伊藤方程

伊藤方程(Ito equation)是一个五阶非线性偏微分方程：^[1]

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

解析解

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

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

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

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

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

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

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

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

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

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

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

行波图

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