伯格斯-赫胥黎方程

伯格斯-赫胥黎方程（Burgers-Huxley equation) 是一个模拟物理学、生物学、经济学和生态学等领域非线性波动现象的非线性偏微分方程

$$pde1 := u[t]-nu*u[x, x]+a*u*u[x] = b*u*(1-u)*(u-c)$$

其中 u=u(x,t),ut=$\frac{\partial u}{\partial t}$ 等等。

## 解析解

### 特解

以

```
  :{a = 1, b = 1, c = 1.5, nu = 1}
  :{a = 1, b = 1, c = 2, nu = 1}
  :{a = -1, b = 1, c = 2.3, nu = 1}
```

代人伯格斯-赫胥黎方程后求解得：

$$p[1] := .50000000000000000000+.50000000000000000000*tanh(3+.25000000000000000000*x-.62500000000000000000*t)$$

$$p[2] := 1-tanh(3+x-t)$$

$$p[3] := .50000000000000000000+.50000000000000000000*coth(3+.50000000000000000000*x-.65000000000000000000*t)$$

### 通解

伯格斯-赫胥黎方程有tanh展开行波解，不存在csch展开行波解$$sol6 := u = 1/2+(1/2)*tanh(_C1+(1/8)*(-a+sqrt(a^2+8*b*nu))*x/nu+(1/8)*(-a^14*b-2880*b^6*a^4*c^3*nu^5+4196*b^6*a^4*c*nu^5+7840*b^6*a^4*c^2*nu^5+64*c^5*b^6*a^4*nu^5+96*a^10*b^3*c*nu^2+8*a^10*b^3*c^2*nu^2+208*a^8*nu^3*b^4*c^2+840*a^8*nu^3*b^4*c+4*b^2*a^12*c*nu-32*a^8*nu^3*b^4*c^3-16*a^6*nu^4*b^5*c^4+3152*a^6*b^5*nu^4*c+1952*a^6*b^5*nu^4*c^2-544*a^6*nu^4*b^5*c^3+11880*b^7*a^2*nu^6*c^2+648*c^5*b^7*a^2*nu^6-2160*c^4*b^7*a^2*nu^6-4536*c^3*b^7*a^2*nu^6-352*b^6*a^4*c^4*nu^5-432*b^7*a^2*nu^6*c-6081*a^6*b^5*nu^4-2064*a^8*nu^3*b^4-3348*b^7*a^2*nu^6-1296*nu^7*b^8*c+3240*nu^7*b^8*c^2-3240*c^4*b^8*nu^7+1296*c^5*b^8*nu^7-8106*b^6*a^4*nu^5-354*a^10*b^3*nu^2-30*b^2*a^12*nu+(1/4)*(-a+sqrt(a^2+8*b*nu))*a^15/nu+(8*(-a+sqrt(a^2+8*b*nu)))*b*a^13-584*nu^3*(-a+sqrt(a^2+8*b*nu))*b^4*a^7*c^2-540*nu^6*(-a+sqrt(a^2+8*b*nu))*a*b^7*c^5-3456*nu^6*(-a+sqrt(a^2+8*b*nu))*a*b^7*c^2+4*nu^3*(-a+sqrt(a^2+8*b*nu))*b^4*a^7*c^4-192*nu^5*(-a+sqrt(a^2+8*b*nu))*b^6*a^3*c^5+696*nu^5*(-a+sqrt(a^2+8*b*nu))*b^6*a^3*c^4-16*nu^4*(-a+sqrt(a^2+8*b*nu))*b^5*a^5*c^5+96*nu^4*(-a+sqrt(a^2+8*b*nu))*b^5*a^5*c^4+1512*nu^6*(-a+sqrt(a^2+8*b*nu))*a*b^7*c^4-2760*nu^4*(-a+sqrt(a^2+8*b*nu))*b^5*a^5*c^2+2160*nu^5*(-a+sqrt(a^2+8*b*nu))*c^3*b^6*a^3+972*nu^6*(-a+sqrt(a^2+8*b*nu))*c^3*a*b^7+152*nu^3*(-a+sqrt(a^2+8*b*nu))*c^3*b^4*a^7+960*nu^4*(-a+sqrt(a^2+8*b*nu))*c^3*b^5*a^5+8*nu^2*(-a+sqrt(a^2+8*b*nu))*c^3*b^3*a^9-1128*nu^3*(-a+sqrt(a^2+8*b*nu))*b^4*a^7*c-2089*nu^4*(-a+sqrt(a^2+8*b*nu))*b^5*a^5*c-26*nu*(-a+sqrt(a^2+8*b*nu))*b^2*a^11*c-254*nu^2*(-a+sqrt(a^2+8*b*nu))*b^3*a^9*c-726*nu^5*(-a+sqrt(a^2+8*b*nu))*b^6*a^3*c+864*nu^6*(-a+sqrt(a^2+8*b*nu))*a*b^7*c-2*nu*(-a+sqrt(a^2+8*b*nu))*b^2*a^11*c^2-56*nu^2*(-a+sqrt(a^2+8*b*nu))*b^3*a^9*c^2-5610*nu^5*(-a+sqrt(a^2+8*b*nu))*b^6*a^3*c^2-(-a+sqrt(a^2+8*b*nu))*b*a^13*c+(9193/4)*nu^3*(-a