﻿ KdV与BBM方程孤立波完全相互作用近似解析研究
 舰船科学技术  2022, Vol. 44 Issue (5): 76-79    DOI: 10.3404/j.issn.1672-7649.2022.05.015 PDF
KdV与BBM方程孤立波完全相互作用近似解析研究

1. 中国石油大学(北京) 石油工程学院，北京 102249;
2. 华中科技大学 船舶与海洋工程学院，湖北 武汉 430074;
3. 内蒙古工业大学 理学院，内蒙古 呼和浩特 010051

Analytical study on the interaction of solitary waves between KdV and BBM equation
YOU Xiang-cheng1, LIU Zeng2, CUI Ji-feng3
1. College of Petroleum Engineering, China University of Petroleum-Beijing, Beijing 102249, China;
2. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
3. College of Sciences, Inner Mongolia University of Technology, Hohhot, 010051
Abstract: This paper uses the conserved quantities of mass, momentum and energy to approximate and analytically study the complete interaction of solitary waves in KdV equation and BBM equation. The conserved quantities are applied to calculate the merged wave without solving nonlinear partial differential equations. The numerical results of KdV equation and BBM equation and the analytical results of approximate merged wave are comprehensively compared. It is considered that using the conserved quantities to study the solitary wave interaction has a good approximate analytical model and a small amount of calculation. The results show that this method has good engineering approximation accuracy and maybe suitable for predicting the interaction between waves and ocean, coastal and canal structures.
Key words: KdV equation     BBM equation     solitary wave     interaction     conserved quantities
0 引　言

1 数学模型

 ${u_t} + {u_x} + u{u_x} + {u_{xxx}} = 0 ,$ (1)
 ${u_t} + {u_x} + u{u_x}{\text{ - }}{u_{xxt}} = 0。$ (2)

KdV方程和BBM的孤立波解可以通过应用一般的孤立波形状来求得：

 ${u_{{\text{KdV}}}} = a{\text{sec}}{{\text{h}}^{\text{2}}}\left[ {\sqrt {\frac{a}{{12}}} \left( {x - ct} \right)} \right],$ (3)
 ${u_{{\text{BBM}}}} = a{\text{sec}}{{\text{h}}^{\text{2}}}\left[ {\sqrt {\frac{a}{{12}}} \frac{{\left( {x - ct} \right)}}{{\sqrt c }}} \right]。$ (4)

KdV方程和BBM方程的质量，动量和能量守恒量[18]可分别用下式计算：

 ${I_1} = \int_{ - \infty }^{ + \infty } {u{\rm{d}}x},$ (5)
 ${I_2} = \int_{ - \infty }^{ + \infty } {{u^2}dx} ,{I_2} = \int_{ - \infty }^{ + \infty } {{u^2} + u_x^2{\rm{d}}x},$ (6)
 ${I_3} = \int_{ - \infty }^{ + \infty } {{u^3}{\text{ - 3}}u_x^2{\rm{d}}x}。$ (7)

2 计算结果与讨论

2个输入孤立波的振幅为 ${a_1}$ ${a_2}$ ，如表1所示。KdV方程2个输入孤立波 ${a_1} = 6.75$ ${a_2} = 1.687\;50$ ，BBM方程2个输入孤立波 ${a_1} = 6.75$ ${a_2} = 0.944\;50$ 相互作用三维图形如图1所示。可以看出，双孤立波完全相互作用最大合并瞬间，KdV方程最大合并波高要低于BBM方程最大合并波高。而KdV方程和BBM方程空间剖面分布如图2所示。如图2(a)所示，当 $t = 0{\text{s}}$ 时，左侧的2条黑色实线代表KdV方程输入的2个孤立波，点线是当 $t = 13{\text{s}}$ 时的合并形状，当 $t = 25{\text{s}}$ 时点划线表示KdV方程输入孤立波完全相互作用后输出的孤立波。当然，振幅较高的输入波比振幅较小的输入波传播得快得多，输出波也是如此。如图2(b)所示，当 $t = 0\;{\text{s}}$ 时，左侧的2条黑色实线代表BBM方程输入的2个孤立波，点线是当 $t = 11\;{\text{s}}$ 时的合并形状，当 $t = 25\;{\text{s}}$ 时点划线表示BBM方程输入孤立波近似完全相互作用后输出的孤立波。从图2可以看出，BBM相互作用比KdV相互作用发生的快，例如当 $t = 9\;{\text{s}}$ 时虚线波形。合并的波形 $U\left( x \right)$ 可定义为：

