非线性偏微分方程列表

非线性偏微分方程的在物理学、气动力学、流体力学、大气物理、海洋物理、爆炸物理、化学、生理学、生物学、生态学等领域都有重要的应用。非线性偏微分方程的研究，是当前微分方程研究的中心。求解非线性偏微分方程比求解线性偏微分方程，难度大的多，大多数非线性偏微分方程只能依靠数值解法。但多年来数学家们发现了一些行之有效的求解非线性偏微分方程的构造性解法，如反散射法、达布变换法，tanh、雅可比函数展开法等，得出非线性偏微分方程的解析解。解非线性偏微分方程，过程复杂，多数得力于Maple、Mathematica、Matlab等商用计算机代数系統。

已知的非线性偏微分方程，数目不下3000余种，但有名的不过一百多种，多以发现者命名。

分离变数法

反散射法

达布变换

混合指数法

齐次平衡法

Tanh 函数展开法

主条目：Tanh 函数展开法

Tanh 函数展开法是求解非线性偏微分方程行波解的重要方法。

设一个非线性偏微分方程可以用下列表述：

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

作变数代换：

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

${\displaystyle \xi =k*(x-c*t)}$ 

得到：

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

1992年数学家 Malfliet 首先应用 tanh 展开法^[1]

对称分析

Lax 可积系统

Equation	中文	方程	图
BBM	班傑明-小野方程	:${\displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0.\,}$
Belousov-Zhabotinsky	别洛乌索夫-扎伯廷斯基方程	${\displaystyle u_{t}=d*u_{xx}+u*(1-r*u-u)}$ , ${\displaystyle v_{t}=v_{xx}-s*u*v}$
(Benjamin-Ono equation	本杰明-小野方程	${\displaystyle u_{tt}+\alpha *(u^{2})_{xx}-\beta *u_{xxxx}=0}$
Bogoyavlenski-Konoplechenko	波格雅夫连斯基-科譳普勒琛科方程	${\displaystyle u_{xt}+\alpha *u_{xxxx}+\beta *u_{xxxy}}$ ${\displaystyle +6*\alpha *u_{xx}*u_{x}+4*\beta *u_{xy}*u_{x}+\beta *u_{xx}*u_{y}=0}$
Born-Infeld	玻恩-英费尔德方程	${\displaystyle \displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0}$
Boussinesq	博欣内斯克方程	${\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}-{\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u^{2}}{\partial ^{2}y^{2}}}+{\frac {\partial ^{4}u}{\partial x^{4}}}=0}$
Boussinesa type	博欣内斯克型方程	${\displaystyle u_{tt}-u_{xx}-2*\alpha *(u*u_{x})_{x}-\beta *u_{xxtt}=0}$
Unnormalized Boussinesq	非规范博欣内斯克方程	${\displaystyle u_{tt}-\alpha *(u*u_{x})_{x}-\beta *u_{xxxx}=0}$
Broer-Kaup	布罗尔-库普方程组	${\displaystyle u_{y,t}+(2*u*u_{x})_{x}+2*v_{xx}-u_{xxy}=0}$  ${\displaystyle v_{t}+2*(vu)_{x}+v_{xx}=0}$
Burgers	伯格斯方程	${\displaystyle {\frac {\partial u(x,t)}{\partial t}}+u(x,t)*{\frac {\partial (u(x,t)}{\partial x}}-nu*{\frac {\partial ^{2}(u(x,t)}{\partial x^{2}}}=0}$
Burgers-Fisher	伯格斯-费希尔 方程	:${\displaystyle {\frac {\partial u}{\partial t}}+u^{2}*{\frac {\partial u}{\partial x}}-{\frac {\partial ^{2}u}{\partial u^{2}}}=u*(1-u^{2})}$
Modified Burgers	变形伯格斯方程	${\displaystyle u_{t}+{\frac {k}{t}}*u+b*u*u_{x}=a*u_{xx}}$
Unnormalized Burgers	非规范伯格斯方程	${\displaystyle u_{t}-\alpha *u_{xx}-\beta *u*u_{x}=0}$
Generalized Burgers	广义伯格斯方程	${\displaystyle }$
