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

Abstract

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.

Review on long time asymptotics of large data in some nonintegrable dispersive models

“…Some fine commutator estimates are needed, and the liminf is obtained since we need a particular modification of the argument in [104] that allows us to control the energy norm as time tends to infinity. More applications: The method of proof above is quite general and it can be applied to several other dispersive models of importance: BBM equations (Kwak-M. [70]), abcd type models [68,69] (small data case only, large data seems quite complicated), Camassa-Holm type equations (Alejo-Cortez-Kwak-M. [3]), KdV on the half line [12], the quite involved Zakharov-Rubenchik / Benney-Roskes system (María Martínez and José Palacios [91]), and the ILW model (M.-Ponce-Saut [106]).…”