非規範變形KdV方程

非規範變形KdV方程 (Unnormalized modified KdV equation)是一個非線性偏微分方程：^[1]

$u_{t}+u_{xxx}+\alpha *u^{2}*u_{x}=0$

解析解

{u(x,t)={\sqrt {(}}6)*_{C}2*sech(_{C}1+_{C}2*x-_{C}2^{3}*t)/{\sqrt {(}}\alpha )}

u(x,t)={\sqrt {(}}6)*_{C}3*JacobiDN(_{C}2+_{C}3*x+(-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1)/{\sqrt {(}}\alpha )

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

u(x,t)={\sqrt {(}}6)*_{C}1*_{C}3*JacobiCN(-_{C}2-_{C}3*x-(-2*_{C}3^{3}*_{C}1^{2}+_{C}3^{3})*t,_{C}1)/{\sqrt {(}}\alpha )

{u(x,t)={\sqrt {(}}6)*{\sqrt {(}}\alpha *(-1+_{C}1^{2}))*_{C}3*JacobiNC(-_{C}2-_{C}3*x-(-2*_{C}3^{3}*_{C}1^{2}+_{C}3^{3})*t,_{C}1)/\alpha }

u(x,t)=-6*_{C}3*JacobiNS(_{C}2+_{C}3*x+(_{C}3^{3}*_{C}1^{2}+_{C}3^{3})*t,_{C}1)/{\sqrt {(}}-6*\alpha )

u(x,t)=-{\sqrt {(}}6)*_{C}3*JacobiDN(_{C}2+_{C}3*x+(-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1)/{\sqrt {(}}\alpha )

^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS,（《非線性偏微分方程手冊》） SECOND EDITION p870 CRC PRESS

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