布罗尔-库普方程（Broer-Kaup equation）是一二元个非线性偏微分方程组：

u_{y,t}+(2uu_{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)_C4(2_C3+_C2)/(_C2(_C3+2_C2))+_C2cot(_C1+_C2x+_C3y+_C4t), v(x, t) = -(1/4)(2_C2^4_C3^3+10_C2^5_C3^2+16_C2^6_C3+8_C2^7-_C2_C4^2_C3^2-_C2^2_C4^2_C3+_C2^3_C4^2+_C4^2_C3^3)/(_C2^3(_C3^2+4_C3_C2+4_C2^2))+(1/2)_C4(-_C3^2+_C2^2)cot(_C1+_C2x+_C3y+_C4t)/(_C2(_C3+2_C2))+(-(1/2)_C3_C2-(1/2)_C2^2)cot(_C1+_C2x+_C3y+_C4t)^2
:u(x, y, t) = -(1/2)_C4(2_C3+_C2)/(_C2(_C3+2_C2))+_C2coth(_C1+_C2x+_C3y+_C4t), v(x, t) = (1/4)(2_C2^4_C3^3+10_C2^5_C3^2+16_C2^6_C3+8_C2^7+_C2_C4^2_C3^2+_C2^2_C4^2_C3-_C2^3_C4^2-_C4^2_C3^3)/(_C2^3(_C3^2+4_C3_C2+4_C2^2))+(1/2)_C4(-_C3^2+_C2^2)coth(_C1+_C2x+_C3y+_C4t)/(_C2(_C3+2_C2))+(-(1/2)_C3_C2-(1/2)_C2^2)coth(_C1+_C2x+_C3y+_C4t)^2
:u(x, y, t) = -(1/2)_C4(2_C3+_C2)/(_C2(_C3+2_C2))-tan(_C1+_C2x+_C3y+_C4t)_C2, v(x, t) = -(1/4)(2_C2^4_C3^3+10_C2^5_C3^2+16_C2^6_C3+8_C2^7-_C2_C4^2_C3^2-_C2^2_C4^2_C3+_C2^3_C4^2+_C4^2_C3^3)/(_C2^3(_C3^2+4_C3_C2+4_C2^2))-(1/2)_C4(-_C3^2+_C2^2)tan(_C1+_C2x+_C3y+_C4t)/(_C2(_C3+2_C2))+(-(1/2)_C3_C2-(1/2)_C2^2)tan(_C1+_C2x+_C3y+_C4t)^2
:u(x, y, t) = -(1/2)_C4(2_C3+_C2)/(_C2(_C3+2_C2))+tanh(_C1+_C2x+_C3y+_C4t)_C2, v(x, t) = (1/4)(2_C2^4_C3^3+10_C2^5_C3^2+16_C2^6_C3+8_C2^7+_C2_C4^2_C3^2+_C2^2_C4^2_C3-_C2^3_C4^2-_C4^2_C3^3)/(_C2^3(_C3^2+4_C3_C2+4_C2^2))+(1/2)_C4(-_C3^2+_C2^2)tanh(_C1+_C2x+_C3y+_C4t)/(_C2(_C3+2_C2))+(-(1/2)_C3_C2-(1/2)_C2^2)tanh(_C1+_C2x+_C3y+_C4t)^2
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行波图
布罗尔-库普方程具有亮孤波、暗孤波和扭型孤波。

参考文献

*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

李志斌编著 《非线性数学物理方程的行波解》 科学出版社

#王东明著 《消去法及其应用》 科学出版社 2002

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#Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press

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David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004

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