五阶KdV方程

**五階 KdV 方程**(Fifth order KdV equation) 是一個非線性偏微分方程，簡稱 fKdV 方程： $u_{t}+\alpha *u^{2}*u_{x}+\beta *u_{x}*u_{xx}+\gamma *u*u_{xxx}+\delta *u_{xxxxx}=0$

## 解析解

　
: $u(x,t)=6*_{C}3^{2}*(-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\gamma ^{2}/\alpha +(60*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\delta *\beta ^{2})*JacobiND(_{C}2+_{C}3*x-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*_{C}3^{5}*t/\alpha ,{\sqrt {(}}2))^{2}/(\beta *(6*\beta ^{2}-120*\delta *\alpha +12*\gamma ^{2}+12*\beta *\gamma +6*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))$

　
: $u(x,t)=6*_{C}3^{2}*(-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\gamma ^{2}/\alpha +(60*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\delta *\beta ^{2})*JacobiNS(_{C}2+_{C}3*x-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*_{C}3^{5}*t/\alpha ,I)^{2}/(\beta *(6*\beta ^{2}-120*\delta *\alpha +12*\gamma ^{2}+12*\beta *\gamma +6*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))$

　
: $u(x,t)=-3*_{C}3^{2}*(-(3/2)*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2}))*\gamma ^{2}/\alpha +(15*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -18*\delta *\gamma ^{2}+180*\delta ^{2}*\alpha -30*\delta *\beta ^{2})*JacobiCN(_{C}2+_{C}3*x-(3/2)*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma -{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2}))*_{C}3^{5}*t/\alpha ,(1/2)*{\sqrt {(}}2))^{2}/(\beta *((3/2)*\beta ^{2}-30*\delta *\alpha +3*\gamma ^{2}+3*\beta *\gamma +(3/2)*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))$

　
: $u(x,t)=-6*_{C}3^{2}*(-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma +{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\gamma ^{2}/\alpha +(60*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma +{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*\delta -72*\delta *\gamma ^{2}+720*\delta ^{2}*\alpha -120*\delta *\beta ^{2})*JacobiDN(_{C}2+_{C}3*x-(6*(-12*\delta *\alpha +\beta ^{2}+2*\beta *\gamma +{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))*_{C}3^{5}*t/\alpha ,{\sqrt {(}}2))^{2}/(\beta *(6*\beta ^{2}-120*\delta *\alpha +12*\gamma ^{2}+12*\beta *\gamma -6*{\sqrt {(}}-40*\delta *\alpha *\beta ^{2}+\beta ^{4}+4*\beta ^{3}*\gamma +4*\beta ^{2}*\gamma ^{2})))$

　
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## 行波圖

## 參考文獻

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