Long time solutions for 1D cubic dispersive equations, Part II: the focusing case

TL;DR

This paper investigates long-time solutions for 1D focusing cubic dispersive equations, establishing existence and dispersive estimates on long time scales proportional to the initial data size, extending previous defocusing results.

Contribution

It provides the first long-time existence and dispersive estimate results for focusing cubic dispersive equations, with solutions valid up to time scales of ^{-8} for initial data of size .

Findings

Solutions exist up to time ^{-8} for small initial data

Solutions satisfy global Strichartz estimates and bilinear bounds on these long time scales

First result achieving such long-time dispersive estimates in the focusing case

Abstract

This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small initial data yields global, scattering solutions. Then this conjecture was proved in the case of a Schr\"odinger dispersion relation. In terms of scattering, our global solutions were proved to satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. Notably, no localization assumption is made on the initial data. In this article we consider the focusing scenario. There potentially one may have small solitons, so one cannot hope to have global scattering solutions in general. Instead, we look for long time solutions, and ask what is the time-scale on which the solutions exist and satisfy good dispersive estimates. Our main result, which also applies in the case of the Schr\"odinger dispersion relation, asserts that for initial data of size $\epsilon$, the solutions exist on the time-scale $\epsilon^{-8}$, and satisfy the desired $L^6$ Strichartz estimates and bilinear $L^2$ bounds on the time-scale $\epsilon^{-6}$. To the best of our knowledge, this is the first result to reach such a threshold.