非线性偏微分方程列表
Lax 可积系统

| Equation | 中文 | 方程 | 图 |  |  |
| --- | --- | --- | --- | --- | --- |
| BBM | 班傑明-小野方程 |  : u_{t}+u_{x}+uu_{x}-u_{xxt}=0.\, |  |  |  |
| 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  |  |  |  |
| (Benjamin-Ono equation | 本杰明-小野方程 |  u_{tt}+\alpha *(u^{2})_{xx}-\beta *u_{xxxx}=0  |  |  |  |
| 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  |  |  |  |
| Born-Infeld | 玻恩-英费尔德方程 |  \displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0 |  |  |  |
| 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 |  |  |  |
| Boussinesa type | 博欣内斯克型方程 |  u_{tt}-u_{xx}-2*\alpha *(u*u_{x})_{x}-\beta *u_{xxtt}=0  |  |  |  |
| Unnormalized Boussinesq | 非规范博欣内斯克方程 |  u_{tt}-\alpha *(u*u_{x})_{x}-\beta *u_{xxxx}=0 |  |  |  |
| 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  |  |  |  |
| 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 |  |  |  |
| 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}) |  |  |  |
| Modified Burgers | 变形伯格斯方程 |  u_{t}+{\frac {k}{t}}*u+b*u*u_{x}=a*u_{xx} |  |  |  |
| Unnormalized Burgers | 非规范伯格斯方程 |  u_{t}-\alpha *u_{xx}-\beta *u*u_{x}=0 |  |  |  |
| 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) |  |  |  |
| Bretherton | 布雷瑟顿方程 |  u_{tt}+u_{xx}+u_{xxxx}-\alpha *u^{3}=0 |  |  |  |
| 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}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}  |  |  |  |
| Chaffee-Infante | 查菲 - 堙方特方程 |  u_{t}-u_{xx}+\lambda *(u^{3}-u)=0 |  |  |  |
| Chaplygin | 查普里金方程 |  0.5*u_{tt}+u_{x}*u_{xt}-u_{t}*u_{xx}=0 |  |  |  |
| Davey–Stewartson | 戴维-斯图尔森方程组 | : iu_{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}.\,  |  |  |  |
| Degasperis-Procesi | DP 方程 |  \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}  |  |  |  |
| 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}} |  |  |  |
| Dodd-Bullough-Mikhailov | 多德-布洛-米哈伊洛夫方程 | [[ u_{xt}+\alpha *e^{u}+\gamma *e^{-2*u}=0 |  |  |  |
| Nonlinear Diffusion | 非线性扩散方程 |  u_{t}=\alpha *u_{xx}-\beta *u^{3}-\gamma *u^{2} |  |  |  |
| Harry Dym | 迪姆方程 | : u_{t}=u^{3}u_{xxx}.\, |  |  |  |
| 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 |  |  |  |
| Eikonal | 程函方程 |  sys:=(u(x,t)_{t}))^{2}+(u(x,t)_{x})^{2}-4=0 |  |  |  |
| 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  |  |  |  |
| 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) |  |  |  |
| Fisher | 费希尔方程 |  u_{t}=u_{xx}+a*u*(1-u) |  |  |  |
| 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}}}.\, |  |  |  |
| Fujita-Storm | 藤田-斯托姆方程 |  u_{t}=a*(u^{-2}*u_{x})_{x} |  |  |  |
| 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 |  |  |  |
| Gibbons-Tsarev | 吉本斯-查理夫方程 |  u_{t}*u_{xt}-u_{x}*u_{tt}+u_{xx}+1=0 |  |  |  |
| 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  |  |  |  |
| 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 |  |  |  |
| Hunt-Saxton | 亨特 - 萨克斯顿方程 | : (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2} |  |  |  |
| Ito | 伊藤方程 |  U_{t}+((6*U^{5}+10*\alpha *(U^{2}*U_{xx}+U*U_{x}^{2})+U_{xxxx})_{x}=0 |  |  |  |
| KdV | KdV方程 | : \partial _{t}\phi +6\phi \partial _{x}\phi +\partial _{x}^{3}\phi =0  |  |  |  |
| Modified KdV | MKdV方程 |  u_{t}+\alpha *u^{2}*u_{x}+u_{xxx}=0  |  |  |  |
| KdV-mKdV | KdV-mKdV方程 |  u_{t}+6*\alpha *u*u_{x}+6*\beta *u^{2}*u_{}+\gamma *U_{xxx}=0 |  |  |  |
| KdV-Burgers | KdV-Burgers方程 |  u_{t}+u*u_{x}-\alpha *u_{xx}-\beta *u_{xxx}=0 |  |  |  |
| Modified KdV-Burgers | 变形KdV-Burgers方程 |  u_{t}+u_{xxx}-\alpha *u^{2}*u_{x}-\beta *u_{xx}=0  |  |  |  |
| 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 |  |  |  |
| Fifth order dispersion KdV | 五阶色散KdV方程 |  u_{t}+\alpha *u*u_{x}+\beta *u_{xxx}+u_{xxxxx}=0 |  |  |  |
| 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 |  |  |  |
| 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 |  |  |  |
| Unnormalized KdV equation | 非规范KdV方程 |  u_{t}+\alpha *u_{xxx}+\beta *u*u_{x}=0  |  |  |  |
| 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 |  |  |  |
| Unnormalized modified  KdV | 非规范变形KdV方程 |  u_{t}+u_{xxx}+\alpha *u^{2}*u_{x}=0 |  |  |  |
| 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  |  |  |  |