Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric Potential

Abstract

A two component nonlocal vector nonlinear Schrödinger equation (VNLSE) is considered with a self-induced $\mathcal{P}\mathcal{T}$ symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of these charges being non-hermitian. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is also mentioned. An inhomogeneous version of this VNLSE with space -time modulated nonlinear interaction term is also considered and a mapping of this Eq. with standard VNLSE through similarity transformation is used to generate its solutions.

19 pages

References in corpus (13)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PT-Symmetric Nonlinear Equations
• Breathers in PT-symmetric optical couplers
• Stable dark solitons in PT-symmetric dual-core waveguides
• Bragg solitons in nonlinear PT-symmetric periodic potentials
• On Symmetries and Exact Solutions of a Class of Non-local Non-linear Schrodinger Equations with Self-induced PT-symmetric Potential
• A return to observability near exceptional points in a schematic PT-symmetric model
• Exactly Solvable Quasi-hermitian Transverse Ising Model
• Level statistics of a pseudo-Hermitian Dicke model
• Nonlinear multi-core waveguiding structures with balanced gain and loss
• Quantum Phase Transition in a Pseudo-hermitian Dicke model
• Deconstructing non-Dirac-hermitian supersymmetric quantum systems

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• Discrete Symmetries and Nonlocal Reductions
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• Soliton solutions of the shifted nonlocal NLS and MKdV equations
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• Classical Hamiltonian Systems with Balanced loss and gain
• Hirota bilinear forms of the AKNS($N$) systems
• Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I Symmetry preserving and symmetry breaking solutions