Complete integrability of Nonlocal Nonlinear Schr\"odinger equation

TL;DR

This paper proves the complete integrability of a subset of nonlocal nonlinear Schrödinger equations by analyzing their spectral properties, soliton solutions, and Hamiltonian structures using inverse scattering and symplectic basis methods.

Contribution

It extends the inverse scattering method and Hamiltonian formalism to nonlocal NLS equations, demonstrating their complete integrability and spectral characteristics.

Findings

Existence of two types of soliton solutions.

Spectral properties indicate complete integrability.

Construction of hierarchy Hamiltonian structures.

Abstract

Based on the completeness relation for the squared solutions of the Lax operator $L$ we show that a subset of nonlocal equations from the hierarchy of nonlocal nonlinear Schr\"odinger equations (NLS) is a completely integrable system. The spectral properties of the Lax operator indicate that there are two types of soliton solutions. The relevant action-angle variables are parametrized by the scattering data of the Lax operator. The notion of the symplectic basis, which directly maps the variations of the potential of $L$ to the variations of the action-angle variables has been generalized to the nonlocal case. We also show that the inverse scattering method can be viewed as a generalized Fourier transform. Using the trace identities and the symplectic basis we construct the hierarchy Hamiltonian structures for the nonlocal NLS equations.