非线性偏微分方程

非线性偏微分方程是具有非线性项的偏微分方程。起源于各种应用科学中，如固体力学、流体力学、声学、非线性光学、等离子体物理学、量子场论等学科。它们描述了许多不同的物理系统，并被用于解决数学问题，如庞加莱猜想和卡拉比猜想。它们很难研究：几乎没有通用的技术可以用于所有这样的方程，通常每个单独的方程都必须作为一个单独的问题进行研究。通常，线性和非线性偏微分方程的区别是基于定义偏微分方程本身的算子的属性。

函数关系 F(,x_1,X_2..x_n,u,u_x1,u_x2..u_xn,u_x1x2,u_x1x3...)=0 是一个广义的偏微分方程，如果 u，v 是此微分方程的两个解，而（au+bv) 也是此微分方程的解，则此偏微分方程称为线性偏微分方程，否则称为非线性偏微分方程。

## 常见的非线性偏微分方程

Equation｜中文｜方程｜图

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

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Belousov-Zhabotinsky｜别洛乌索夫-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｜本杰明-小野方程｜$u_{tt}+\alpha*(u^2)_{xx}-\beta*u_{xxxx}=0$｜

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Bogoyavlenski-Konoplechenko｜波格雅夫连斯基-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-英费尔德方程｜$\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0$｜

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

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

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Unnormalized Boussinesq｜非规范博欣内斯克方程｜$u_{tt}-\alpha*(u*u_{x})_{x}-\beta*u_{xxxx}=0$｜

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Broer-Kaup｜布罗尔-库普方程组｜$u_{y,t}+(2*u*u_{x})_{x}+2*v_{xx}-u_{xxy}=0$ $v_{t}+2*(vu)_{x}+v_{xx}=0$｜

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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-费希尔 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｜变形伯格斯方程｜$u_{t}+\frac{k}{t}*u+b*u*u_{x}=a*u_{xx}$｜

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Unnormalized Burgers｜非规范伯格斯方程｜$u_{t}-\alpha*u_{xx}-\beta*u*u_{x}=0$｜

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Generalized Burgers｜广义伯格斯方程

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

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Cahn-Hilliard｜卡恩-希利亚德方程

Cassama-Holm｜卡马萨-霍尔姆方程｜:$u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}$｜

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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｜戴维-斯图尔森方程组｜:$i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \phi_x,\,$ :$\phi_{xx} + c_3 \phi_{yy} = ( |u|^2 )_x.\,$｜

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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｜非线性扩散方程｜$u_{t}=\alpha*u_{xx}-\beta*u^3-\gamma*u^2$｜

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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｜埃斯特韦斯-曼斯菲尔德-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 - 南云方程｜$\frac{\partial u}{\partial t}=D*\frac{\partial^2 u}{\partial^2 x^2}-u*(1-u)*(a-u)$｜

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Fisher｜费希尔方程｜$u_{t}=u_{xx}+a*u*(1-u)$｜

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

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

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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｜金兹堡－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｜广田-萨摩方程组｜:$u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0$ :$v_{t}+v_{xxx}-3uv_{x}=0$ :$w_{t}+w_{xxx}-3uw_{x}=0$｜

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

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

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Fifth order KdV｜五阶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｜五阶色散KdV方程｜$u_{t}+\alpha*u*u_{x}+\beta*u_{xxx}+u_{xxxxx}=0$｜

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Seventh order KdV｜七阶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｜九阶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｜非规范KdV方程｜$u_{t}+\alpha*u_{xxx}+\beta*u*u_{x}=0$｜

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Generalized Burgers-KdV｜广义伯格斯-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｜非规范变形KdV方程｜$u_{t}+u_{xxx}+\alpha*u^2*u_{x}=0$｜

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

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## 参考文献

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