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The Analytical Solution and Application of a Coupled Shallow Water Wave Equation
DOI: 10.12677/AAM.2022.115268, PDF, HTML, XML, 下载: 41  浏览: 63  国家自然科学基金支持

Abstract: In this paper, solutions of a set of coupled nonlinear shallow water wave equations are solved based on the analytical method, and the results are discussed. The behavior of the equation solution un-der different parameters is studied. We concluded that the dispersion coefficient can significantly affect the dynamics of the shallow water wave. Based on the equations given in this paper, some ex-tended applications under special parameters are discussed. The result presented in this paper may give insights to the relevant studies of the shallow water wave equation.

1. 引言

${\eta }_{t}+{\left(u\eta \right)}_{x}+\alpha {u}_{xxx}-\beta {\eta }_{xx}=0,$ (1a)

${u}_{t}+{\eta }_{x}+u{u}_{x}+\beta {u}_{xx}=0,$ (1b)

2. 方法介绍

$P\left(u,{u}_{t},{u}_{x},{u}_{xx},{u}_{tt},{u}_{xt},\cdots \right)=0.$ (2)

$\xi =kx-\omega t,$ (3)

$P\left(u,{u}^{\prime },{u}^{″},\cdots \right)=0.$ (4)

$u\left(\xi \right)=\underset{n=0}{\overset{N}{\sum }}{a}_{n}{Q}^{n}\left(\xi \right),$ (5)

$Q\left(\xi \right)=\frac{1}{1+q{A}^{\xi }}.$ (6)

${Q}^{\prime }=\left({Q}^{2}-Q\right)\mathrm{ln}A.$ (7)

$\text{Deg}\left[{u}^{m}\left(\xi \right){\left(\frac{{\text{d}}^{s}{u}^{r}\left(\xi \right)}{\text{d}{\xi }^{s}}\right)}^{q}\right]=np+q\left(mr+s\right).$ (8)

$P\left(Q\left(\xi \right)\right)=0.$ (9)

3. 浅水波的行波解

$-\omega {u}^{\prime }+ku{u}^{\prime }+\beta {k}^{2}{u}^{″}+k{\eta }^{\prime }=0,$ (10a)

$-\omega {\eta }^{\prime }-\beta {k}^{2}{\eta }^{″}+k{\left(u\eta \right)}^{\prime }+\alpha {k}^{3}{u}^{‴}=0,$ (10b)

$-\omega u+\frac{1}{2}k{u}^{2}+\beta {k}^{2}{u}^{\prime }+k\eta +C=0.$ (11)

$\eta =\frac{\omega }{k}u-\frac{1}{2}{u}^{2}-\beta k{u}^{\prime }+C.$ (12)

$\left(\alpha +{\beta }^{2}\right){k}^{3}{u}^{‴}-\frac{3}{2}k{u}^{2}{u}^{\prime }+3\omega u{u}^{\prime }+\left(Ck-\frac{{\omega }^{2}}{k}\right){u}^{\prime }=0.$ (13)

$6{k}^{3}\left(\alpha +{\beta }^{2}\right){\mathrm{ln}}^{3}A{a}_{1}-\frac{3}{2}k\mathrm{ln}A{a}_{1}=0,$ (14a)

$\frac{3}{2}k\mathrm{ln}A{a}_{1}^{3}+3\left(\omega -k{a}_{0}\right)\mathrm{ln}A{a}_{1}^{2}-12\left(\alpha +{\beta }^{2}\right){k}^{3}{\mathrm{ln}}^{3}A{a}_{1}=0,$ (14b)

$3\left(k{a}_{0}-\omega \right)\mathrm{ln}A{a}_{1}^{2}+7\left(\alpha +{\beta }^{2}\right){k}^{3}{\mathrm{ln}}^{3}A{a}_{1}+3\left(\omega -\frac{k}{2}{a}_{0}^{2}\right){a}_{1}+\left(Ck-\frac{{\omega }^{2}}{k}\right)\mathrm{ln}A{a}_{1}=0,$ (14c)

$\left(\alpha +{\beta }^{2}\right){k}^{3}\mathrm{ln}A{a}_{1}-3\left(\omega -\frac{k}{2}{a}_{0}^{2}\right)\mathrm{ln}A{a}_{1}+\left(Ck-\frac{{\omega }^{2}}{k}\right)\mathrm{ln}A{a}_{1}=0.$ (14d)

${a}_{0}=\frac{-2\left(\alpha +{\beta }^{2}\right)k\mathrm{ln}A+{R}_{l}{R}_{s}}{{R}_{s}},$ (15a)

${a}_{1}=k\mathrm{ln}A{R}_{s},$ (15b)

$k=k,$ (15c)

$\omega =k{R}_{l}.$ (15d)

