布罗尔-库普方程

**布羅爾 - 庫普方程**（Broer-Kaup equation）是一二元個非線性偏微分方程組：

$$
u_{y,t}+(2*u*u_{x})_{x}+2*v_{xx}-u_{xxy}=0
$$

$$
v_{t}+2*(vu)_{x}+v_{xx}=0
$$

## 解析解

　
: $u(x,y,t)=-(1/2)*_{C}4*(2*_{C}3+_{C}2)/(_{C}2*(_{C}3+2*_{C}2))+_{C}2*cot(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t),v(x,t)=-(1/4)*(2*_{C}2^{4}*_{C}3^{3}+10*_{C}2^{5}*_{C}3^{2}+16*_{C}2^{6}*_{C}3+8*_{C}2^{7}-_{C}2*_{C}4^{2}*_{C}3^{2}-_{C}2^{2}*_{C}4^{2}*_{C}3+_{C}2^{3}*_{C}4^{2}+_{C}4^{2}*_{C}3^{3})/(_{C}2^{3}*(_{C}3^{2}+4*_{C}3*_{C}2+4*_{C}2^{2}))+(1/2)*_{C}4*(-_{C}3^{2}+_{C}2^{2})*cot(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)/(_{C}2*(_{C}3+2*_{C}2))+(-(1/2)*_{C}3*_{C}2-(1/2)*_{C}2^{2})*cot(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)^{2}$

　
: $u(x,y,t)=-(1/2)*_{C}4*(2*_{C}3+_{C}2)/(_{C}2*(_{C}3+2*_{C}2))+_{C}2*coth(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t),v(x,t)=(1/4)*(2*_{C}2^{4}*_{C}3^{3}+10*_{C}2^{5}*_{C}3^{2}+16*_{C}2^{6}*_{C}3+8*_{C}2^{7}+_{C}2*_{C}4^{2}*_{C}3^{2}+_{C}2^{2}*_{C}4^{2}*_{C}3-_{C}2^{3}*_{C}4^{2}-_{C}4^{2}*_{C}3^{3})/(_{C}2^{3}*(_{C}3^{2}+4*_{C}3*_{C}2+4*_{C}2^{2}))+(1/2)*_{C}4*(-_{C}3^{2}+_{C}2^{2})*coth(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)/(_{C}2*(_{C}3+2*_{C}2))+(-(1/2)*_{C}3*_{C}2-(1/2)*_{C}2^{2})*coth(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)^{2}$

　
: $u(x,y,t)=-(1/2)*_{C}4*(2*_{C}3+_{C}2)/(_{C}2*(_{C}3+2*_{C}2))-tan(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)*_{C}2,v(x,t)=-(1/4)*(2*_{C}2^{4}*_{C}3^{3}+10*_{C}2^{5}*_{C}3^{2}+16*_{C}2^{6}*_{C}3+8*_{C}2^{7}-_{C}2*_{C}4^{2}*_{C}3^{2}-_{C}2^{2}*_{C}4^{2}*_{C}3+_{C}2^{3}*_{C}4^{2}+_{C}4^{2}*_{C}3^{3})/(_{C}2^{3}*(_{C}3^{2}+4*_{C}3*_{C}2+4*_{C}2^{2}))-(1/2)*_{C}4*(-_{C}3^{2}+_{C}2^{2})*tan(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)/(_{C}2*(_{C}3+2*_{C}2))+(-(1/2)*_{C}3*_{C}2-(1/2)*_{C}2^{2})*tan(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)^{2}$

　
: $u(x,y,t)=-(1/2)*_{C}4*(2*_{C}3+_{C}2)/(_{C}2*(_{C}3+2*_{C}2))+tanh(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)*_{C}2,v(x,t)=(1/4)*(2*_{C}2^{4}*_{C}3^{3}+10*_{C}2^{5}*_{C}3^{2}+16*_{C}2^{6}*_{C}3+8*_{C}2^{7}+_{C}2*_{C}4^{2}*_{C}3^{2}+_{C}2^{2}*_{C}4^{2}*_{C}3-_{C}2^{3}*_{C}4^{2}-_{C}4^{2}*_{C}3^{3})/(_{C}2^{3}*(_{C}3^{2}+4*_{C}3*_{C}2+4*_{C}2^{2}))+(1/2)*_{C}4*(-_{C}3^{2}+_{C}2^{2})*tanh(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)/(_{C}2*(_{C}3+2*_{C}2))+(-(1/2)*_{C}3*_{C}2-(1/2)*_{C}2^{2})*tanh(_{C}1+_{C}2*x+_{C}3*y+_{C}4*t)^{2}$

　
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## 行波圖

布羅爾 - 庫普方程具有亮孤波、暗孤波和扭型孤波。

## 參考文獻

1. \* 谷超豪 《孤立子理論中的達布變換及其幾何應用》 上海科學技術出版社
2. 李志斌編著 《非線性數學物理方程的行波解》 科學出版社
3. 王東明著 《消去法及其應用》 科學出版社 2002
4. \* 何青 王麗芬編著 《Maple 教程》 科學出版社 2010 ISBN 9787030177445
5. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
6. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
7. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
8. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
9. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
10. Dongming Wang, Elimination Practice,Imperial College Press 2004
11. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
12. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759