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

TL;DR

This paper reviews recent advances in understanding the long-term behavior of large data solutions in nonintegrable dispersive models, highlighting new virial estimate techniques for local energy decay.

Contribution

It presents a comprehensive review and introduces new virial estimate methods to prove local energy decay in dispersive models like KdV and KP.

Findings

Virial estimates enable proving local energy decay.

Analysis covers KdV and KP models.

Results show decay in regions without solitons or lumps.

Abstract

In this short note, we review recent results concerning the long time dynamics of large data solutions to several dispersive models. Starting with the KdV case and ending with the KP models, we review the literature and present new results where virial estimates allow one to prove local energy decay in different regions of space where solitons, lumps or solitary waves are not present.