# (2 + 1)维Hirota-Satsuma-Ito方程的呼吸波与不同非线性波之间的态转换State Transformations between the Breather Wave and Different Nonlinear Waves for the (2 + 1)-Dimensional Hirota-Satsuma-Ito Equation

Abstract: In this paper, the (2 + 1)-dimensional Hirota-Satsuma-Ito equation is studied based on the Hirota bilinear method. Under certain circumstances, the breather wave can be transformed into other types of nonlinear waves, including W-type, M-type, oscillating-W-type, oscillating-M-type and quasi-periodic-type waves, and the dynamic characteristics of these nonlinear waves are analyzed. Based on the characteristic line analysis, the conversion condition between the breather wave and other nonlinear waves is obtained. The results enrich the dynamic characteristics of the (2 + 1)-dimensional nonlinear waves.

1. 引言

${u}_{x}\left(3\int {u}_{t}\text{d}x+\gamma \right)+3u{u}_{t}+{u}_{xxt}+\int {u}_{yt}\text{d}x=0,$ (1)

2. 呼吸波解和态转换条件

$u=2{\left[\mathrm{ln}f\left(x,y,t\right)\right]}_{xx},$ (2)

$\left({D}_{t}{D}_{x}^{3}+\gamma {D}_{x}^{2}+{D}_{t}{D}_{y}\right)f\cdot f=0,$ (3)

${D}_{x}^{p}{D}_{y}^{q}f\cdot g={\left(\frac{\partial }{\partial x}-\frac{\partial }{\partial {x}^{\prime }}\right)}^{p}{\left(\frac{\partial }{\partial y}-\frac{\partial }{\partial {y}^{\prime }}\right)}^{q}f\left(x,y\right)\cdot g\left({x}^{\prime },{y}^{\prime }\right)|{}_{\left(x,y\right)=\left({x}^{\prime },{y}^{\prime }\right)}.$ (4)

${f}_{2}=1+{\text{e}}^{{\theta }_{1}}+{\text{e}}^{{\theta }_{2}}+{\text{e}}^{{\theta }_{1}+{\theta }_{2}}{d}_{12}.$ (5)

${\theta }_{i}={k}_{i}x+{l}_{i}y+{\omega }_{i}t+{\delta }_{i},\left(i=1,2\right)$，其中

${\omega }_{i}=-\frac{\gamma {k}_{i}^{2}}{{k}_{i}^{3}+{l}_{i}},\left(i=1,2\right),{d}_{12}=-\frac{\left({\omega }_{1}-{\omega }_{2}\right){\left({k}_{1}-{k}_{2}\right)}^{3}+\left({\omega }_{1}-{\omega }_{2}\right)\left({l}_{1}-{l}_{2}\right)+\gamma {\left({k}_{1}-{k}_{2}\right)}^{2}}{\left({\omega }_{1}+{\omega }_{2}\right){\left({k}_{1}+{k}_{2}\right)}^{3}+\left({\omega }_{1}+{\omega }_{2}\right)\left({l}_{1}+{l}_{2}\right)+\gamma {\left({k}_{1}+{k}_{2}\right)}^{2}},$ (6)

${k}_{i},{l}_{i},{\delta }_{i}\left(i=1,2\right)$ 是一些自由参量，我们对这些参量进行复数化，即：

${k}_{1}={a}_{1}+i{b}_{1}={k}_{2}^{*},{l}_{1}={c}_{1}+i{d}_{1}={l}_{2}^{*},{\delta }_{1}=\mathrm{ln}\frac{{\lambda }_{1}}{2}+{\beta }_{1}+i{\eta }_{1}={\delta }_{2}^{*},$ (7)

*代表共轭， ${a}_{1},{b}_{1},{c}_{1},{d}_{1},{\lambda }_{1}\left(>0\right),{\beta }_{1},{\eta }_{1}$ 都是实常数。将式(6)和(7)代入到式(5)中，可以得到

${f}_{2}~{\lambda }_{1}\mathrm{cos}\left({\Lambda }_{1}\right)+2\sqrt{{\lambda }_{2}}\mathrm{cosh}\left({\xi }_{1}+\frac{1}{2}\mathrm{ln}{\lambda }_{2}\right),$ (8)

