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.

arXiv comment6 pages, 1 figure, submitted on Jan. 25, 2015 (corrected version)

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• A new nonlocal nonlinear Schroedinger equation and its soliton solutions
• General rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger equation
• Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized PT-symmetric Scarff-II potentials
• Mixed soliton solutions of the defocusing nonlocal nonlinear Schrodinger equation
• Multi-place nonlocal systems
• Solitons and their stability in the nonlocal nonlinear Schroedinger equation with PT-symmetric potentials
• General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric system
• Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval
• On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation
• Coherent structure of Alice-Bob modified Korteweg de-Vries Equation
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• Attraction centers and PT-symmetric delta-functional dipoles in critical and supercritical self-focusing media
• Solutions of local and nonlocal equations reduced from the AKNS hierarchy
• General $N$-Dark Soliton Solutions of the Multi-Component Mel'nikov System
• Complete integrability of Nonlocal Nonlinear Schrödinger equation
• Rational solutions of the defocusing nonlocal nonlinear Schrodinger equation: Asymptotic analysis and soliton interactions
• Discrete Symmetries and Nonlocal Reductions
• Transformations between nonlocal and local integrable equations
• Nonlocal Coupled HI-MKdV Systems
• Dynamics of Rogue Waves in the Partially PT-symmetric Nonlocal Davey-Stewartson Systems
• Riemann-Hilbert approach for the NLSLab equation with nonzero boundary conditions
• Multi-component AKNS systems
• Symmetries and reductions of integrable nonlocal partial differential equations
• Solutions to nonlocal nonisospectral (2+1)-dimensional breaking soliton equations
• An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval
• Soliton solutions of the shifted nonlocal NLS and MKdV 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
• Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations
• Dbar-approach to coupled nonlocal NLS equation and general nonlocal Reduction
• A unified inverse scattering transform and soliton solutions of the nonlocal modified KdV equation with non-zero boundary conditions
• Local and nonlocal complex discrete and semi-discrete sine-Gordon equations and solutions