布罗尔-库普方程组

布罗尔-库普方程（Broer-Kaup equation）是一二元个非线性偏微分方程组：^[1]

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

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

解析解

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

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

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

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

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行波图

布罗尔-库普方程具有亮孤波、暗孤波和扭型孤波。

Broer-Kaup equation traveling wave plot

Broer-Kaup equation traveling wave plot

Broer-Kaup equation traveling wave plot

Broer-Kaup equation traveling wave plot

Broer-Kaup equation traveling wave plot

Broer-Kaup equation traveling wave plot

Broer-Kaup equation traveling wave plot

Broer-Kaup equation traveling wave plot

参考文献

1. 阎振亚著 《复杂非线性波的构造性理论及其应用》 第65页 科学出版社 2007年

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2. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
3. 王东明著 《消去法及其应用》 科学出版社 2002
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8. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
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