非線性偏微分方程

非線性偏微分方程是具有非線性項的偏微分方程。起源於各種應用科學中，如固體力學、流體力學、聲學、非線性光學、等離子體物理學、量子場論等學科。它們描述了許多不同的物理系統，並被用於解決數學問題，如龐加萊猜想和卡拉比猜想。它們很難研究：幾乎沒有通用的技術可以用於所有這樣的方程，通常每個單獨的方程都必須作為一個單獨的問題進行研究。通常，線性和非線性偏微分方程的區別是基於定義偏微分方程本身的算子的屬性。

函數關係 F(,x_1,X_2..x_n,u,u_x1,u_x2..u_xn,u_x1x2,u_x1x3...)=0 是一個廣義的偏微分方程，如果 u，v 是此微分方程的兩個解，而（au+bv) 也是此微分方程的解，則此偏微分方程稱為線性偏微分方程，否則稱為非線性偏微分方程。

## 常見的非線性偏微分方程

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

- *谷超豪 《孤立子理論中的達布變換及其幾何應用》上海科學技術出版社
- *閻振亞著 《複雜非線性波的構造性理論及其應用》科學出版社 2007年
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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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