非線性偏微分方程列表

非線性偏微分方程的在物理學、氣動力學、流體力學、大氣物理、海洋物理、爆炸物理、化學、生理學、生物學、生態學等領域都有重要的應用。非線性偏微分方程的研究，是當前微分方程研究的中心。求解非線性偏微分方程比求解線性偏微分方程，難度大的多，大多數非線性偏微分方程只能依靠數值解法。但多年來數學家們發現了一些行之有效的求解非線性偏微分方程的構造性解法，如反散射法、達布變換法，tanh、雅可比函數展開法等，得出非線性偏微分方程的解析解。解非線性偏微分方程，過程複雜，多數得力於Maple、Mathematica、Matlab等商用電腦代數系統。

已知的非線性偏微分方程，數目不下3000餘種，但有名的不過一百多種，多以發現者命名。

## 分離變數法

可以藉代數來將方程式重新編排，讓方程式的一部分只含有一個變量，而剩餘部分則跟此變量無關。這樣，隔離出的兩個部分的值，都分別等於常數，而兩個部分的值的代數和等於零。

## Tanh 函數展開法

Tanh 函數展開法是求解非線性偏微分方程行波解的重要方法。

設一個非線性偏微分方程可以用下列表述：

$$\psi(u,u_{t},u_{x},u_{tt},u_{xx},u_{tx})=0$$

作變數代換：

$u(x,t)$-> $U(\xi)$

$$\xi=k*(x-c*t)$$

得到：

$\psi(U(\psi),-kc*\frac{\partial U}{\partial \psi},k*\frac{\partial U}{\partial \psi},$ $k^2*c^2*\frac{\partial^2 U}{\partial \psi^2}$ $,k^2*\frac{\partial^2 U}{\partial \psi^2},-k^3*c^3*\frac{\partial^3 U}{\partial \psi^3},k^3*\frac{\partial^3 U}{\partial \psi^3})=0$

1992年數學家 Malfliet 首先應用 tanh 展開法

## Lax 可積系統

Equation｜中文｜方程｜圖

BBM｜班傑明-小野方程｜:$u_t+u_x+uu_x-u_{xxt}=0.\,$｜

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Belousov-Zhabotinsky｜別洛烏索夫-扎伯廷斯基方程｜$u_{t}=d*u_{xx}+u*(1-r*u-u)$, $v_{t}=v_{xx}-s*u*v$｜

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(Benjamin-Ono equation｜班傑明-小野方程｜$u_{tt}+\alpha*(u^2)_{xx}-\beta*u_{xxxx}=0$｜

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Bogoyavlenski-Konoplechenko｜波格雅夫連斯基-科譳普勒琛科方程｜$u_{xt}+\alpha*u_{xxxx}+\beta*u_{xxxy}$ $+6*\alpha*u_{xx}*u_{x}+4*\beta*u_{xy}*u_{x}+\beta*u_{xx}*u_{y}=0$｜

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Born-Infeld｜玻恩-英費爾德方程｜$\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0$｜

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Boussinesq｜博欣內斯克方程｜$\frac{\partial^2 u }{\partial t^2}-\frac{\partial^2 u }{\partial x^2 }-\frac{\partial^2 u^2 }{\partial^2 y^2 }+\frac{\partial^4 u }{\partial x^4}=0$｜

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Boussinesa type｜博欣內斯克型方程｜$u_{tt}-u_{xx}-2*\alpha*(u*u_{x})_{x}-\beta*u_{xxtt}=0$｜

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Unnormalized Boussinesq｜非規範博欣內斯克方程｜$u_{tt}-\alpha*(u*u_{x})_{x}-\beta*u_{xxxx}=0$｜

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Broer-Kaup｜布羅爾-庫普方程組｜$u_{y,t}+(2*u*u_{x})_{x}+2*v_{xx}-u_{xxy}=0$ $v_{t}+2*(vu)_{x}+v_{xx}=0$｜

