**正则长波方程**(Regularized long wave equation)是一个非线性偏微分方程：

{\displaystyle u\_{t}+u\_{x}+\alpha \*u\*u\_{x}-\mu \*u\_{txx}=0;}

当 α=1，μ=1，正则长波方程即本杰明-博纳-马奥尼方程。

解析解
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{\displaystyle {u(x,t)=-(\_{C}5+\_{C}4)/(\alpha \*\_{C}4)+12\*\mu \*\_{C}5\*\_{C}4\*WeierstrassP(\_{C}3+\_{C}4\*x+\_{C}5\*t,\_{C}2,\_{C}1)/\alpha }}
{\displaystyle {u(x,t)=(4\*\mu \*\_{C}3\*\_{C}2^{2}-\_{C}3-\_{C}2)/(\alpha \*\_{C}2)+12\*\mu \*\_{C}3\*\_{C}2\*csch(\_{C}1+\_{C}2\*x+\_{C}3\*t)^{2}/\alpha }}
{\displaystyle {u(x,t)=(4\*\mu \*\_{C}3\*\_{C}2^{2}-\_{C}3-\_{C}2)/(\alpha \*\_{C}2)-12\*\mu \*\_{C}3\*\_{C}2\*sech(\_{C}1+\_{C}2\*x+\_{C}3\*t)^{2}/\alpha }}
{\displaystyle {u(x,t)=-(4\*\mu \*\_{C}3\*\_{C}2^{2}+\_{C}3+\_{C}2)/(\alpha \*\_{C}2)+12\*\mu \*\_{C}3\*\_{C}2\*csc(\_{C}1+\_{C}2\*x+\_{C}3\*t)^{2}/\alpha }}
{\displaystyle {u(x,t)=(-4\*\mu \*\_{C}4\*\_{C}3^{2}-\_{C}4-\_{C}3+8\*\mu \*\_{C}4\*\_{C}3^{2}\*\_{C}1^{2})/(\alpha \*\_{C}3)-12\*\mu \*\_{C}4\*\_{C}3\*\_{C}1^{2}\*JacobiCN(\_{C}2+\_{C}3\*x+\_{C}4\*t,\_{C}1)^{2}/\alpha }}
{\displaystyle {u(x,t)=(-4\*\mu \*\_{C}4\*\_{C}3^{2}-\_{C}4-\_{C}3+8\*\mu \*\_{C}4\*\_{C}3^{2}\*\_{C}1^{2})/(\alpha \*\_{C}3)-12\*\mu \*\_{C}4\*\_{C}3\*(-1+\_{C}1^{2})\*JacobiNC(\_{C}2+\_{C}3\*x+\_{C}4\*t,\_{C}1)^{2}/\alpha }}
{\displaystyle {u(x,t)=-(4\*\mu \*\_{C}4\*\_{C}3^{2}\*\_{C}1^{2}+\_{C}4+\_{C}3-8\*\mu \*\_{C}4\*\_{C}3^{2})/(\alpha \*\_{C}3)-12\*\mu \*\_{C}4\*\_{C}3\*JacobiDN(\_{C}2+\_{C}3\*x+\_{C}4\*t,\_{C}1)^{2}/\alpha }}
{\displaystyle {u(x,t)=-(4\*\mu \*\_{C}4\*\_{C}3^{2}\*\_{C}1^{2}+\_{C}4+\_{C}3-8\*\mu \*\_{C}4\*\_{C}3^{2})/(\alpha \*\_{C}3)+12\*\mu \*\_{C}4\*\_{C}3\*(-1+\_{C}1^{2})\*JacobiND(\_{C}2+\_{C}3\*x+\_{C}4\*t,\_{C}1)^{2}/\alpha }}
{\displaystyle {u(x,t)=-(4\*\mu \*\_{C}4\*\_{C}3^{2}\*\_{C}1^{2}+4\*\mu \*\_{C}4\*\_{C}3^{2}+\_{C}4+\_{C}3)/(\alpha \*\_{C}3)+12\*\mu \*\_{C}4\*\_{C}3\*JacobiNS(\_{C}2+\_{C}3\*x+\_{C}4\*t,\_{C}1)^{2}/\alpha }}
{\displaystyle {u(x,t)=-(4\*\mu \*\_{C}4\*\_{C}3^{2}\*\_{C}1^{2}+4\*\mu \*\_{C}4\*\_{C}3^{2}+\_{C}4+\_{C}3)/(\alpha \*\_{C}3)+12\*\mu \*\_{C}4\*\_{C}3\*\_{C}1^{2}\*JacobiSN(\_{C}2+\_{C}3\*x+\_{C}4\*t,\_{C}1)^{2}/\alpha }}

行波图
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| --- | --- | --- | --- |
| 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 |

|  |  |  |  |
| --- | --- | --- | --- |
| 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 |

|  |  |  |  |
| --- | --- | --- | --- |
| 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 |

|  |  |  |  |
| --- | --- | --- | --- |
| 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 | 正则长波方程行波图 |

|  |  |
| --- | --- |
| 正则长波方程行波图 | 正则长波方程行波图 |

参考文献
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1. ↑  Griffith, chapter 13, p239-260

1. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
2. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
3. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
4. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
5. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
6. Dongming Wang, Elimination Practice,Imperial College Press 2004
7. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
8. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

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| * 
* 论
* 编
非线性偏微分方程理论与解法 |
| --- |
|  |
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* 阿多米安分解法
* 贝克隆德变换
* C-K直接约化法
* 达布变换
* Domion结构
* 高阶约束流
* 反散射法
* 非古典李对称
* 分離變數法
* 加德纳变换
* 古典李对称
* 广田法
* 霍普夫-科尔变换
* 换位表示
* 混合指数法
* Lax 对
* Liouville可积
* 龙格-库塔法
* Miura变换
* 潘勒韦分析
* 齐次平衡法
* Riccati方程展开法
* 摄动理论
* Tanh函数展开法
* 特征线法
* 屠格式理论
* 有限差分法
* 线条法

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