Integrable $\mathcal{P}T$-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model

Abstract

We introduce a new unified two-parameter $\{({\epsilon }_x,{\epsilon }_t)|{\epsilon }_{x,t}=\pm 1\}$ wave model (simply called ${\mathcal{Q}}_{{\epsilon }_x,{\epsilon }_t}^{(n)}$ model), connecting integrable local and nonlocal vector nonlinear Schrödinger equations. The two-parameter $({\epsilon }_x,{\epsilon }_t)$ family also brings insight into a one-to-one connection between four points $({\epsilon }_x,{\epsilon }_t)$ (or complex numbers ${\epsilon }_x+i{\epsilon }_t$) with $\{\mathcal{I},\mathcal{P},\mathcal{T},\mathcal{P}T\}$ symmetries for the first time. The ${\mathcal{Q}}_{{\epsilon }_x,{\epsilon }_t}^{(n)}$ model with $({\epsilon }_x,{\epsilon }_t)=(\pm 1,1)$ is shown to possess a Lax pair and infinite number of conservation laws, and to be $\mathcal{P}T$ symmetric. Moreover, the Hamiltonians with self-induced potentials are shown to be $\mathcal{P}T$ symmetric only for ${\mathcal{Q}}_{-1,-1}^{(n)}$ model and to be $\mathcal{T}$ symmetric only for ${\mathcal{Q}}_{+1,-1}^{(n)}$ model. The multi-linear form and some self-similar solutions are also given for the ${\mathcal{Q}}_{{\epsilon }_x,{\epsilon }_t}^{(n)}$ model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.

6 pages, 1 figure, submitted on Jan. 25, 2015 (corrected version)

References in corpus (8)

• Making Sense of Non-Hermitian Hamiltonians
• Coherent Perfect Absorbers: Time-reversed Lasers
• Visualization of Branch Points in PT-Symmetric Waveguides
• PT-Symmetric Phonon Laser
• Gain-Driven Discrete Breathers in PT-Symmetric Nonlinear Metamaterials
• Stable dark solitons in PT-symmetric dual-core waveguides
• Nonlinear localized modes in PT-symmetric Rosen-Morse potential well
• Spontaneous Parity--Time Symmetry Breaking and Stability of Solitons in Bose-Einstein Condensates

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• General $N$-Dark Soliton Solutions of the Multi-Component Mel'nikov System
• Dynamics of Rogue Waves in the Partially PT-symmetric Nonlocal Davey-Stewartson Systems
• Riemann-Hilbert approach for the NLSLab equation with nonzero boundary conditions
• Symmetries and reductions of integrable nonlocal partial differential equations
• Families of rational and semi-rational solutions of the partial reverse space-time nonlocal Mel'nikov equation
• Hirota bilinear forms of the AKNS($N$) systems
• General mixed multi-soliton solution to the multi-component Maccari system
• Local and nonlocal complex discrete and semi-discrete sine-Gordon equations and solutions