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Burgers｜伯格斯方程｜$\frac{\partial u(x, t)}{\partial t}+u(x, t)*\frac{\partial (u(x, t)}{\partial x}-nu*\frac{\partial^2 (u(x, t)}{\partial x^2} = 0$｜

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Burgers-Fisher｜伯格斯-費希爾 方程｜:$\frac{\partial u}{\partial t}+u^2*\frac{\partial u}{\partial x}-\frac{\partial^2 u}{\partial u^2}=u*(1-u^2)$｜

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Modified Burgers｜變形伯格斯方程｜$u_{t}+\frac{k}{t}*u+b*u*u_{x}=a*u_{xx}$｜

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Unnormalized Burgers｜非規範伯格斯方程｜$u_{t}-\alpha*u_{xx}-\beta*u*u_{x}=0$｜

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Generalized Burgers｜廣義伯格斯方程

Burgers-Huxley｜伯格斯-赫胥黎方程｜$u[t]-nu*u[x, x]+a*u*u[x] = b*u*(1-u)*(u-c)$｜

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Bretherton｜布雷瑟頓方程｜$u_{tt}+u_{xx}+u_{xxxx}-\alpha*u^3=0$｜

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Cahn-Hilliard｜卡恩-希利亞德方程

Cassama-Holm｜卡馬薩-霍爾姆方程｜:$u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}$｜

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Chaffee-Infante｜查菲 - 堙方特方程｜$u_{t}-u_{xx}+\lambda*(u^3-u)=0$｜

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Chaplygin｜查普里金方程｜$0.5*u_{tt}+u_{x}*u_{xt}-u_{t}*u_{xx}=0$｜

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Davey–Stewartson｜戴維-斯圖爾森方程組｜:$i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \phi_x,\,$ :$\phi_{xx} + c_3 \phi_{yy} = ( |u|^2 )_x.\,$｜

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Degasperis-Procesi｜DP 方程｜$\displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx}$｜

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Drinfeld-Solokov-Wilson｜DSW 方程｜$\frac{\partial u}{\partial t}+3*v*\frac{\partial v}{\partial x}=0$ $\frac{\partial v}{\partial t}-2*\frac{\partial^3 v}{\partial x^3}+\frac{\partial u}{\partial x}*v+2u*\frac{\partial v}{\partial x}$｜

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Dodd-Bullough-Mikhailov｜多德-布洛-米哈伊洛夫方程｜$u_{xt}+\alpha*e^u+\gamma*e^{-2*u} = 0$｜

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Nonlinear Diffusion｜非線性擴散方程｜$u_{t}=\alpha*u_{xx}-\beta*u^3-\gamma*u^2$｜

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Harry Dym｜迪姆方程｜:$u_t = u^3u_{xxx}.\,$｜

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Eckhaus｜艾克豪斯方程｜$u(x, y, t)_t+v(x, t)_x+1.0*u(x, y, t)*(u(x, y, t)_x) = 0$ $v(x, t)_{xt}+u(x, y, t)_{xx}*v(x, t)+$ $2*u(x, y, t)_x*v(x, t)_x+u(x, y, t)*(v(x, t)_{xx}+$ $u(x, y, t)_{xx}+u(x, y, t)_{xxxx}+u(x, y, t)_{yy} = 0$｜

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Eikonal｜程函方程｜$sys := (u(x, t)_t))^2+(u(x, t)_x)^2-4 = 0$｜

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Estevez-Mansfield-Clarkson｜埃斯特韋斯-曼斯菲爾德-克拉克森方程｜$u_{tyyy}+\beta*u_{y}*u_{yt}+\beta*u_{yy}*u_{t}+u_{tt}=0$｜

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Fitzhugh-Nagumo｜菲茨休 - 南雲方程｜$\frac{\partial u}{\partial t}=D*\frac{\partial^2 u}{\partial^2 x^2}-u*(1-u)*(a-u)$｜

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Fisher｜費希爾方程｜$u_{t}=u_{xx}+a*u*(1-u)$｜

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Fisher-Kolmogorov｜費希爾-柯爾莫哥洛夫方程｜::$\frac{\partial u}{\partial t}=\frac{\alpha}{k}*u(1-u^q)+\frac{\partial^2 u}{\partial x^2}.\,$｜

