# On classical solutions of the KdV equation

## TL;DR

This paper proves the existence and uniqueness of global classical solutions for the KdV equation under specific initial conditions, using a determinant formula, extending previous results in the field.

## Contribution

It establishes the most general conditions known to date for the existence of classical solutions to the KdV equation, including non-decaying initial profiles.

## Key findings

- Unique global classical solutions exist under broad initial conditions.
- Solutions can be explicitly constructed using a determinant formula.
- The results extend previous known conditions for well-posedness.

## Abstract

\begin{abstract} We show that if the initial profile $q\left( x\right) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $\int^{\infty }x^{5/2}\left\vert q\left( x\right) \right\vert dx<\infty,$ (no decay at $-\infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. \end{abstract}