Self-similar transformation and excitation of rogue waves for (2+1)-dimensional Zakharov equation

Abstract

The search for the excitation of two-dimensional rogue wave in a (2+1)-dimensional nonlinear evolution model is a research hotspot. In this paper, the self-similar transformation of the (2+1)-dimensional Zakharov equation is established, and this equation is transformed into the (1+1)-dimensional nonlinear Schrödinger equation. Based on the similarity transformation and the rational formal solution of the (1+1)-dimensional nonlinear Schrödinger equation, the rogue wave excitation of the (2+1)-dimensional Zakharov equation is obtained by selecting appropriate parameters. We can see that the shape and amplitude of the rogue waves can be effectively controlled. Finally, the propagation characteristics of line rogue waves are diagrammed visually. We also find that the line-type characteristics of two-dimensional rogue wave are present in the <i>x-y</i> plane when the parameter <inline-formula><tex-math id="M5">\begin{document}$ \gamma = 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M5.png"/></alternatives></inline-formula>. The line rogue wave is converted into discrete localized rogue wave in the <i>x-y</i> plane when the parameter <inline-formula><tex-math id="M6">\begin{document}$ \gamma \ne 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M6.png"/></alternatives></inline-formula>. The spatial localized rogue waves with short-life can be obtained in the required <i>x-y</i> plane region. This is similar to the Peregrine soliton (PS) first discovered by Peregrine in the (1+1)-dimensional NLS equation, which is the limit case of the “Kuznetsov-Ma soliton” (KMS) or “Akhmediev breather” (AB). The proposed approach to constructing the line rogue waves of the (2+1) dimensional Zakharov equation can serve as a potential physical mechanism to excite two-dimensional rogue waves, and can be extended to other (2+1)-dimensional nonlinear systems.