On the convexity of the KdV Hamiltonian

TL;DR

This paper proves that the nonlinear part of the KdV Hamiltonian, when expressed in action variables, extends analytically and is strictly concave near zero, implying local diffeomorphism properties.

Contribution

It establishes the real analyticity and strict concavity of the nonlinear KdV Hamiltonian in action variables, revealing new geometric properties.

Findings

The nonlinear part of the KdV Hamiltonian extends analytically to the positive quadrant.

The nonlinear Hamiltonian is strictly concave near zero.

The differential of the nonlinear Hamiltonian defines a local diffeomorphism near zero.

Abstract

We prove that the nonlinear part $H^{*}$ of the KdV Hamiltonian $H^{kdv}$, when expressed in action variables $I = (I_{n})_{n\ge 1}$, extends to a real analytic function on the positive quadrant $\ell^2_+(\mathbb N)$ of $\ell^{2}(\mathbb N)$ and is strictly concave near $0$. As a consequence, the differential of $H^{*}$ defines a local diffeomorphism near $0$ of $\ell_{\mathbb C}^{2}(\mathbb N)$.