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Fujita-Storm｜藤田-斯托姆方程｜$u_{t}=a*(u^{-2}*u_{x})_{x}$｜

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Gardner｜加德納方程｜$\frac{\partial u}{\partial t}+(2*a*u-3*b*u^2)*\frac{\partial u}{\partial x }+\frac{\partial^3 u}{\partial x^3}=0$｜

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Gibbons-Tsarev｜吉本斯-查理夫方程｜$u_{t}*u_{xt}-u_{x}*u_{tt}+u_{xx}+1=0$｜

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Ginzburg-Landau｜金茲堡－朗道方程｜$\frac{\partial u}{\partial t}-a*u*\frac{\partial^2 u }{\partial x^2}-b*u+c*|u|^2*u=0$｜

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Hirota Satsuma｜廣田-薩摩方程組｜:$u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0$ :$v_{t}+v_{xxx}-3uv_{x}=0$ :$w_{t}+w_{xxx}-3uw_{x}=0$｜

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Hunt-Saxton｜亨特 - 薩克斯頓方程｜:$(u_t + u u_x)_x = \frac{1}{2} \, u_x^2$｜

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Ito｜伊藤方程｜$U_{t}+((6*U^5+10*\alpha*(U^2*U_{xx}+U*U_{x}^2)+U_{xxxx})_{x}=0$｜

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KdV｜KdV方程｜:$\partial_t\phi+6\phi\partial_x\phi+\partial^3_x\phi=0$｜

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Modified KdV｜MKdV方程｜$u_{t}+\alpha*u^2*u_{x}+u_{xxx}=0$｜

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KdV-mKdV｜KdV-mKdV方程｜$u_{t}+6*\alpha*u*u_{x}+6*\beta*u^2*u_{}+\gamma*U_{xxx}=0$｜

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KdV-Burgers｜KdV-Burgers方程｜$u_{t}+u*u_{x}-\alpha*u_{xx}-\beta*u_{xxx}=0$｜

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Modified KdV-Burgers｜變形KdV-Burgers方程｜$u_{t}+u_{xxx}-\alpha*u^2*u_{x}-\beta*u_{xx}=0$｜

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Fifth order KdV｜五階KdV方程｜$u_{t}+\alpha*u^2*u_{x}+\beta*u_{x}*u_{xx}+\gamma*u*u_{xxx}+\delta*u_{xxxxx}=0$｜

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Fifth order dispersion KdV｜五階色散KdV方程｜$u_{t}+\alpha*u*u_{x}+\beta*u_{xxx}+u_{xxxxx}=0$｜

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Seventh order KdV｜七階KdV方程｜$U[t]+6*U*U[x]+U[x,x,x]-U[x,x,x,x,x]+\alpha*U[x,x,x,x,x,x,x,x,x]=0$｜

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Nineth order KdV｜九階KdV方程｜$U[t]+6*U*U[x]+U[x,x,x]-U[x,x,x,x,x]+\alpha*U[x,x,x,x,x,x,x]+\beta*U[x,x,x,x,x,x,x,x,x]=0$｜

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Unnormalized KdV equation｜非規範KdV方程｜$u_{t}+\alpha*u_{xxx}+\beta*u*u_{x}=0$｜

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Generalized Burgers-KdV｜廣義伯格斯-KdV方程｜$U[t]-\alpha*\frac{\partial^n u(x,t)}{\partial x^n}-\beta*u(x,t)*\frac{\partial u(x,t)}{\partial x}=0$｜

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Unnormalized modified KdV｜非規範變形KdV方程｜$u_{t}+u_{xxx}+\alpha*u^2*u_{x}=0$｜

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von Karman｜馮·卡門方程｜$\Delta\Delta(u)=a((w_{xy})^2-w_{xx}w_{yy})$ $\Delta\Delta(w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c$｜

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## 參考文獻

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- 王東明著 《消去法及其應用》科學出版社 2002
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