Burgers-Huxley	伯格斯-赫胥黎方程	${\displaystyle u[t]-nu*u[x,x]+a*u*u[x]=b*u*(1-u)*(u-c)}$
Bretherton	布雷瑟顿方程	${\displaystyle u_{tt}+u_{xx}+u_{xxxx}-\alpha *u^{3}=0}$
Cahn-Hilliard	卡恩-希利亚德方程
Cassama-Holm	卡马萨-霍尔姆方程	:${\displaystyle u_{t}+2\kappa u_{x}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}}$
Chaffee-Infante	查菲 - 堙方特方程	${\displaystyle u_{t}-u_{xx}+\lambda *(u^{3}-u)=0}$
Chaplygin	查普里金方程	${\displaystyle 0.5*u_{tt}+u_{x}*u_{xt}-u_{t}*u_{xx}=0}$
Davey–Stewartson	戴维-斯图尔森方程组	:${\displaystyle iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi _{x},\,}$  ${\displaystyle \phi _{xx}+c_{3}\phi _{yy}=(|u|^{2})_{x}.\,}$
Degasperis-Procesi	DP 方程	${\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}}$
Drinfeld-Solokov-Wilson	DSW 方程	${\displaystyle {\frac {\partial u}{\partial t}}+3*v*{\frac {\partial v}{\partial x}}=0}$  ${\displaystyle {\frac {\partial v}{\partial t}}-2*{\frac {\partial ^{3}v}{\partial x^{3}}}+{\frac {\partial u}{\partial x}}*v+2u*{\frac {\partial v}{\partial x}}}$
Dodd-Bullough-Mikhailov	多德-布洛-米哈伊洛夫方程	[[${\displaystyle u_{xt}+\alpha *e^{u}+\gamma *e^{-2*u}=0}$
Nonlinear Diffusion	非线性扩散方程	${\displaystyle u_{t}=\alpha *u_{xx}-\beta *u^{3}-\gamma *u^{2}}$
Harry Dym	迪姆方程	:${\displaystyle u_{t}=u^{3}u_{xxx}.\,}$
Eckhaus	艾克豪斯方程	${\displaystyle u(x,y,t)_{t}+v(x,t)_{x}+1.0*u(x,y,t)*(u(x,y,t)_{x})=0}$  ${\displaystyle v(x,t)_{xt}+u(x,y,t)_{xx}*v(x,t)+}$ ${\displaystyle 2*u(x,y,t)_{x}*v(x,t)_{x}+u(x,y,t)*(v(x,t)_{xx}+}$ ${\displaystyle u(x,y,t)_{xx}+u(x,y,t)_{xxxx}+u(x,y,t)_{yy}=0}$
Eikonal	程函方程	${\displaystyle sys:=(u(x,t)_{t}))^{2}+(u(x,t)_{x})^{2}-4=0}$
Estevez-Mansfield-Clarkson	埃斯特韦斯-曼斯菲尔德-克拉克森方程	${\displaystyle u_{tyyy}+\beta *u_{y}*u_{yt}+\beta *u_{yy}*u_{t}+u_{tt}=0}$
Fitzhugh-Nagumo	菲茨休 - 南云方程	${\displaystyle {\frac {\partial u}{\partial t}}=D*{\frac {\partial ^{2}u}{\partial ^{2}x^{2}}}-u*(1-u)*(a-u)}$
Fisher	费希尔方程	${\displaystyle u_{t}=u_{xx}+a*u*(1-u)}$
Fisher-Kolmogorov	费希尔-柯尔莫哥洛夫方程	::${\displaystyle {\frac {\partial u}{\partial t}}={\frac {\alpha }{k}}*u(1-u^{q})+{\frac {\partial ^{2}u}{\partial x^{2}}}.\,}$
Fujita-Storm	藤田-斯托姆方程	${\displaystyle u_{t}=a*(u^{-2}*u_{x})_{x}}$
Gardner	加德纳方程	${\displaystyle {\frac {\partial u}{\partial t}}+(2*a*u-3*b*u^{2})*{\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0}$
Gibbons-Tsarev	吉本斯-查理夫方程	${\displaystyle u_{t}*u_{xt}-u_{x}*u_{tt}+u_{xx}+1=0}$
