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

TL;DR

This paper investigates the long-time decay and integrability properties of solutions to the one-dimensional Zakharov-Rubenchik/Benney-Roskes system, showing decay in local energy norms and time-integrability in expanding intervals.

Contribution

It establishes decay and integrability results for solutions without restrictions on initial data size or parity conditions, advancing understanding of the system's long-term behavior.

Findings

Time-integrability in intervals growing as t^r, r<2/3, around characteristic curves

Decay of local energy-norm in far-field regions

Results are independent of initial data size and parity conditions

Abstract

We study decay properties for solutions to the initial value problem associated with the one-dimensional Zakharov-Rubenchik/Benney-Roskes 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.