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。^[1]

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	:${\displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0.\,}$
Belousov-Zhabotinsky	別洛烏索夫-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	本傑明-小野方程	${\displaystyle u_{tt}+\alpha *(u^{2})_{xx}-\beta *u_{xxxx}=0}$
Bogoyavlenski-Konoplechenko	波格雅夫連斯基-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-英費爾德方程	${\displaystyle \displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0}$
Boussinesq	博欣內斯克方程	${\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	博欣內斯克型方程	${\displaystyle u_{tt}-u_{xx}-2*\alpha *(u*u_{x})_{x}-\beta *u_{xxtt}=0}$
Unnormalized Boussinesq	非規範博欣內斯克方程	${\displaystyle u_{tt}-\alpha *(u*u_{x})_{x}-\beta *u_{xxxx}=0}$
Broer-Kaup	布羅爾-庫普方程組	${\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-費希爾 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	變形伯格斯方程	${\displaystyle u_{t}+{\frac {k}{t}}*u+b*u*u_{x}=a*u_{xx}}$
Unnormalized Burgers	非規範伯格斯方程	${\displaystyle u_{t}-\alpha *u_{xx}-\beta *u*u_{x}=0}$
Generalized Burgers	廣義伯格斯方程	${\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	布雷瑟頓方程	${\displaystyle u_{tt}+u_{xx}+u_{xxxx}-\alpha *u^{3}=0}$
Cahn-Hilliard	卡恩-希利亞德方程
Cassama-Holm	卡馬薩-霍爾姆方程	:${\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	戴維-斯圖爾森方程組	:${\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	非線性擴散方程	${\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	埃斯特韋斯-曼斯菲爾德-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 - 南雲方程	${\displaystyle {\frac {\partial u}{\partial t}}=D*{\frac {\partial ^{2}u}{\partial ^{2}x^{2}}}-u*(1-u)*(a-u)}$
Fisher	費希爾方程	${\displaystyle u_{t}=u_{xx}+a*u*(1-u)}$
Fisher-Kolmogorov	費希爾-柯爾莫哥洛夫方程	::${\displaystyle {\frac {\partial u}{\partial t}}={\frac {\alpha }{k}}*u(1-u^{q})+{\frac {\partial ^{2}u}{\partial x^{2}}}.\,}$
Fujita-Storm	藤田-斯托姆方程	${\displaystyle u_{t}=a*(u^{-2}*u_{x})_{x}}$
Gardner	加德納方程	${\displaystyle {\frac {\partial u}{\partial t}}+(2*a*u-3*b*u^{2})*{\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0}$
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	金茲堡－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	廣田-薩摩方程組	:${\displaystyle u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0}$ ${\displaystyle v_{t}+v_{xxx}-3uv_{x}=0}$ ${\displaystyle w_{t}+w_{xxx}-3uw_{x}=0}$
Hunt-Saxton	亨特 - 薩克斯頓方程	:${\displaystyle (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}}$
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	變形KdV-Burgers方程	${\displaystyle u_{t}+u_{xxx}-\alpha *u^{2}*u_{x}-\beta *u_{xx}=0}$
Fifth order KdV	五階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	五階色散KdV方程	${\displaystyle u_{t}+\alpha *u*u_{x}+\beta *u_{xxx}+u_{xxxxx}=0}$
Seventh order KdV	七階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	九階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	非規範KdV方程	${\displaystyle u_{t}+\alpha *u_{xxx}+\beta *u*u_{x}=0}$
Generalized Burgers-KdV	廣義伯格斯-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	非規範變形KdV方程	${\displaystyle u_{t}+u_{xxx}+\alpha *u^{2}*u_{x}=0}$
von Karman	馮·卡門方程	${\displaystyle \Delta \Delta (u)=a((w_{xy})^{2}-w_{xx}w_{yy})}$ ${\displaystyle \Delta \Delta (w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c}$