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。^[1]

常见的非线性偏微分方程

Equation	中文	方程
BBM	班杰明-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	$u_{t}=du_{xx}+u(1-ru-u)$ , $v_{t}=v_{xx}-su*v$
(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	$u_{tt}-u_{xx}-2\alpha (uu_{x})_{x}-\beta u_{xxtt}=0$
Unnormalized Boussinesq	非规范博欣内斯克方程	$u_{tt}-\alpha (uu_{x})_{x}-\beta *u_{xxxx}=0$
Broer-Kaup	布罗尔-库普方程组	$u_{y,t}+(2uu_{x})_{x}+2v_{xx}-u_{xxy}=0$ $v_{t}+2(vu)_{x}+v_{xx}=0$
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	$u_{t}+{\frac {k}{t}}u+buu_{x}=au_{xx}$
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	$u[t]-nuu[x,x]+auu[x]=bu(1-u)(u-c)$
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	$0.5u_{tt}+u_{x}u_{xt}-u_{t}*u_{xx}=0$
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	${\frac {\partial u}{\partial t}}+3v{\frac {\partial v}{\partial x}}=0$ ${\frac {\partial v}{\partial t}}-2{\frac {\partial ^{3}v}{\partial x^{3}}}+{\frac {\partial u}{\partial x}}v+2u*{\frac {\partial v}{\partial x}}$
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	$u(x,y,t)_{t}+v(x,t)_{x}+1.0u(x,y,t)(u(x,y,t)_{x})=0$ $v(x,t)_{xt}+u(x,y,t)_{xx}v(x,t)+$ $2u(x,y,t)_{x}v(x,t)_{x}+u(x,y,t)(v(x,t)_{xx}+$ $u(x,y,t)_{xx}+u(x,y,t)_{xxxx}+u(x,y,t)_{yy}=0$
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	$u_{t}=u_{xx}+au(1-u)$
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	${\frac {\partial u}{\partial t}}+(2au-3bu^{2})*{\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0$
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	${\frac {\partial u}{\partial t}}-au{\frac {\partial ^{2}u}{\partial x^{2}}}-bu+c\|u\|^{2}*u=0$
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	$U_{t}+((6U^{5}+10\alpha (U^{2}U_{xx}+U*U_{x}^{2})+U_{xxxx})_{x}=0$
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	$u_{t}+6\alpha uu_{x}+6\beta u^{2}u_{}+\gamma *U_{xxx}=0$
KdV-Burgers	KdV-Burgers方程	$u_{t}+uu_{x}-\alpha u_{xx}-\beta *u_{xxx}=0$
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	$u_{t}+\alpha u^{2}u_{x}+\beta u_{x}u_{xx}+\gamma uu_{xxx}+\delta *u_{xxxxx}=0$
Fifth order dispersion KdV	五阶色散KdV方程	$u_{t}+\alpha uu_{x}+\beta *u_{xxx}+u_{xxxxx}=0$
Seventh order KdV	七阶KdV方程	$U[t]+6UU[x]+U[x,x,x]-U[x,x,x,x,x]+\alpha *U[x,x,x,x,x,x,x,x,x]=0$
Nineth order KdV	九阶KdV方程	$U[t]+6UU[x]+U[x,x,x]-U[x,x,x,x,x]+\alpha U[x,x,x,x,x,x,x]+\beta U[x,x,x,x,x,x,x,x,x]=0$
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	$\Delta \Delta (u)=a((w_{xy})^{2}-w_{xx}w_{yy})$ $\Delta \Delta (w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c$

参考文献

^ Inna Shingareva Carlos Lizarrga Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, Springer, ISBN 9783709105160

*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
*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
李志斌编著《非线性数学物理方程的行波解》科学出版社
王东明著《消去法及其应用》科学出版社 2002
*何青王丽芬编著《Maple 教程》科学出版社 2010 ISBN 9787030177445
Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION CRC PRESS
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Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
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Dongming Wang, Elimination Practice,Imperial College Press 2004
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[1] Inna Shingareva Carlos Lizarrga Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, Springer, ISBN 9783709105160

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