Complete integrability of Nonlocal Nonlinear Schr\"odinger equation

By V. Gerdjikov, Avadh Saxena. 2015

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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 Schrodinger 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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/133b58c4733852f1c487bc4de86b5794020af6af