On the extension of the frequency maps of the KdV and the KdV2 equations

Abstract

In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on $H^s$ the KdV2 equation is $C^0$-wellposed if $s\ge 0$ and illposed (in a strong sense) if $s<0$.

76 pages

References in corpus (1)

• On Two-Sided Estimates for the Nonlinear Fourier Transform of KdV

Cited by in corpus (6)

• Sharp well-posedness results of the Benjamin-Ono equation in $H^s(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution
• Global well-posedness for the fifth-order KdV equation in $H^{-1}(\mathbb{R})$
• On the flow map of the Benjamin-Ono equation on the torus
• On the stability of periodic multi-solitons of the KdV equation
• Normal form coordinates for the KdV equation having expansions in terms of pseudodifferential operators
• On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$