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## 留言板

Fokas系统的怪波激发

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## Excitation of rogue waves of Fokas system

Zhang Jie-Fang, Jin Mei-Zhen
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• #### 摘要

Fokas系统是最简单的二维空间非线性演化模型. 本文首先研究一种相似变换将该系统转换为长波-短波共振模型形式; 然后基于该相似变换和已知的长波-短波共振模型的有理形式解, 通过选择空间变量y的待定函数为Hermite函数, 得到了Fokas系统的一个有理函数表示的严格解析解; 进而选定合适自由参数给出了Fokas系统丰富的二维怪波激发, 并可对二维怪波的形状和幅度进行有效控制; 最后借助图示展现了二维怪波的传播特征. 本文提出的构造Fokas系统二维怪波的途径可以作为一种激发二维怪波现象的潜在物理机制, 并推广应用于其他(2 + 1)维非线性局域或非局域模型.

#### Abstract

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: ${\rho _0},\;n,\;k,\;{\rm{and}}\;\omega \left( {{\rm{or}}\;\alpha } \right)$. The parameter ${\rho _0}$ controls directly the amplitude of the two-dimensional rogue wave, and the larger the value of ${\rho _0}$, 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 $(x,\;y)$ and $(y,\;t)$ plane depends on merely the parameter n but not on the parameter k. When $n = 0,\;1,\;2, \cdots$, only single peak appears in the $(x,\;t)$ plane, but single peak, two peaks to three peaks appear in the $(x,\;y)$ and $(y,\;t)$ 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 $(x,\;t)$ plane, but the two-dimensional rogue wave appears from the line solitons in the $(x,\;y)$ plane and $(y,\;t)$ 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.

#### 作者及机构信息

###### 通信作者: 张解放, Zhangjief@cuz.edu.cn
• 基金项目: 国家自然科学基金(批准号: 61877053)资助的课题

#### Authors and contacts

###### Corresponding author: Zhang Jie-Fang, Zhangjief@cuz.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61877053)

#### 施引文献

• 图 1  ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 0$, $k = 0$时, 由(22)式所确定的二维怪波激发　(a), (d) $y = 0$时在$\left( {x, t} \right)$平面上的演化图和相应的等高线图; (b), (e) $t = 0$时在$\left( {x, y} \right)$平面上的演化图和相应的等高线图; (c), (f) $x = 0$时在$\left( {y, t} \right)$平面的演化图和相应的等高线图

Fig. 1.  Cross-sections of two-dimensional rogue wave propagations (top row) and contour plots (bottom row) for the density distribution ${\left| {u\left( {x, y, t} \right)} \right|^2}$ given in Eq. (22) for ${\rho _0} = 1.5,\; \alpha =$ ${\left| \omega \right|^{ - 1}} = 1,\; n = 0$ with $k = 0$: (a), (d) $\left( {x, t} \right)$-plan with $y = 0$; (b), (e) $\left( {x, y} \right)$-space with $t = 0$; (c), (f) $\left( {y, t} \right)$-plane with $x = 0$.

图 2  ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 0$, $k = 1$时, 由(22)式所确定的二维怪波激发　(a), (d) $y = 0$时在$\left( {x, t} \right)$平面上的演化图和相应的等高线图; (b), (e) $t = 0$时在$\left( {x, y} \right)$平面上的演化图和相应的等高线图; (c), (f) $x = 0$时在$\left( {y, t} \right)$平面的演化图和相应的等高线图

Fig. 2.  Cross-sections of two-dimensional rogue wave propagations (top row) and contour plots (bottom row) for the density distribution ${\left| {u\left( {x, y, t} \right)} \right|^2}$ given in Eq. (22) for ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 0$ with $k = 1$: (a), (d) $\left( {x, t} \right)$-plane with $y = 0$; (b), (e) $\left( {x, y} \right)$-space with $t = 0$; (c), (f) $\left( {y, t} \right)$-plane with $x = 0$.

