Qualitative features of periodic solutions of KdV

TL;DR

This paper investigates the qualitative behavior of solutions to the KdV equation on the circle, revealing Fourier coefficient expansions and approximation estimates that deepen understanding of their oscillatory nature.

Contribution

It introduces new qualitative features of KdV solutions, including WKB-type expansions of Fourier coefficients and approximation bounds by trigonometric polynomials.

Findings

Fourier coefficients admit a WKB type expansion with oscillating phases.

Solutions can be approximated by trigonometric polynomials with explicit error estimates.

Provides new insights into the oscillatory structure of KdV solutions.

Abstract

In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree.