Excitation of rogue waves of Fokas system

Abstract

<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>