伊藤方程

伊藤方程(Ito equation)是一個五階非線性偏微分方程：

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

解析解

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

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

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

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

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

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

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

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

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

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

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

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

參考文獻

1. *谷超豪 《孤立子理論中的達布變換及其幾何應用》 上海科學技術出版社
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10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
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