2015/10/02 by Vladimir S. Gerdjikov, Avadh Saxena, Gerdjikov, V. S. +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.1510.00480
openalex publication_date 2015/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Schrödinger 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.