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

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

解析解
当 n=7， 有下列特解：
: u(x, t) = (71280\alpha_C4^7_C1+_C5)/(\beta_C4)-665280\alpha_C4^6WeierstrassP(_C3+_C4x+_C5*t, 0, _C1)^3/\beta
: u(x, t) = (_C4-42240\alpha_C3^7+84480\alpha_C3^7(-(1/2)\sqrt(3)-1/2I)^2)/(\beta_C3)-665280\alpha_C3^6(-1+(-(1/2)\sqrt(3)-1/2I)^2)JacobiDN(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^2/\beta+665280\alpha_C3^6((-(1/2)\sqrt(3)-1/2I)^2-2)JacobiDN(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^4/\beta+665280\alpha_C3^6JacobiDN(_C2+_C3x+_C4t, (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)-1/2I)^2)/(\beta_C3)-665280\alpha_C3^6(-1+(-(1/2)\sqrt(3)-1/2I)^2)JacobiNC(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^2/\beta+665280\alpha_C3^6((-(1/2)\sqrt(3)-1/2I)^2-2)JacobiNC(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^4/\beta+665280\alpha_C3^6JacobiNC(_C2+_C3x+_C4t, (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)-1/2I)^2)/(\beta_C3)-665280_C3^6\alpha(-(1/2)\sqrt(3)-1/2I)^2JacobiND(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^2/\beta+665280\alpha_C3^6(1+(-(1/2)\sqrt(3)-1/2I)^2)JacobiND(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^4/\beta-665280\alpha_C3^6JacobiND(_C2+_C3x+_C4t, (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)+1/2I)^2)/(\beta_C3)-665280_C3^6\alpha(-(1/2)\sqrt(3)+1/2I)^2JacobiND(_C2+_C3x+_C4t, (1/2)\sqrt(3)-1/2I)^2/\beta+665280\alpha_C3^6(1+(-(1/2)\sqrt(3)+1/2I)^2)JacobiND(_C2+_C3x+_C4t, (1/2)\sqrt(3)-1/2I)^4/\beta-665280\alpha_C3^6JacobiND(_C2+_C3x+_C4t, (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)+1/2I)^2)/(\beta_C3)-665280\alpha_C3^6(-1+((1/2)\sqrt(3)+1/2I)^2)JacobiSN(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^2/\beta+665280\alpha_C3^6(((1/2)\sqrt(3)+1/2I)^2-2)JacobiSN(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2I)^4/\beta+665280\alpha_C3^6JacobiSN(_C2+_C3x+_C4t, (1/2)\sqrt(3)+1/2*I)^6/\beta

广义伯格斯-KdV方程之部分通解为：

: u(x, t) = C1^(n-1)(x+C1)(C1x+bC1C2t+C3)/(b*(C2+t))+C2
: u(x, t) = C1^(n-1)(x+C1)(C1x+bC1C2t+C3)/(b*(C2+t))+C2
: u(x, t) = C1^(n-1)((-1)^na(2n-1)!/(b(n-1)!(x+bC1t+C2)^(n-1))+C1)(C1x+bC1C2*t+C3)+C2
: u(x, t) =C1^(n-1)((-1)^na(2n-1)!/(b(n-1)!(x+bC1t+C2)^(n-1))+C1)(C1x+bC1C2*t+C3)+C2
: u(x, t) =C1^(n-1)(x+C1)(C1^nt+C4)/(b(C2+t))+C2
: u(x, t) =C1^(n-1)(x+C1)(C1^nt+C4)/(b(C2+t))+C2
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行波图
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