+sqrt(a^2+8*b*nu))*b^4*a^7+3931*nu^4*(-a+sqrt(a^2+8*b*nu))*b^5*a^5+(205/2)*nu*(-a+sqrt(a^2+8*b*nu))*b^2*a^11+667*nu^2*(-a+sqrt(a^2+8*b*nu))*b^3*a^9+2673*nu^5*(-a+sqrt(a^2+8*b*nu))*b^6*a^3+324*nu^6*(-a+sqrt(a^2+8*b*nu))*a*b^7)*t/(nu*(-a^12*b-8*a^8*b^3*c*nu^2+8*a^8*b^3*c^2*nu^2+144*nu^3*b^4*a^6*c^2-144*nu^3*b^4*a^6*c-16*nu^4*a^4*b^5*c^4-848*a^4*b^5*nu^4*c+832*a^4*b^5*nu^4*c^2-1728*c*b^6*nu^5*a^2+32*nu^4*a^4*b^5*c^3-162*a^2*b^6*c^4*nu^5+324*a^2*b^6*c^3*nu^5+1566*a^2*b^6*c^2*nu^5+324*b^7*nu^6*c^2-324*c^4*b^7*nu^6+648*c^3*b^7*nu^6-254*a^8*b^3*nu^2-26*a^10*nu*b^2-648*nu^6*b^7*c-2217*b^5*a^4*nu^4-1350*b^6*a^2*nu^5-1136*nu^3*a^6*b^4+7*a^11*b*(-a+sqrt(a^2+8*b*nu))+(1/4)*a^13*(-a+sqrt(a^2+8*b*nu))/nu-2*b^2*a^9*nu*(-a+sqrt(a^2+8*b*nu))*c^2-272*b^4*a^5*nu^3*(-a+sqrt(a^2+8*b*nu))*c^2-40*b^3*a^7*nu^2*(-a+sqrt(a^2+8*b*nu))*c^2-8*b^4*a^5*nu^3*(-a+sqrt(a^2+8*b*nu))*c^3-459*nu^5*(-a+sqrt(a^2+8*b*nu))*a*b^6*c^2-270*nu^5*(-a+sqrt(a^2+8*b*nu))*a*b^6*c^3+135*nu^5*(-a+sqrt(a^2+8*b*nu))*a*b^6*c^4+4*b^4*a^5*nu^3*c^4*(-a+sqrt(a^2+8*b*nu))+2*b^2*a^9*nu*(-a+sqrt(a^2+8*b*nu))*c+594*nu^5*(-a+sqrt(a^2+8*b*nu))*a*b^6*c+744*nu^4*(-a+sqrt(a^2+8*b*nu))*a^3*b^5*c+276*b^4*a^5*nu^3*(-a+sqrt(a^2+8*b*nu))*c+40*b^3*a^7*nu^2*(-a+sqrt(a^2+8*b*nu))*c-696*nu^4*(-a+sqrt(a^2+8*b*nu))*a^3*b^5*c^2-96*nu^4*(-a+sqrt(a^2+8*b*nu))*a^3*b^5*c^3+48*nu^4*(-a+sqrt(a^2+8*b*nu))*a^3*b^5*c^4+(151/2)*b^2*a^9*nu*(-a+sqrt(a^2+8*b*nu))+162*nu^5*(-a+sqrt(a^2+8*b*nu))*a*b^6+918*nu^4*(-a+sqrt(a^2+8*b*nu))*a^3*b^5+(3809/4)*b^4*a^5*nu^3*(-a+sqrt(a^2+8*b*nu))+389*b^3*a^7*nu^2*(-a+sqrt(a^2+8*b*nu)))))$$

代人参数params1 := {a = 1, b = 1, c = 1.5, nu = 1} 得

$$q2 := .50000000000000000000-.50000000000000000000*tanh((.12500000000000000000-.33071891388307382381*I)*x+(1.1875000000000000000+.16535945694153691190*I)*t)$$

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## 参考文献

- *谷超豪 《孤立子理论中的达布变换及其几何应用》上海科学技术出版社
- *阎振亚著 《复杂非线性波的构造性理论及其应用》科学出版社 2007年
- 李志斌编著 《非线性数学物理方程的行波解》科学出版社
- 王东明著 《消去法及其应用》科学出版社 2002
- *何青 王丽芬编著 《Maple 教程》科学出版社 2010 ISBN 9787030177445
- Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
- Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
- Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
- Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
- Dongming Wang, Elimination Practice,Imperial College Press 2004
- David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
- George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759