Excitation of rogue waves of Fokas system
Authors: Zhang Jie-Fang , Jin Mei-Zhen ,   Institute of Intelligent Media Technology, Communication University of Zhejiang, Hangzhou 310018, China,   Zhejiang Provincial Key Laboratory of Film and Television Media Technology, Communication University of Zhejiang, Hangzhou 310018, China,   Network and Data Center, Communication University of Zhejiang, Hangzhou 310018, China
Published: December 2020

<sec>Rogue wave (RW) is one of the most fascinating phenomena in nature and has been observed recently in nonlinear optics and water wave tanks. It is considered as a large and spontaneous nonlinear wave and seems to appear from nowhere and disappear without a trace. </sec><sec>The Fokas system is the simplest two-dimensional nonlinear evolution model. In this paper, we firstly study a similarity transformation for transforming the system into a long wave-short wave resonance model. Secondly, based on the similarity transformation and the known rational form solution of the long-wave-short-wave resonance model, we give the explicit expressions of the rational function form solutions by means of an undetermined function of the spatial variable <i>y</i>, which is selected as the Hermite function. Finally, we investigate the rich two-dimensional rogue wave excitation and discuss the control of its amplitude and shape, and reveal the propagation characteristics of two-dimensional rogue wave through graphical representation under choosing appropriate free parameter. </sec><sec>The results show that the two-dimensional rogue wave structure is controlled by four parameters: <inline-formula><tex-math id="M1">\begin{document}${\rho _0},\;n,\;k,\;{\rm{and}}\;\omega \left( {{\rm{or}}\;\alpha } \right)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M1.png"/></alternatives></inline-formula>. The parameter <inline-formula><tex-math id="M2">\begin{document}$ {\rho _0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M2.png"/></alternatives></inline-formula> controls directly the amplitude of the two-dimensional rogue wave, and the larger the value of <inline-formula><tex-math id="M3">\begin{document}$ {\rho _0}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M3.png"/></alternatives></inline-formula>, the greater the amplitude of the amplitude of the two-dimensional rogue wave is. The peak number of the two-dimensional rogue wave in the <inline-formula><tex-math id="M4">\begin{document}$(x,\;y)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M4.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$(y,\;t)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M5.png"/></alternatives></inline-formula> plane depends on merely the parameter <i>n</i> but not on the parameter <i>k</i>. When <inline-formula><tex-math id="M6">\begin{document}$n = 0,\;1,\;2, \cdots$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M6.png"/></alternatives></inline-formula>, only single peak appears in the <inline-formula><tex-math id="M7">\begin{document}$(x,\;t)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M7.png"/></alternatives></inline-formula> plane, but single peak, two peaks to three peaks appear in the <inline-formula><tex-math id="M8">\begin{document}$(x,\;y)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M8.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$(y,\;t)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M9.png"/></alternatives></inline-formula> plane, respectively, for the two-dimensional rogue wave of Fokas system. We can find that the two-dimensional rogue wave occurs from the zero background in the <inline-formula><tex-math id="M10">\begin{document}$(x,\;t)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M10.png"/></alternatives></inline-formula> plane, but the two-dimensional rogue wave appears from the line solitons in the <inline-formula><tex-math id="M11">\begin{document}$(x,\;y)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M11.png"/></alternatives></inline-formula> plane and <inline-formula><tex-math id="M12">\begin{document}$(y,\;t)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20200710_M12.png"/></alternatives></inline-formula> plane.</sec><sec>It is worth pointing out that the rogue wave obtained here can be used to describe the possible physical mechanism of rogue wave phenomenon, and may have potential applications in other (2 + 1)-dimensional nonlinear local or nonlocal models.</sec>