非规范KdV方程

非规范KdV方程（Unnormalized KdV equation）是一个非线性偏微分方程：^[1]

${\displaystyle u_{t}+\alpha *u_{xxx}+\beta *u*u_{x}=0}$

解析解

${\displaystyle u(x,t)=-_{C}5/(\beta *_{C}4)-12*\alpha *_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,_{C}2,_{C}1)/\beta }$

${\displaystyle u(x,t)=(4*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=(4*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=-(4*\alpha *_{C}2^{3}+_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=-(4*\alpha *_{C}2^{3}+_{C}3)/(\beta *_{C}2)+12*\alpha *_{C}2^{2}*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=(8*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=(8*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }$

${\displaystyle u(x,t)=(-8*\alpha *_{C}3^{3}+4*\alpha *_{C}3^{3}*_{C}1^{2}-_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }$

${\displaystyle u(x,t)=(-8*\alpha *_{C}3^{3}+4*\alpha *_{C}3^{3}*_{C}1^{2}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*(-1+_{C}1^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }$

${\displaystyle u(x,t)=(4*\alpha *_{C}3^{3}*_{C}1^{2}+4*\alpha *_{C}3^{3}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }$

${\displaystyle u(x,t)=(4*\alpha *_{C}3^{3}*_{C}1^{2}+4*\alpha *_{C}3^{3}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }$

${\displaystyle u(x,t)=-(8*\alpha *_{C}3^{3}*_{C}1^{2}-4*\alpha *_{C}3^{3}+_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }$

${\displaystyle u(x,t)=-(8*\alpha *_{C}3^{3}*_{C}1^{2}-4*\alpha *_{C}3^{3}+_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*(-1+_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }$

行波图

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

Unnormalized KdV equation traveling wave plot

参考文献

1. Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS

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