${u}_{1}=\frac{\sqrt{\left(\alpha +2{\beta }^{2}\right)\left[\left(\alpha +{\beta }^{2}\right){k}^{2}{\mathrm{ln}}^{2}A-2C\right]}}{\sqrt{\alpha +{\beta }^{2}}}-\frac{\left(\alpha +{\beta }^{2}\right)k\mathrm{ln}A}{\sqrt{\alpha +{\beta }^{2}}}+\frac{2\sqrt{\alpha +{\beta }^{2}}k\mathrm{ln}A}{1+q{A}^{\xi }},$ (16a)

${\eta }_{1}=\frac{2\left(\alpha +{\beta }^{2}+\beta \sqrt{\alpha +{\beta }^{2}}\right){k}^{2}q{A}^{\xi }{\mathrm{ln}}^{2}A}{{\left(1+q{A}^{\xi }\right)}^{2}}.$ (16b)

${u}_{2}=\frac{-\sqrt{\left(\alpha +2{\beta }^{2}\right)\left[\left(\alpha +{\beta }^{2}\right){k}^{2}{\mathrm{ln}}^{2}A-2C\right]}}{\sqrt{\alpha +{\beta }^{2}}}+\frac{\left(\alpha +{\beta }^{2}\right)k\mathrm{ln}A}{\sqrt{\alpha +{\beta }^{2}}}-\frac{2\sqrt{\alpha +{\beta }^{2}}k\mathrm{ln}A}{1+q{A}^{\xi }},$ (17a)

${\eta }_{2}=\frac{2\left(\alpha +{\beta }^{2}-\beta \sqrt{\alpha +{\beta }^{2}}\right){k}^{2}q{A}^{\xi }{\mathrm{ln}}^{2}A}{{\left(1+q{A}^{\xi }\right)}^{2}}.$ (17b)

${u}_{3}=\frac{-\sqrt{\left(\alpha +2{\beta }^{2}\right)\left[\left(\alpha +{\beta }^{2}\right){k}^{2}{\mathrm{ln}}^{2}A-2C\right]}}{\sqrt{\alpha +{\beta }^{2}}}-\frac{\left(\alpha +{\beta }^{2}\right)k\mathrm{ln}A}{\sqrt{\alpha +{\beta }^{2}}}+\frac{2\sqrt{\alpha +{\beta }^{2}}k\mathrm{ln}A}{1+q{A}^{\xi }},$ (18a)

${\eta }_{3}=\frac{2\left(\alpha +{\beta }^{2}+\beta \sqrt{\alpha +{\beta }^{2}}\right){k}^{2}q{A}^{\xi }{\mathrm{ln}}^{2}A}{{\left(1+q{A}^{\xi }\right)}^{2}}.$ (18b)

${u}_{4}=\frac{-\sqrt{\left(\alpha +2{\beta }^{2}\right)\left[\left(\alpha +{\beta }^{2}\right){k}^{2}{\mathrm{ln}}^{2}A-2C\right]}}{\sqrt{\alpha +{\beta }^{2}}}+\frac{\left(\alpha +{\beta }^{2}\right)k\mathrm{ln}A}{\sqrt{\alpha +{\beta }^{2}}}-\frac{2\sqrt{\alpha +{\beta }^{2}}k\mathrm{ln}A}{1+q{A}^{\xi }},$ (19a)

${\eta }_{4}=\frac{2\left(\alpha +{\beta }^{2}-\beta \sqrt{\alpha +{\beta }^{2}}\right){k}^{2}q{A}^{\xi }{\mathrm{ln}}^{2}A}{{\left(1+q{A}^{\xi }\right)}^{2}}.$ (19b)

(a) (b)

Figure 1. The evolutions of velocity field $u\left(x,t\right)$ and $\eta \left(x,t\right)$ for solution groups 1(a) and 2(b). Here $\alpha =2$, $\beta =2$, $A=2$, $q=2$, $C=1/2$ and $k=1$ are chosen

4. 讨论

Figure 2. The cross-section at t = 0 of the horizontal velocity $u\left(x,t\right)$ and height deviation $\eta \left(x,t\right)$ for solutions with ${R}_{s}>0$ (Group 1) and ${R}_{s}<0$ (Group 4), respectively. The left two panels are for case 1 and right two panels for case 2 with $\alpha =2$, $\beta =2$, $C=1/2$ and $k=1$. Solid line ( $A=3,q=1$ ), dashed line ( $A=2,q=2$ ) and dotted line ( $A=5/2,q=4$ ) are shown here

(a) (b)

Figure 3. The evolution behavior of $u\left(x,t\right)$ and $\eta \left(x,t\right)$ for classical long wave equation if $\alpha =0$ (a) and variant Boussinesq equation if $\alpha =1$ and $\beta =0$ (b)

5. 总结

NOTES

*第一作者。

#通讯作者。

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