伊藤方程

伊藤方程(Ito equation)是一個五階非線性偏微分方程：^[1]

${\displaystyle U_{t}+((6*U^{5}+10*\alpha *(U^{2}*U_{xx}+U*U_{x}^{2})+U_{xxxx})_{x}=0}$

解析解

${\displaystyle u(x,t)=_{C}2*sech(_{C}1+_{C}2*x-_{C}2^{5}*t)}$

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

${\displaystyle u(x,t)={\sqrt {(}}2)*_{C}3*JacobiCN(_{C}2+_{C}3*x-7*_{C}3^{5}*t,(1/2)*sqrt(2))}$

${\displaystyle u(x,t)=-I*_{C}2*cot(_{C}1+_{C}2*x-6*_{C}2^{5}*t)}$

${\displaystyle u(x,t)=-I*_{C}2*tan(_{C}1+_{C}2*x-6*_{C}2^{5}*t)}$

${\displaystyle u(x,t)=-I*_{C}3*JacobiNS(-_{C}2-_{C}3*x-(-_{C}3^{5}*_{C}1^{4}-_{C}3^{5}-4*_{C}3^{5}*_{C}1^{2})*t,_{C}1)}$

${\displaystyle u(x,t)=I*_{C}2*csc(_{C}1+_{C}2*x-_{C}2^{5}*t)}$

${\displaystyle u(x,t)=(2*I)*_{C}3*JacobiND(_{C}2+_{C}3*x-28*_{C}3^{5}*t,sqrt(2))}$

${\displaystyle u(x,t)=-I*sqrt(2)*_{C}3*JacobiNC(_{C}2+_{C}3*x-7*_{C}3^{5}*t,(1/2)*sqrt(2))}$

${\displaystyle u(x,t)=-2*_{C}3*JacobiSN(_{C}2+_{C}3*x-28*_{C}3^{5}*t,I)}$

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

行波圖

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot

Ito equation traveling wave plot