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

参考文献

1. 李志斌编著 《非线性数学物理方程的行波解》 138页 科学出版社 2008

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