Complete integrability of nonlocal nonlinear Schrödinger equation

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...

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Bibliographic Details
Published in:Journal of mathematical physics 2017-01, Vol.58 (1)
Main Authors: Gerdjikov, V. S., Saxena, A.
Format: Article
Language:English
Subjects:
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Fourier transforms
Inverse scattering
Mathematical analysis
MATHEMATICS AND COMPUTING
Nonlinear equations
Physics
Schrodinger equation
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Summary: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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4974018