KdV Hamiltonian as function of actions

TL;DR

This paper proves that the nonlinear part of the KdV Hamiltonian, expressed via actions, is a continuous convex function on bcl^2, with bounds derived using quasimomentum and conformal mapping theory.

Contribution

It introduces a new representation of the KdV Hamiltonian in terms of quasimomentum and analyzes its convexity and bounds using conformal mapping techniques.

Findings

The nonlinear KdV Hamiltonian is a continuous convex function on bcl^2.

Derived explicit lower and upper bounds for the Hamiltonian.

Established a new representation of the Hamiltonian using quasimomentum.

Abstract

We prove that the non-linear part of the Hamiltonian of the KdV equation on the circle, written as a function of the actions, defines a continuous convex function on the $\ell^2$ space and derive for it lower and upper bounds in terms of some functions of the $\ell^2$-norm. The proof is based on a new representation of the Hamiltonian in terms of the quasimomentum and its analysis using the conformal mapping theory.