**库普-库珀施密特方程**(Kaup-Kupershmidt Equation)是一个非线性偏微分方程：

  

{\displaystyle {\frac {\partial ^{4}u(x,t)}{\partial x^{4}}}+{\frac {\partial u(x,t)}{\partial x}}+45({\frac {\partial u(x,t)}{\partial x}}\*u(x,t)^{2}-(75/2)\*{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}\*{\frac {\partial u(x,t)}{\partial x}}-15\*u(x,t)\*{\frac {\partial ^{3}u(x,t)}{\partial x^{3}}}}

行波解
---

利用Maple软件包TWSolution，随所选定展开函数不同，可得多种行波解

tanh 展开
{\displaystyle g[2]:={u(x,t)=-(2/3)\*(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*tanh(\_{C}1+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[3]:={u(x,t)=-(2/3)\*(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}+(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*tanh(\_{C}1+(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[4]:={u(x,t)=-(2/3)\*((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}+((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*tanh(\_{C}1+((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[5]:={u(x,t)=-(2/3)\*((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}+((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*tanh(\_{C}1+((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[6]:={u(x,t)=-(4/3)\*(-(1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*(-(1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*tanh(\_{C}1+(-(1/44)\*sqrt(2)\*11^{(}3/4)-(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[7]:={u(x,t)=-(4/3)\*(-(1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*(-(1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*tanh(\_{C}1+(-(1/44)\*sqrt(2)\*11^{(}3/4)+(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[8]:={u(x,t)=-(4/3)\*((1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*((1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*tanh(\_{C}1+((1/44)\*sqrt(2)\*11^{(}3/4)-(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[9]:={u(x,t)=-(4/3)\*((1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*((1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*tanh(\_{C}1+((1/44)\*sqrt(2)\*11^{(}3/4)+(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}

* Kaup Kupershmidt eq tanh method animation2Kaup Kupershmidt eq tanh method animation2
* Kaup Kupershmidt eq tanh method animation7Kaup Kupershmidt eq tanh method animation7
* Kaup Kupershmidt eq tanh method animation8Kaup Kupershmidt eq tanh method animation8

JacobiSN 展开
{\displaystyle g[2]:={u(x,t)=-(1/2)\*\_{C}3^{2}-(1/6)\*sqrt(-3\*\_{C}3^{4}-4)+((1/2)\*\_{C}3^{2}+(1/2)\*sqrt(-3\*\_{C}3^{4}-4))\*JacobiSN(\_{C}2+\_{C}3\*x+\_{C}4\*t,(1/2)\*sqrt(2\*\_{C}3^{2}+2\*sqrt(-3\*\_{C}3^{4}-4))/\_{C}3)^{2}}}
{\displaystyle g[3]:={u(x,t)=-(1/2)\*\_{C}3^{2}+(1/6)\*sqrt(-3\*\_{C}3^{4}-4)+((1/2)\*\_{C}3^{2}-(1/2)\*sqrt(-3\*\_{C}3^{4}-4))\*JacobiSN(\_{C}2+\_{C}3\*x+\_{C}4\*t,(1/2)\*sqrt(2\*\_{C}3^{2}-2\*sqrt(-3\*\_{C}3^{4}-4))/\_{C}3)^{2}}}
{\displaystyle g[4]:={u(x,t)=-4\*\_{C}3^{2}-(2/33)\*sqrt(-1452\*\_{C}3^{4}-11)+(4\*\_{C}3^{2}+(2/11)\*sqrt(-1452\*\_{C}3^{4}-11))\*JacobiSN(\_{C}2+\_{C}3\*x+\_{C}4\*t,(1/22)\*sqrt(242\*\_{C}3^{2}+11\*sqrt(-1452\*\_{C}3^{4}-11))/\_{C}3)^{2}}}
{\displaystyle g[5]:={u(x,t)=-4\*\_{C}3^{2}+(2/33)\*sqrt(-1452\*\_{C}3^{4}-11)+(4\*\_{C}3^{2}-(2/11)\*sqrt(-1452\*\_{C}3^{4}-11))\*JacobiSN(\_{C}2+\_{C}3\*x+\_{C}4\*t,(1/22)\*sqrt(242\*\_{C}3^{2}-11\*sqrt(-1452\*\_{C}3^{4}-11))/\_{C}3)^{2}}}

