Construction of KdV flow

TL;DR

This paper constructs a KdV flow on a spectral space that includes decaying and ergodic functions, allowing smooth almost periodic functions as initial data for the KdV equation.

Contribution

It introduces a new spectral space framework for the KdV flow that encompasses a broader class of initial data, including almost periodic functions.

Findings

Constructed KdV flow on a spectral space including ergodic functions.

Extended initial data to include smooth almost periodic functions.

Demonstrated the applicability of the spectral approach to broader initial conditions.

Abstract

A KdV flow is constructed on a space whose structure is described in terms of the spectrum of the underlying Schr\"odinger operators. The space includes the conventional decaying functions and ergodic ones. Especially any smooth almost periodic function can be initial data for the KdV equation.