广义伯格斯-KdV方程

广义伯格斯-KdV方程 (Generalized Burgers-KdV equation)是一个非线性偏微分方程：^[1]

${\displaystyle U[t]-\alpha *{\frac {\partial ^{n}u(x,t)}{\partial x^{n}}}-\beta *u(x,t)*{\frac {\partial u(x,t)}{\partial x}}=0}$

解析解

当 n=7， 有下列特解：

${\displaystyle u(x,t)=(71280*\alpha *_{C}4^{7}*_{C}1+_{C}5)/(\beta *_{C}4)-665280*\alpha *_{C}4^{6}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,0,_{C}1)^{3}/\beta }$

${\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*((-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}-2)*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }$

${\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*((-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}-2)*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }$

${\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})/(\beta *_{C}3)-665280*_{C}3^{6}*\alpha *(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(1+(-(1/2)*{\sqrt {(}}3)-1/2*I)^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta -665280*\alpha *_{C}3^{6}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }$

${\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2})/(\beta *_{C}3)-665280*_{C}3^{6}*\alpha *(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(1+(-(1/2)*{\sqrt {(}}3)+1/2*I)^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{4}/\beta -665280*\alpha *_{C}3^{6}*JacobiND(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)-1/2*I)^{6}/\beta }$

${\displaystyle u(x,t)=(_{C}4-42240*\alpha *_{C}3^{7}+84480*\alpha *_{C}3^{7}*((1/2)*{\sqrt {(}}3)+1/2*I)^{2})/(\beta *_{C}3)-665280*\alpha *_{C}3^{6}*(-1+((1/2)*{\sqrt {(}}3)+1/2*I)^{2})*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{2}/\beta +665280*\alpha *_{C}3^{6}*(((1/2)*{\sqrt {(}}3)+1/2*I)^{2}-2)*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{4}/\beta +665280*\alpha *_{C}3^{6}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*{\sqrt {(}}3)+1/2*I)^{6}/\beta }$

广义伯格斯-KdV方程之部分通解为：

${\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1*x+b*C1*C2*t+C3)/(b*(C2+t))+C2}$

${\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1*x+b*C1*C2*t+C3)/(b*(C2+t))+C2}$

${\displaystyle u(x,t)=C1^{(}n-1)*((-1)^{n}*a*(2*n-1)!/(b*(n-1)!*(x+b*C1*t+C2)^{(}n-1))+C1)*(C1*x+b*C1*C2*t+C3)+C2}$

${\displaystyle u(x,t)=C1^{(}n-1)*((-1)^{n}*a*(2*n-1)!/(b*(n-1)!*(x+b*C1*t+C2)^{(}n-1))+C1)*(C1*x+b*C1*C2*t+C3)+C2}$

${\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1^{n}*t+C4)/(b*(C2+t))+C2}$

${\displaystyle u(x,t)=C1^{(}n-1)*(x+C1)*(C1^{n}*t+C4)/(b*(C2+t))+C2}$

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行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

7阶广义伯格斯-KdV方程行波图

10阶广义伯格斯-KdV方程行波图

10阶广义伯格斯-KdV方程行波图

10阶广义伯格斯-KdV方程行波图

10阶广义伯格斯-KdV方程行波图

10阶广义伯格斯-KdV方程行波图

10阶广义伯格斯-KdV方程行波图

10阶广义伯格斯-KdV方程行波图

参考文献

1. Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS,（《非线性偏微分方程手册》） SECOND EDITION p1045 CRC PRESS

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