Hamiltonian and small action variables for periodic dNLS

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.