* Kaup Kupershmidt JacobiSN method animation2Kaup Kupershmidt JacobiSN method animation2
* Kaup Kupershmidt JacobiSN method animation3Kaup Kupershmidt JacobiSN method animation3
* Kaup Kupershmidt JacobiSN method animation4Kaup Kupershmidt JacobiSN method animation4

sech 展开
{\displaystyle g[2]:={u(x,t)=(1/3)\*(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}-(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*sech(\_{C}1+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[3]:={u(x,t)=(1/3)\*(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}-(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*sech(\_{C}1+(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[4]:={u(x,t)=(1/3)\*((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}-((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*sech(\_{C}1+((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[5]:={u(x,t)=(1/3)\*((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}-((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*sech(\_{C}1+((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[6]:={u(x,t)=(2/3)\*(-(1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}-2\*(-(1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sech(\_{C}1+(-(1/44)\*sqrt(2)\*11^{(}3/4)-(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[7]:={u(x,t)=(2/3)\*(-(1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}-2\*(-(1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sech(\_{C}1+(-(1/44)\*sqrt(2)\*11^{(}3/4)+(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[8]:={u(x,t)=(2/3)\*((1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}-2\*((1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sech(\_{C}1+((1/44)\*sqrt(2)\*11^{(}3/4)-(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[9]:={u(x,t)=(2/3)\*((1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}-2\*((1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sech(\_{C}1+((1/44)\*sqrt(2)\*11^{(}3/4)+(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}

* Kaup Kupershmidt sech method animation2Kaup Kupershmidt sech method animation2
* Kaup Kupershmidt sech method animation4Kaup Kupershmidt sech method animation4
* Kaup Kupershmidt sech method animation8Kaup Kupershmidt sech method animation8

sec、coth 展开
{\displaystyle g[2]:={u(x,t)=-(1/3)\*(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*sec(\_{C}1+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[3]:={u(x,t)=-(1/3)\*(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}+(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*sec(\_{C}1+(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[4]:={u(x,t)=-(1/3)\*((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}+((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*sec(\_{C}1+((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[5]:={u(x,t)=-(1/3)\*((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}+((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*sec(\_{C}1+((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[6]:={u(x,t)=-(2/3)\*(-(1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*(-(1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sec(\_{C}1+(-(1/44)\*sqrt(2)\*11^{(}3/4)-(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[7]:={u(x,t)=-(2/3)\*(-(1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*(-(1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sec(\_{C}1+(-(1/44)\*sqrt(2)\*11^{(}3/4)+(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[8]:={u(x,t)=-(2/3)\*((1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*((1/22)\*sqrt(2)\*11^{(}3/4)-(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sec(\_{C}1+((1/44)\*sqrt(2)\*11^{(}3/4)-(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[9]:={u(x,t)=-(2/3)\*((1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}+2\*((1/22)\*sqrt(2)\*11^{(}3/4)+(1/22\*I)\*sqrt(2)\*11^{(}3/4))^{2}\*sec(\_{C}1+((1/44)\*sqrt(2)\*11^{(}3/4)+(1/44\*I)\*sqrt(2)\*11^{(}3/4))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[10]:={u(x,t)=-(2/3)\*(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*coth(\_{C}1+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}

* Kaup Kupershmidt sec method animation3Kaup Kupershmidt sec method animation3
* Kaup Kupershmidt sec method animation5Kaup Kupershmidt sec method animation5
* Kaup Kupershmidt sec method animation10Kaup Kupershmidt sec method animation10

csch 展开
{\displaystyle {u(x,t)=\_{C}4}}
{\displaystyle g[2]:={u(x,t)=(1/3)\*(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*csch(\_{C}1+(-(1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[3]:={u(x,t)=(1/3)\*(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}+(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*csch(\_{C}1+(-(1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[4]:={u(x,t)=(1/3)\*((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}+((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))^{2}\*csch(\_{C}1+((1/2)\*sqrt(2)-(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}
{\displaystyle g[5]:={u(x,t)=(1/3)\*((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}+((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))^{2}\*csch(\_{C}1+((1/2)\*sqrt(2)+(1/2\*I)\*sqrt(2))\*x+\_{C}3\*t)^{2}}}

* Kaup Kupershmidt csch method animation2Kaup Kupershmidt csch method animation2
* Kaup Kupershmidt csch method animation3Kaup Kupershmidt csch method animation3
* Kaup Kupershmidt csch method animation5Kaup Kupershmidt csch method animation5

参考文献
----

1. ↑  Qinghua Feng New Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation,2012 International Conference on Computer Technology and Science (ICCTS 2012) IPCSIT vol. 47 (2012)
2. ↑  Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Page 27

1. \*谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社
2. \*阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
3. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
4. 王东明著 《消去法及其应用》 科学出版社 2002
5. \*何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445
6. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
7. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
8. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
9. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
11. Dongming Wang, Elimination Practice,Imperial College Press 2004
12. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
13. 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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