图 3  ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 1$, $k = 0$时, 由(22)式所确定的二维怪波激发　(a), (d) $y = 1$时在$\left( {x, t} \right)$平面上的演化图和相应的等高线图; (b), (e) $t = 0$时在$\left( {x, y} \right)$平面上的演化图和相应的等高线图; (c), (f) $x=0$时在$(y, t)$平面的演化图和相应的等高线图

Fig. 3.  Cross-sections of two-dimensional rogue wave propagations (top row) and contour plots (bottom row) for the density distribution ${\left| {u\left( {x, y, t} \right)} \right|^2}$ given in Eq. (22) for ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 1$ with $k = 0$: (a), (d) $\left( {x, t} \right)$-plane with $y = 1$; (b), (e) $\left( {x, y} \right)$-space with $t = 0$; (c), (f) $\left( {y, t} \right)$-plane with $x = 0$.

图 4  ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1$, $n = 1$, $k = 1$时, 由(22)式所确定的二维怪波激发　(a), (d) $y = 1$时在$\left( {x, t} \right)$平面上的演化图和相应的等高线图; (b), (e) $t = 0$时在$\left( {x, y} \right)$平面上的演化图和相应的等高线图; (c), (f) $x = 0$时在$\left( {y, t} \right)$平面的演化图和相应的等高线图

Fig. 4.  Cross-sections of two-dimensional rogue wave propagations (top row) and contour plots (bottom row) for the density distribution ${\left| {u\left( {x, y, t} \right)} \right|^2}$ given in Eq. (22) for ${\rho _0} = 1.5$, $\alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 1$ with $k = 1$: (a), (d) $\left( {x, t} \right)$-plane with $y = 1$; (b), (e) $\left( {x, y} \right)$-space with $t = 0$; (c), (f) $\left( {y, t} \right)$-plane with $x = 0$.

图 5  ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 2,\; k = 2$时, 由(22)式所确定的二维怪波激发　(a), (d) $y = 0$时在$\left( {x, t} \right)$平面上的演化图和相应的等高线图; (b), (e) $t = 0$时在$\left( {x, y} \right)$平面上的演化图和相应的等高线图; (c), (f) $x = 0$时在$\left( {y, t} \right)$平面的演化图和相应的等高线图

Fig. 5.  Cross-sections of two-dimensional rogue wave propagations (top row) and contour plots (bottom row) for the density distribution ${\left| {u\left( {x, y, t} \right)} \right|^2}$ given in Eq. (22) for ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 2$ with $k = 2$: (a), (d) $\left( {x, t} \right)$-plane with $y = 0$; (b), (e) $\left( {x, y} \right)$-space with $t = 0$; (c), (f) $\left( {y, t} \right)$-plane with $x = 0$.

图 6  由(22)式确定的二维怪波在$\left( {x, t} \right)$平面上的演化(这里取${\rho _0}{{ = 1}}{{.5}}$, $\alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 0$, $k = 3$)　(a) $t = - 7$; (b) $t = - 3$; (c) $t = 0$; (d)$t = 0.5$; (e) $t = 5$; (f) $t = 10$

Fig. 6.  Cross-sections of two-dimensional wave propagations (top row) for the density distribution ${\left| {u\left( {x, y, t} \right)} \right|^2}$ given in Eq. (22) for ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 0$ with $k = 3$: (a) $t = - 7$; (b) $t = - 3$; (c) $t = 0$; (d) $t = 0.5$; (e) $t = 5$; (f) $t = 10$.

图 7  由(22)式确定的二维怪波在$\left( {x, t} \right)$平面上的演化(这里取${\rho _0}{{ = 1}}{{.5}}$, $\alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 0$, $k = 3$)　(a) $y = - 3$; (b) $y = -1$; (c) $y = 0$; (d) $y = 0.5$; (e) $y = 2$; (f) $y = 3$

Fig. 7.  Cross-sections of two-dimensional rogue wave propagations (top row) for the density distribution ${\left| {u\left( {x, y, t} \right)} \right|^2}$ given in Eq. (22) for ${\rho _0} = 1.5,\; \alpha = {\left| \omega \right|^{ - 1}} = 1,\; n = 0$ with $k = 3$: (a) $y = - 3$; (b) $y = - 1$; (c) $y = 0$; (d) $y = 0.5$; (e) $y = 2$; (f) $y = 3$.