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利用Kadomtsev-Petviashvili (KP)系列约束方法和双线性方法, 构造了空间位移宇称–时间反演(
$\mathcal{PT}$ )对称非局域非线性薛定谔方程的高阶怪波解. 任意$N$ 阶怪波解的解析表达式是通过舒尔多项式表示的. 首先通过分析一阶怪波解的动力学行为, 发现怪波的最大振幅可以大于背景平面三倍的任意高度. 分析了对称非局域非线性薛定谔方程中的空间位移因子$x_0$ 在一阶怪波解中的影响, 结果表明其仅改变怪波中心的位置. 另外，研究了二阶怪波解的动力学行为以及怪波模式, 然后给出了$N$ 阶怪波模式与$N$ 阶怪波解的解析表达式中参数之间的关系, 进一步展示了高阶怪波的不同模式.General higher-order rogue wave solutions to the space-shifted$\mathcal{PT}$ -symmetric nonlocal nonlinear Schrödinger equation are constructed by employing the Kadomtsev-Petviashvili hierarchy reduction method. The analytical expressions for rogue wave solutions of any Nth-order are given through Schur polynomials. We first analyze the dynamics of the first-order rogue waves, and find that the maximum amplitude of the rogue waves can reach any height larger than three times of the constant background amplitude. The effects of the space-shifted factor$x_0$ of the$\mathcal{PT}$ -symmetric nonlocal nonlinear Schrödinger equation in the first-order rogue wave solutions are studied, which only changes the center positions of the rogue waves. The dynamical behaviours and patterns of the second-order rogue waves are also analytically investigated. Then the relationships between Nth-order rogue wave patterns and the parameters in the analytical expressions of the rogue wave solutions are given, and the several different patterns of the higher-order rogue waves are further shown.-
Keywords:
- rogue waves/
- $\mathcal{PT}$-symmetric nonlocal nonlinear Schrödinger equation/
- Kadomtsev-Petviashvili hierarchy reduction method
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