﻿ (2 + 1)维变系数Broer-Kaup方程的精确解与局域激发

# (2 + 1)维变系数Broer-Kaup方程的精确解与局域激发Exact Solution and Local Excitation of (2 + 1)-Dimensional Variable Coefficient Broer-Kaup Equation

Abstract: The extended Riccati mapping method is used to analyze the 2 + 1 dimensional variable coefficient Broer-Kaup equation, and the exact solution containing any function is obtained. By selecting the arbitrary function, the relevant local excitation structure is obtained. This method is also widely used in solving other nonlinear partial differential equations.

1. 引言

${u}_{yt}-\alpha \left(t\right)\left[{u}_{xxy}-2{\left(u{u}_{x}\right)}_{y}-2{v}_{xx}\right]=0$

${v}_{t}+\alpha \left(t\right)\left[{v}_{xx}+2{\left(uv\right)}_{x}\right]=0$ (1.1)

2. 拓展的Riccati展开法解法概述

$P\left(u,{u}_{t},{u}_{x},{u}_{xx}{u}_{xt},\cdots \right)$ (2.1)

$u=\underset{i=-n}{\overset{n}{\sum }}{a}_{i}\left(x\right){\varphi }^{i}\left(q\right)$ (2.2)

${\varphi }^{\prime }=\sigma +{\varphi }^{2}$ (2.3)

(2.3)解的情况如下

$\begin{array}{l}\varphi =\sqrt{\sigma }\mathrm{tan}\left(\sqrt{\sigma }q\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma >0\\ \varphi =-\sqrt{\sigma }\mathrm{cot}\left(\sqrt{\sigma }q\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma >0\end{array}$

$\begin{array}{l}\varphi =-\sqrt{-\sigma }\mathrm{tanh}\left(\sqrt{-\sigma }q\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma <0\\ \varphi =-\sqrt{-\sigma }\mathrm{coth}\left(\sqrt{-\sigma }q\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma <0\end{array}$

$\varphi =-1/q\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sigma =0$ (2.4)

3. 2 + 1维变系数BK方程的精确解

$v={u}_{y}$ (3.1)

${u}_{yt}+\alpha \left(t\right)\left[2{\left(u{u}_{x}\right)}_{y}+{u}_{xxy}\right]=0$ (3.2)

$u=f+g\varphi \left(q\right)+h{\varphi }^{-1}\left(q\right)$ (3.3)

$f=-\frac{\alpha \left(t\right){q}_{xx}+{q}_{t}}{2{q}_{x}\alpha \left(t\right)}$$g=-{q}_{x}$$h={q}_{x}\sigma$ (3.4)

$q\left(x,y,t\right)=\chi \left(x,t\right)+\phi \left(y\right)$ (3.5)

${u}_{1}=-\frac{\alpha \left(t\right){\chi }_{xx}+{\chi }_{t}}{2{\chi }_{x}\alpha \left(t\right)}+{\chi }_{x}\cdot \sqrt{-\sigma }\mathrm{tanh}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)-{\chi }_{x}\sigma \cdot {\left(\sqrt{-\sigma }\mathrm{tanh}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)\right)}^{-1}$ (3.6)

${v}_{1}=-{\chi }_{x}\sigma {\phi }^{\prime }\left({\text{sech}}^{2}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)-\frac{{\text{sech}}^{2}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)}{{\mathrm{tanh}}^{2}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)}\right)$ (3.7)

${u}_{2}=-\frac{\alpha \left(t\right){\chi }_{xx}+{\chi }_{t}}{2{\chi }_{x}\alpha \left(t\right)}+{\chi }_{x}\cdot \sqrt{-\sigma }\cdot \mathrm{coth}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)-{\chi }_{x}\sigma \cdot {\left(\sqrt{-\sigma }\cdot \mathrm{coth}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)\right)}^{-1}$ (3.8)

${v}_{2}={\chi }_{x}\sigma {\phi }^{\prime }\left({\text{csch}}^{2}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)-\frac{{\text{csch}}^{2}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)}{{\mathrm{coth}}^{2}\left(\sqrt{-\sigma }\left(\chi +\phi \right)\right)}\right)$ (3.9)

${u}_{3}=-\frac{\alpha \left(t\right){\chi }_{xx}+{\chi }_{t}}{2{\chi }_{x}\alpha \left(t\right)}-{\chi }_{x}\cdot \sqrt{\sigma }\mathrm{tan}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)+{\chi }_{x}\sigma \cdot {\left(\sqrt{\sigma }\mathrm{tan}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)\right)}^{-1}$ (3.10)

${v}_{3}=-{\chi }_{x}\sigma {\phi }^{\prime }\left({\mathrm{sec}}^{2}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)+\frac{{\mathrm{sec}}^{2}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)}{{\mathrm{tan}}^{2}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)}\right)$ (3.11)

${u}_{4}=-\frac{\alpha \left(t\right){\chi }_{xx}+{\chi }_{t}}{2{\chi }_{x}\alpha \left(t\right)}+{\chi }_{x}\cdot \sqrt{\sigma }\mathrm{cot}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)-{\chi }_{x}\sigma \cdot {\left(\sqrt{\sigma }\mathrm{cot}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)\right)}^{-1}$ (3.12)

${v}_{4}=-{\chi }_{x}\sigma {\phi }^{\prime }\left({\mathrm{csc}}^{2}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)+\frac{{\mathrm{csc}}^{2}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)}{{\mathrm{cot}}^{2}\left(\sqrt{\sigma }\left(\chi +\phi \right)\right)}\right)$ (3.13)

${u}_{5}=-\frac{\alpha \left(t\right){\chi }_{xx}+{\chi }_{t}}{2{\chi }_{x}\alpha \left(t\right)}+{\chi }_{x}\cdot \frac{1}{\chi +\phi }-{\chi }_{x}\sigma \cdot \left(\chi +\phi \right)$ (3.14)

${v}_{5}=-{\chi }_{x}\cdot \frac{{\phi }^{\prime }}{{\left(\chi +\phi \right)}^{2}}-{\chi }_{x}\sigma \cdot {\phi }^{\prime }$ (3.15)

4. 局域激发与分形结构

$\chi \left(x,t\right)=-{x}^{2}+{t}^{2}$$\phi \left(y\right)=-{y}^{2}$ (3.16)

Figure 1. The ring soliton structure is obtained when $t=4$, $\sigma =1$

Figure 2. Take $t=0$, $\sigma =1$ Take to get the light and dark domion

Figure 3. Figure 2 takes $x,y\in \left[-1×{10}^{-6},1×{10}^{-6}\right]$

Figure 4. Figure 2 takes $x,y\in \left[-1×{10}^{-9},1×{10}^{-9}\right]$

5. 结论

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