Hamiltonian and small action variables for periodic dNLS

TL;DR

This paper introduces a new method using conformal mapping and L"owner equations to analyze the Hamiltonian of the defocussing NLS equation with small initial data, revealing asymptotics and frequency relations.

Contribution

It develops a novel approach linking Hamiltonian analysis to conformal mapping theory for the NLS equation, providing asymptotic and gradient formulas for small action variables.

Findings

Derived asymptotics of the Hamiltonian for small action variables.

Determined the gradient of the Hamiltonian with respect to action variables.

Connected the Hamiltonian analysis to conformal mapping and quasimomentum theory.

Abstract

We consider the defocussing NLS equation with small periodic initial condition. A new approach to study the Hamiltonian as a function of action variables is demonstrated. The problems for the NLS equation is reformulated as the problem of conformal mapping theory corresponding to quasimomentum of the Zakharov-Shabat operator. The main tool is the L\"owner type equation for the quasimomentum. In particular, we determine the asymptotics of the Hamiltonian for small action variables. Moreover, we determine the gradient of Hamiltonian with respect to action variables. This gives so called frequencies and determines how the angles variables depend on the time.