Complete integrability of Nonlocal Nonlinear Schrödinger equation

V. S. Gerdjikov; A. Saxena

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arXiv · 1510.00480

Complete integrability of Nonlocal Nonlinear Schrödinger equation

V. S. Gerdjikov·A. Saxena

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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 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.

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BibTeX RIS

V. S. Gerdjikov, A. Saxena. 2016-06-14. Complete integrability of Nonlocal Nonlinear Schrödinger equation. https://arxiv.org/abs/1510.00480

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