 图 1 双孤立波完全相互作用三维图 $({a_1} = 6.75)$ Fig. 1 Three dimensional diagram of perfect merging of two solitons $({a_1} = 6.75)$

 图 2 双孤立波完全相互作用的空间剖面 $({a_1} = 6.75)$ Fig. 2 Spatial profiles of perfect merging of two solitons $({a_1} = 6.75)$
 $U\left( x \right) = A{\text{sec}}{{\text{h}}^{\text{2}}}\left( {Bx} \right) 。$ (8)

 ${I_{1{\text{KdV}}}} = 4\sqrt 3 \left( {{a_1}^{1/2} + {a_2}^{1/2}} \right) = \frac{{2A}}{B},$ (9)
 ${I_{2{\text{KdV}}}} = \frac{8}{{\sqrt 3 }}\left( {{a_1}^{3/2} + {a_2}^{3/2}} \right) = \frac{{4{A^2}}}{{3B}} ,$ (10)
 $\begin{split}{I_{3{\text{KdV}}}} = & \frac{{8\sqrt 3 }}{5}\left( {{a_1}^{5/2} + {a_2}^{5/2}} \right) =\\& \frac{{16{A^2}\left( {A - 3{B^2}} \right)}}{{15B}}, \end{split}$ (11)
 ${I_{1{\text{BBM}}}} = 4\left( {{a_1}\sqrt {1 + \frac{3}{{{a_1}}}} + {a_2}\sqrt {1 + \frac{3}{{{a_2}}}} } \right) = \frac{{2A}}{B},$ (12)
 $\begin{split} {I_{2{\text{BBM}}}} = &\frac{{16}}{5}\left[ {\frac{{a_1^2\left( {5 + 2{a_1}} \right)}}{{6 + 2{a_1}}}\sqrt {1 + \frac{3}{{{a_1}}}} + \frac{{a_2^2\left( {5 + 2{a_2}} \right)}}{{6 + 2{a_2}}}\sqrt {1 + \frac{3}{{{a_2}}}} } \right] =\\& \frac{{4{A^2}\left( {5 + 4{B^2}} \right)}}{{15B}}, \\[-15pt]\end{split}$ (13)
 $\begin{split}{I_{3{\text{BBM}}}} = & \frac{{16}}{{15}}\left[ {\frac{{a_1^3\left( {9 + 4{a_1}} \right)}}{{6 + 2{a_1}}}\sqrt {1 + \frac{3}{{{a_1}}}} + \frac{{a_2^3\left( {9 + 4{a_2}} \right)}}{{6 + 2{a_2}}}\sqrt {1 + \frac{3}{{{a_2}}}} } \right] =\\& \frac{{16{A^2}\left( {A - 3{B^2}} \right)}}{{15B}}。\\[-15pt]\end{split}$ (14)

2个输入孤立波的振幅 ${a_2}$ $A$ $B$ 3个参数是未知的，用Mathematica或者Maple软件就可以求解2个方程，然后得到相对应的解。合并波的不变量与两个输入波的总和进行比较，在所有情况下，都使用类似的方法来求解合并波参数的联立方程，但是当组合形状由更多参数定义或者输入孤立波未知时，求解将变得更加复杂困难。已知一个输入孤立波振幅 ${a_1}$ ，根据质量（ ${I_1}$ ），动量（ ${I_2}$ ）和能量（ ${I_3}$ ）守恒量等式进行计算得到，KdV方程和BBM方程的完全合并波形数据，如表1所示，可以找到未知的输入波 ${a_2}$ ，从而在不损失质量、动量或能量的情况下，得到近似完全相互作用的合并孤立波形。当 ${a_1} = 6.75$ 时，KdV方程的完全合并波形为 ${U_{{\text{KdV}}}} = {\text{5}}{\text{.10972}}sec{h^2}\left( {0.35850x} \right)$ 。BBM方程的完全合并波形为 ${U_{{\text{BBM}}}} = {\text{5}}{\text{.97487}}sec{h^2}\left( {0.37010x} \right)$ 。KdV方程和BBM方程数值计算结果和合并波形计算结果，如表2所示，绝对误差非常小，说明合并形状近似非常好。同时KdV方程拟合精度要高于BBM方程拟合精度。KdV完全合并波形 ${U_{{\text{KdV}}}}$ 与BBM完全合并波形 ${U_{{\text{BBM}}}}$ ，如图3所示，在靠近波高部分差值较大，最大波高处差值为0.865151。

 图 3 KdV与BBM双孤立波相互作用完全合并波形 $({a_1} = 6.75)$ Fig. 3 KdV and BBM perfect merged wave shape $({a_1} = 6.75)$
3 结　语

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