On long-time behavior of solutions of the Zakharov–Rubenchik/Benney–Roskes system

We study decay properties for solutions to the initial value problem associated with the one-dimensional Zakharov–Rubenchik/Benney–Roskes (ZR/BR) system. We prove time-integrability in growing compact intervals of size t r , r < 2/3, centered on some characteristic curves coming from the underlying transport equations associated with the ZR/BR system. Additionally, we prove decay to zero of the local energy-norm in so-called far-field regions. Our results are independent of the size of the initial data and do not require any parity condition.