非規範KdV方程

非規範KdV方程（Unnormalized KdV equation）是一個非線性偏微分方程：

$u_{t}+\alpha *u_{xxx}+\beta *u*u_{x}=0$

解析解

$u(x,t)=-_{C}5/(\beta *_{C}4)-12*\alpha *_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,_{C}2,_{C}1)/\beta$

$u(x,t)=(4*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta$

$u(x,t)=(4*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta$

$u(x,t)=-(4*\alpha *_{C}2^{3}+_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta$

$u(x,t)=-(4*\alpha *_{C}2^{3}+_{C}3)/(\beta *_{C}2)+12*\alpha *_{C}2^{2}*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta$

$u(x,t)=(8*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta$

$u(x,t)=(8*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta$

$u(x,t)=(-8*\alpha *_{C}3^{3}+4*\alpha *_{C}3^{3}*_{C}1^{2}-_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta$

$u(x,t)=(-8*\alpha *_{C}3^{3}+4*\alpha *_{C}3^{3}*_{C}1^{2}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*(-1+_{C}1^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta$

$u(x,t)=(4*\alpha *_{C}3^{3}*_{C}1^{2}+4*\alpha *_{C}3^{3}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta$

$u(x,t)=(4*\alpha *_{C}3^{3}*_{C}1^{2}+4*\alpha *_{C}3^{3}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta$

$u(x,t)=-(8*\alpha *_{C}3^{3}*_{C}1^{2}-4*\alpha *_{C}3^{3}+_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta$

$u(x,t)=-(8*\alpha *_{C}3^{3}*_{C}1^{2}-4*\alpha *_{C}3^{3}+_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*(-1+_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta$

行波圖

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

參考文獻

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