General higher-order rogue waves in the space-shifted <inline-formula><tex-math id="M2">\begin{document}$\mathcal{PT}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M2.png"/></alternatives></inline-formula>-symmetric nonlocal nonlinear Schrödinger equation

Abstract

General higher-order rogue wave solutions to the space-shifted <inline-formula><tex-math id="M8">\begin{document}$\mathcal{PT}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M8.png"/></alternatives></inline-formula>-symmetric nonlocal nonlinear Schrödinger equation are constructed by employing the Kadomtsev-Petviashvili hierarchy reduction method. The analytical expressions for rogue wave solutions of any <i>N</i>th-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 <inline-formula><tex-math id="M10">\begin{document}$x_0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M10.png"/></alternatives></inline-formula> of the <inline-formula><tex-math id="M11">\begin{document}$\mathcal{PT}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222298_M11.png"/></alternatives></inline-formula>-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 <i>N</i>th-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.