Large data local well-posedness for a class of KdV-type equations

TL;DR

This paper extends local well-posedness results for a class of KdV-type equations with large initial data to translation invariant Sobolev spaces, adapting methods from quasilinear Schrödinger equations.

Contribution

It establishes local well-posedness in translation invariant Sobolev spaces for KdV-type equations with large data, building on and adapting previous quasilinear Schrödinger techniques.

Findings

Proves local well-posedness in a new functional setting.

Adapts techniques from quasilinear Schrödinger equations.

Handles large initial data for KdV-type equations.

Abstract

In this article we consider the Cauchy problem with large initial data for an equation of the form (\partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms. Local well-posedness was established in weighted Sobolev spaces by Kenig-Ponce-Vega. In this paper we prove local well-posedness in a translation invariant subspace of H^s by adapting the result of Marzuola-Metcalfe-Tataru on quasilinear Schrodinger equations.