Excitation of rogue waves of Fokas system

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. 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 y , 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. The results show that the two-dimensional rogue wave structure is controlled by four parameters: \begin{document}${\rho _0},\;n,\;k,\;{\rm{and}}\;\omega \left( {{\rm{or}}\;\alpha } \right)$\end{document}. The parameter \begin{document}$ {\rho _0}$\end{document}controls directly the amplitude of the two-dimensional rogue wave, and the larger the value of \begin{document}$ {\rho _0}$\end{document}, the greater the amplitude of the amplitude of the two-dimensional rogue wave is. The peak number of the two-dimensional rogue wave in the \begin{document}$(x,\;y)$\end{document}and \begin{document}$(y,\;t)$\end{document}plane depends on merely the parameter n but not on the parameter k . When \begin{document}$n = 0,\;1,\;2, \cdots$\end{document}, only single peak appears in the \begin{document}$(x,\;t)$\end{document}plane, but single peak, two peaks to three peaks appear in the \begin{document}$(x,\;y)$\end{document}and \begin{document}$(y,\;t)$\end{document}plane, respectively, for the two-dimensional rogue wave of Fokas system. We can find that the two-dimensional rogue wave occurs from the zero background in the \begin{document}$(x,\;t)$\end{document}plane, but the two-dimensional rogue wave appears from the line solitons in the \begin{document}$(x,\;y)$\end{document}plane and \begin{document}$(y,\;t)$\end{document}plane. 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.