Ginzburg-Landau	金兹堡－朗道方程	${\displaystyle {\frac {\partial u}{\partial t}}-a*u*{\frac {\partial ^{2}u}{\partial x^{2}}}-b*u+c*|u|^{2}*u=0}$
Hirota Satsuma	广田-萨摩方程组	:${\displaystyle u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0}$  ${\displaystyle v_{t}+v_{xxx}-3uv_{x}=0}$  ${\displaystyle w_{t}+w_{xxx}-3uw_{x}=0}$
Hunt-Saxton	亨特 - 萨克斯顿方程	:${\displaystyle (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}}$
Ito	伊藤方程	${\displaystyle U_{t}+((6*U^{5}+10*\alpha *(U^{2}*U_{xx}+U*U_{x}^{2})+U_{xxxx})_{x}=0}$
KdV	KdV方程	:${\displaystyle \partial _{t}\phi +6\phi \partial _{x}\phi +\partial _{x}^{3}\phi =0}$
Modified KdV	MKdV方程	${\displaystyle u_{t}+\alpha *u^{2}*u_{x}+u_{xxx}=0}$
KdV-mKdV	KdV-mKdV方程	${\displaystyle u_{t}+6*\alpha *u*u_{x}+6*\beta *u^{2}*u_{}+\gamma *U_{xxx}=0}$
KdV-Burgers	KdV-Burgers方程	${\displaystyle u_{t}+u*u_{x}-\alpha *u_{xx}-\beta *u_{xxx}=0}$
Modified KdV-Burgers	变形KdV-Burgers方程	${\displaystyle u_{t}+u_{xxx}-\alpha *u^{2}*u_{x}-\beta *u_{xx}=0}$
Fifth order KdV	五阶KdV方程	${\displaystyle u_{t}+\alpha *u^{2}*u_{x}+\beta *u_{x}*u_{xx}+\gamma *u*u_{xxx}+\delta *u_{xxxxx}=0}$
Fifth order dispersion KdV	五阶色散KdV方程	${\displaystyle u_{t}+\alpha *u*u_{x}+\beta *u_{xxx}+u_{xxxxx}=0}$
Seventh order KdV	七阶KdV方程	${\displaystyle U[t]+6*U*U[x]+U[x,x,x]-U[x,x,x,x,x]+\alpha *U[x,x,x,x,x,x,x,x,x]=0}$
Nineth order KdV	九阶KdV方程	${\displaystyle U[t]+6*U*U[x]+U[x,x,x]-U[x,x,x,x,x]+\alpha *U[x,x,x,x,x,x,x]+\beta *U[x,x,x,x,x,x,x,x,x]=0}$
Unnormalized KdV equation	非规范KdV方程	${\displaystyle u_{t}+\alpha *u_{xxx}+\beta *u*u_{x}=0}$
Generalized Burgers-KdV	广义伯格斯-KdV方程	${\displaystyle U[t]-\alpha *{\frac {\partial ^{n}u(x,t)}{\partial x^{n}}}-\beta *u(x,t)*{\frac {\partial u(x,t)}{\partial x}}=0}$
Unnormalized modified KdV	非规范变形KdV方程	${\displaystyle u_{t}+u_{xxx}+\alpha *u^{2}*u_{x}=0}$
von Karman	冯·卡门方程	${\displaystyle \Delta \Delta (u)=a((w_{xy})^{2}-w_{xx}w_{yy})}$  ${\displaystyle \Delta \Delta (w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c}$

参考文献

1. W. Malfliet, Solitary Wave Solution of Nonlinear wave equation, Am J.of Physics 60(7) 1992,650-654

1. *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
2. *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
3. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
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15. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

外部链接

• 牛津大学非线性偏微分方程研究中心 （页面存档备份，存于互联网档案馆）