${\xi }_{1}={a}_{1}x+{c}_{1}y+{\omega }_{1R}t+{\beta }_{1},{\Lambda }_{1}={b}_{1}x+{d}_{1}y+{\omega }_{1I}t+{\eta }_{1},{\lambda }_{2}=\frac{1}{4}{\lambda }_{1}^{2}{d}_{12},$ (9)

$\begin{array}{l}{\omega }_{1R}=-\frac{\gamma \left({a}_{1}^{5}+2{a}_{1}^{3}{b}_{1}^{2}+{a}_{1}^{2}{c}_{1}-{b}_{1}^{2}{c}_{1}+{a}_{1}\left({b}_{1}^{4}+2{b}_{1}{d}_{1}\right)\right)}{{\left({a}_{1}^{3}-3{a}_{1}{b}_{1}^{2}+{c}_{1}\right)}^{2}+{\left(3{a}_{1}^{2}{b}_{1}-{b}_{1}^{3}+{d}_{1}\right)}^{2}},\\ {\omega }_{1I}=\frac{\gamma \left({a}_{1}^{4}{b}_{1}+{b}_{1}^{5}-2{a}_{1}{b}_{1}{c}_{1}-{b}_{1}^{2}{d}_{1}+{a}_{1}^{2}\left(2{b}_{1}^{3}+{d}_{1}\right)\right)}{{\left({a}_{1}^{3}-3{a}_{1}{b}_{1}^{2}+{c}_{1}\right)}^{2}+{\left(3{a}_{1}^{2}{b}_{1}-{b}_{1}^{3}+{d}_{1}\right)}^{2}}.\end{array}$ (10)

$\begin{array}{c}u\left[x,y,t\right]=\frac{8{a}_{1}{b}_{1}{\lambda }_{1}\sqrt{{\lambda }_{2}}\mathrm{sin}\left({\Lambda }_{1}\right)\mathrm{sinh}\left({\xi }_{1}+\frac{1}{2}\mathrm{ln}{\lambda }_{2}\right)+8{a}_{1}^{2}{\lambda }_{2}-2{b}_{1}^{2}{\lambda }_{1}^{2}}{{\left[{\lambda }_{1}\mathrm{cos}\left({\Lambda }_{1}\right)+2\sqrt{{\lambda }_{2}}\mathrm{cosh}\left({\xi }_{1}+\frac{1}{2}\mathrm{ln}{\lambda }_{2}\right)\right]}^{2}}\\ \text{\hspace{0.17em}}+\frac{4\left({a}_{1}^{2}-{b}_{1}^{2}\right){\lambda }_{1}\sqrt{{\lambda }_{2}}\mathrm{cos}\left({\Lambda }_{1}\right)\mathrm{cosh}\left({\xi }_{1}+\frac{1}{2}\mathrm{ln}{\lambda }_{2}\right)}{{\left[{\lambda }_{1}\mathrm{cos}\left({\Lambda }_{1}\right)+2\sqrt{{\lambda }_{2}}\mathrm{cosh}\left({\xi }_{1}+\frac{1}{2}\mathrm{ln}{\lambda }_{2}\right)\right]}^{2}}.\end{array}$ (11)

$\begin{array}{l}{L}_{1}={a}_{1}x+{c}_{1}y+{\omega }_{1R}t+{\beta }_{1}+\frac{1}{2}\mathrm{ln}{\lambda }_{2}=0,\\ {L}_{2}={b}_{1}x+{d}_{1}y+{\omega }_{1I}t+{\eta }_{1}=0.\end{array}$ (12)

${a}_{1}{d}_{1}-{b}_{1}{c}_{1}=0,$ (13)

3. 转换波的类型

3.1. W型、M型

(a) (b) (c) (d) (e) (f)

Figure 1. Converted waves at different times: (a) t = −19; (b) t = 4.35; (c) t = 21.06; (d), (e) and (f) are the corresponding density maps respectively

Figure 2. Sectional view of three moments

3.2. 振荡W型、振荡M型

(a) (b) (c) (d) (e) (f)

Figure 3. Converted waves at different times: (a) t = −58.6; (b) t = 20.95; (c) t = 92.9; (d), (e) and (f) are the corresponding density maps respectively

Figure 4. Sectional view of three moments

3.3. 准周期型

(a) (b) (c) (d)

Figure 5. Converted waves at different times: (a) t = 37; (b) t = 45; (c) t = 92.9; (c) and (d) are the corresponding density maps respectively

Figure 6. Sectional view of two moments

4. 结论

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