# Sharp well--posedness for the generalized KdV of order three on the half line

## TL;DR

This paper establishes sharp local well-posedness results for the generalized KdV equation with cubic nonlinearity on the half line, filling a gap in the existing mathematical theory.

## Contribution

It proves the sharp local well-posedness for the generalized KdV of order three on the half line, extending previous results and closing an open problem in the field.

## Key findings

- Proved sharp local well-posedness for the generalized KdV with cubic nonlinearity on the half line.
- Closed the gap in the well-posedness theory after previous foundational work.
- Extended the mathematical understanding of boundary value problems for nonlinear dispersive equations.

## Abstract

In this paper we study the generalized Korteweg de Vries (KdV) equation with the nonlinear term of order three: $(u^{3+1})_x$. We prove sharp local well--posedness for the initial and boundary value problem posed on the right half line. We thus close the gap in the well--posedness theory of the generalized KdV which remained open after the seminal work of Colliander and Kenig in \cite{CK}.