博欣内斯克型方程

博欣内斯克型方程（Boussinesq type equation）是一个非线性偏微分方程：^[1]

${\displaystyle u_{tt}-u_{xx}-2*\alpha *(u*u_{x})_{x}-\beta *u_{xxtt}=0}$

解析解

${\displaystyle {u(x,t)=-(1/2)*(-_{C}5^{2}+_{C}4^{2})/(\alpha *_{C}4^{2})+6*\beta *_{C}5^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,_{C}2,_{C}1)/\alpha }}$

${\displaystyle {u(x,t)=-(1/2)*(-_{C}3^{2}+8*\beta *_{C}3^{2}*_{C}2^{2}+_{C}2^{2})/(\alpha *_{C}2^{2})+6*\beta *_{C}3^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }}$

${\displaystyle u(x,t)=-(1/2)*(-_{C}3^{2}+8*\beta *_{C}3^{2}*_{C}2^{2}+_{C}2^{2})/(\alpha *_{C}2^{2})+6*\beta *_{C}3^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }$

${\displaystyle u(x,t)=(1/2)*(_{C}3^{2}-_{C}2^{2}+4*\beta *_{C}3^{2}*_{C}2^{2})/(\alpha *_{C}2^{2})+6*\beta *_{C}3^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }$

${\displaystyle {u(x,t)=(1/2)*(_{C}3^{2}-_{C}2^{2}+4*\beta *_{C}3^{2}*_{C}2^{2})/(\alpha *_{C}2^{2})-6*\beta *_{C}3^{2}*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }}$

${\displaystyle {u(x,t)=(1/2)*(_{C}3^{2}+8*\beta *_{C}3^{2}*_{C}2^{2}-_{C}2^{2})/(\alpha *_{C}2^{2})+6*\beta *_{C}3^{2}*cot(_{C}1+_{C}2*x+_{C}3*t)^{2}/\alpha }}$

${\displaystyle u(x,t)=(1/2)*(-_{C}3^{2}+8*\beta *_{C}4^{2}*_{C}3^{2}*_{C}1^{2}-4*\beta *_{C}4^{2}*_{C}3^{2}+_{C}4^{2})/(\alpha *_{C}3^{2})-6*\beta *_{C}4^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }$

${\displaystyle u(x,t)=(1/2)*(-_{C}3^{2}+8*\beta *_{C}4^{2}*_{C}3^{2}*_{C}1^{2}-4*\beta *_{C}4^{2}*_{C}3^{2}+_{C}4^{2})/(\alpha *_{C}3^{2})-6*\beta *_{C}4^{2}*(-1+_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }$

${\displaystyle {u(x,t)=-(1/2)*(_{C}3^{2}-8*\beta *_{C}4^{2}*_{C}3^{2}-_{C}4^{2}+4*\beta *_{C}4^{2}*_{C}3^{2}*_{C}1^{2})/(\alpha *_{C}3^{2})-6*\beta *_{C}4^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\alpha }}$

行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

博欣内斯克型方程行波图

参考文献

1. Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS,（《非线性偏微分方程手册》） SECOND EDITION p1028 CRC PRESS

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