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Integrable ${\mathcal PT}$-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model
Zhenya Yan
TL;DR
The paper introduces a unified two-parameter model connecting local and nonlocal vector nonlinear Schrödinger equations while clarifying their symmetry relations. It establishes integrability and PT symmetry for part of the family, derives solution forms, and identifies scope boundaries for integrability and extensions.
Problem
The paper addresses the need to explore new PT-symmetric nonlinear waves and integrable PT-symmetric nonlinear wave models, including unified connections between local and nonlocal vector NLS equations.
Method
The authors introduce a two-parameter symmetric reduction of a vector NLS system and analyze its Lax pair, conservation laws, symmetries, multilinear form, self-similar reductions, and exact solutions.
Results
The model connects one local and three nonlocal vector NLS equations; for (ε_x, ε_t)=(±1,1), it has a Lax pair and infinitely many conservation laws, and the family is PT symmetric.
Takeaways & Limitations
The two-parameter construction establishes a one-to-one connection between four parameter points and I, P, T, and PT symmetries, while supplying bright and dark solitons, periodic waves, and multi-rogue waves.
Abstract
from arXiv · showhide
We introduce a new unified two-parameter $\{(ε_x, ε_t)\,|ε_{x,t}=\pm1\}$ wave model (simply called ${\mathcal Q}_{ε_x,ε_t}^{(n)}$ model), connecting integrable local and nonlocal vector nonlinear Schrödinger equations. The two-parameter $(ε_x, ε_t)$ family also brings insight into a one-to-one connection between four points $(ε_x, ε_t)$ (or complex numbers $ε_x+iε_t$) with $\{{\mathcal I}, {\mathcal P}, {\mathcal T}, {\mathcal PT}\}$ symmetries for the first time. The ${\mathcal Q}_{ε_x,ε_t}^{(n)}$ model with $(ε_x, ε_t)=(\pm 1, 1)$ is shown to possess a Lax pair and infinite number of conservation laws, and to be ${\mathcal PT}$ symmetric. Moreover, the Hamiltonians with self-induced potentials are shown to be ${\mathcal PT}$ 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 ${\mathcal Q}_{ε_x,ε_t}^{(n)}$ model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.