relations

Abstract

Consider the initial value problem for systems of cubic derivative nonlinear Schr\"odinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under which the small data solution gains an additional logarithmic decay as $t \to +\infty$ compared with the corresponding free evolution.