# 七阶KdV方程

U[t] 代表 ${\displaystyle {\frac {\partial u(x,t)}{\partial t}}}$

U[x,x,x] 代表 ${\displaystyle {\frac {\partial ^{3}u(x,t)}{\partial x^{3}}}}$

## 解析解

${\displaystyle u(x,t)=6*_{C}3^{2}*((60*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\beta ^{2}*\delta -(6*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\gamma ^{2}/\alpha )*JacobiND(_{C}2+_{C}3*x-(6*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*_{C}3^{5}*t/\alpha ,{\sqrt {(}}2))^{2}/(\beta *(-120*\delta *\alpha +12*\gamma *\beta +12*\gamma ^{2}+6*\beta ^{2}+6*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))}$
${\displaystyle u(x,t)=-3*_{C}3^{2}*(-18*\delta *\gamma ^{2}+180*\delta ^{2}*\alpha +(15*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\delta -30*\beta ^{2}*\delta -(3/2)*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2}))*\gamma ^{2}/\alpha )*JacobiCN(_{C}2+_{C}3*x-(3/2)*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2}))*_{C}3^{5}*t/\alpha ,(1/2)*{\sqrt {(}}2))^{2}/(\beta *(-30*\delta *\alpha +3*\gamma *\beta +3*\gamma ^{2}+(3/2)*\beta ^{2}+(3/2)*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))}$
${\displaystyle u(x,t)=-6*_{C}3^{2}*((60*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\beta ^{2}*\delta -(6*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\gamma ^{2}/\alpha )*JacobiDN(_{C}2+_{C}3*x-(6*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*_{C}3^{5}*t/\alpha ,{\sqrt {(}}2))^{2}/(\beta *(-120*\delta *\alpha +12*\gamma *\beta +12*\gamma ^{2}+6*\beta ^{2}+6*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))}$
${\displaystyle u(x,t)=3*_{C}3^{2}*(-18*\delta *\gamma ^{2}+180*\delta ^{2}*\alpha +(15*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\delta -30*\beta ^{2}*\delta -(3/2)*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2}))*\gamma ^{2}/\alpha )*JacobiNC(_{C}2+_{C}3*x-(3/2)*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2}))*_{C}3^{5}*t/\alpha ,(1/2)*{\sqrt {(}}2))^{2}/(\beta *(-30*\delta *\alpha +3*\gamma *\beta +3*\gamma ^{2}+(3/2)*\beta ^{2}+(3/2)*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))}$
${\displaystyle u(x,t)=6*_{C}3^{2}*((60*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\beta ^{2}*\delta -(6*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\gamma ^{2}/\alpha )*JacobiND(_{C}2+_{C}3*x-(6*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*_{C}3^{5}*t/\alpha ,{\sqrt {(}}2))^{2}/(\beta *(-120*\delta *\alpha +12*\gamma *\beta +12*\gamma ^{2}+6*\beta ^{2}+6*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))}$
${\displaystyle u(x,t)=6*_{C}3^{2}*(-(60*(-2*\gamma *\beta -\beta ^{2}+12*\delta *\alpha +{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\beta ^{2}*\delta +(6*(-2*\gamma *\beta -\beta ^{2}+12*\delta *\alpha +{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\gamma ^{2}/\alpha )*JacobiNS(_{C}2+_{C}3*x+(6*(-2*\gamma *\beta -\beta ^{2}+12*\delta *\alpha +{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*_{C}3^{5}*t/\alpha ,I)^{2}/(\beta *(-120*\delta *\alpha +12*\gamma *\beta +12*\gamma ^{2}+6*\beta ^{2}+6*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))}$
${\displaystyle u(x,t)=-6*_{C}3^{2}*(-(60*(-2*\gamma *\beta -\beta ^{2}+12*\delta *\alpha +{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\beta ^{2}*\delta +(6*(-2*\gamma *\beta -\beta ^{2}+12*\delta *\alpha +{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\gamma ^{2}/\alpha )*JacobiSN(_{C}2+_{C}3*x+(6*(-2*\gamma *\beta -\beta ^{2}+12*\delta *\alpha +{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*_{C}3^{5}*t/\alpha ,I)^{2}/(\beta *(-120*\delta *\alpha +12*\gamma *\beta +12*\gamma ^{2}+6*\beta ^{2}+6*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))}$
${\displaystyle u(x,t)=2*_{C}2^{2}*(-(2*(2*\gamma *\beta +\beta ^{2}-12*\delta *\alpha -{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2})))*\gamma /\alpha -24*\gamma *\delta +40*\beta *\delta )/(2*\beta ^{2}+2*{\sqrt {(}}4*\gamma ^{2}*\beta ^{2}+4*\gamma *\beta ^{3}+\beta ^{4}-40*\delta *\alpha *\beta ^{2}))}$
${\displaystyle u(x,t)=(86625/591361)*sech(5*(x-(180000/591361)*t)/{\sqrt {(}}1538))^{6}}$
${\displaystyle u(x,t)=(86625/591361)*csch(5*(x-(180000/591361)*t)/{\sqrt {(}}1538))^{6}}$
${\displaystyle u(x,t)=(86625/591361)*sec(5*(x-(180000/591361)*t)/{\sqrt {(}}1538))^{6}}$
${\displaystyle u(x,t)=(86625/591361)*csc(5*(x-(180000/591361)*t)/{\sqrt {(}}1538))^{6}}$

## 行波图

 Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot
 Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot
 Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot Seventh order KdV equation traveling wave plot

## 参考文献

1. ^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p1040 CRC PRESS
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