廣義伯格斯-KdV方程

廣義伯格斯-KdV方程 (Generalized Burgers-KdV equation)是一個非線性偏微分方程：

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

## 解析解

當 n=7，有下列特解：

$$u(x, t) = (71280*\alpha*_C4^7*_C1+_C5)/(\beta*_C4)-665280*\alpha*_C4^6*WeierstrassP(_C3+_C4*x+_C5*t, 0, _C1)^3/\beta$$

$$u(x, t) = (_C4-42240*\alpha*_C3^7+84480*\alpha*_C3^7*(-(1/2)*\sqrt(3)-1/2*I)^2)/(\beta*_C3)-665280*\alpha*_C3^6*(-1+(-(1/2)*\sqrt(3)-1/2*I)^2)*JacobiDN(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^2/\beta+665280*\alpha*_C3^6*((-(1/2)*\sqrt(3)-1/2*I)^2-2)*JacobiDN(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^4/\beta+665280*\alpha*_C3^6*JacobiDN(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^6/\beta$$

$$u(x, t) = (_C4-42240*\alpha*_C3^7+84480*\alpha*_C3^7*(-(1/2)*\sqrt(3)-1/2*I)^2)/(\beta*_C3)-665280*\alpha*_C3^6*(-1+(-(1/2)*\sqrt(3)-1/2*I)^2)*JacobiNC(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^2/\beta+665280*\alpha*_C3^6*((-(1/2)*\sqrt(3)-1/2*I)^2-2)*JacobiNC(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^4/\beta+665280*\alpha*_C3^6*JacobiNC(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^6/\beta$$

$$u(x, t) = (_C4-42240*\alpha*_C3^7+84480*\alpha*_C3^7*(-(1/2)*\sqrt(3)-1/2*I)^2)/(\beta*_C3)-665280*_C3^6*\alpha*(-(1/2)*\sqrt(3)-1/2*I)^2*JacobiND(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^2/\beta+665280*\alpha*_C3^6*(1+(-(1/2)*\sqrt(3)-1/2*I)^2)*JacobiND(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^4/\beta-665280*\alpha*_C3^6*JacobiND(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^6/\beta$$

$$u(x, t) = (_C4-42240*\alpha*_C3^7+84480*\alpha*_C3^7*(-(1/2)*\sqrt(3)+1/2*I)^2)/(\beta*_C3)-665280*_C3^6*\alpha*(-(1/2)*\sqrt(3)+1/2*I)^2*JacobiND(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)-1/2*I)^2/\beta+665280*\alpha*_C3^6*(1+(-(1/2)*\sqrt(3)+1/2*I)^2)*JacobiND(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)-1/2*I)^4/\beta-665280*\alpha*_C3^6*JacobiND(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)-1/2*I)^6/\beta$$

$$u(x, t) = (_C4-42240*\alpha*_C3^7+84480*\alpha*_C3^7*((1/2)*\sqrt(3)+1/2*I)^2)/(\beta*_C3)-665280*\alpha*_C3^6*(-1+((1/2)*\sqrt(3)+1/2*I)^2)*JacobiSN(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^2/\beta+665280*\alpha*_C3^6*(((1/2)*\sqrt(3)+1/2*I)^2-2)*JacobiSN(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^4/\beta+665280*\alpha*_C3^6*JacobiSN(_C2+_C3*x+_C4*t, (1/2)*\sqrt(3)+1/2*I)^6/\beta$$

廣義伯格斯-KdV方程之部分通解為：

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

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

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

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

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

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

## 行波圖

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## 參考文獻

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