Action-angle variables for the nonlinear Schr\"{o}dinger equation on the half-line

TL;DR

This paper develops action-angle variables for the integrable nonlinear Schrödinger equation on the half-line, enabling a complete trivialization of its dynamics under certain boundary conditions.

Contribution

It introduces a method to construct action-angle variables for the NLS equation on the half-line, preserving integrability and simplifying the analysis of its dynamics.

Findings

Poisson brackets of scattering data are computed.

Action-angle variables are explicitly constructed.

The dynamics are completely trivialized using these variables.

Abstract

We consider the nonlinear Schr\"{o}dinger (NLS) equation on the half-line subjecting to a class of boundary conditions preserve the integrability of the model. For such a half-line problem, the Poisson brackets of the corresponding scattering data are computed, and the variables of action-angle type are constructed. These action-angle variables completely trivialize the dynamics of the NLS equation on the half-line.