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
BBM	本傑明-小野方程	: $u_{t}+u_{x}+uu_{x}-u_{xxt}=0.\,$
Belousov-Zhabotinsky	別洛烏索夫-扎伯廷斯基方程	$u_{t}=du_{xx}+u(1-ru-u)$ , $v_{t}=v_{xx}-su*v$
(Benjamin-Ono equation	本傑明-小野方程	$u_{tt}+\alpha (u^{2})_{xx}-\beta u_{xxxx}=0$
Bogoyavlenski-Konoplechenko	波格雅夫連斯基-科譳普勒琛科方程	$u_{xt}+\alpha u_{xxxx}+\beta u_{xxxy}$ $+6\alpha u_{xx}u_{x}+4\beta u_{xy}u_{x}+\beta u_{xx}u_{y}=0$
Born-Infeld	玻恩-英費爾德方程	$\displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0$
Boussinesq	博欣內斯克方程	${\frac {\partial ^{2}u}{\partial t^{2}}}-{\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u^{2}}{\partial ^{2}y^{2}}}+{\frac {\partial ^{4}u}{\partial x^{4}}}=0$
Boussinesa type	博欣內斯克型方程	$u_{tt}-u_{xx}-2\alpha (uu_{x})_{x}-\beta u_{xxtt}=0$
Unnormalized Boussinesq	非規範博欣內斯克方程	$u_{tt}-\alpha (uu_{x})_{x}-\beta *u_{xxxx}=0$
Broer-Kaup	布羅爾-庫普方程組	$u_{y,t}+(2uu_{x})_{x}+2v_{xx}-u_{xxy}=0$ $v_{t}+2(vu)_{x}+v_{xx}=0$
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-費雪方程	: ${\frac {\partial u}{\partial t}}+u^{2}{\frac {\partial u}{\partial x}}-{\frac {\partial ^{2}u}{\partial u^{2}}}=u(1-u^{2})$
Modified Burgers	變形伯格斯方程	$u_{t}+{\frac {k}{t}}u+buu_{x}=au_{xx}$
Unnormalized Burgers	非規範伯格斯方程	$u_{t}-\alpha u_{xx}-\beta u*u_{x}=0$
Generalized Burgers	廣義伯格斯方程
Burgers-Huxley	伯格斯-赫胥黎方程	$u[t]-nuu[x,x]+auu[x]=bu(1-u)(u-c)$
Bretherton	布雷瑟頓方程	$u_{tt}+u_{xx}+u_{xxxx}-\alpha *u^{3}=0$
Cahn-Hilliard	卡恩-希利亞德方程
Cassama-Holm	卡馬薩-霍爾姆方程	: $u_{t}+2\kappa u_{x}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}$
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	戴維-斯圖爾森方程組	: $iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}\|u\|^{2}u+c_{2}u\phi _{x},\,$ $\phi _{xx}+c_{3}\phi _{yy}=(\|u\|^{2})_{x}.\,$
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	非線性擴散方程	$u_{t}=\alpha u_{xx}-\beta u^{3}-\gamma *u^{2}$
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	埃斯特韋斯-曼斯菲爾德-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 - 南雲方程	${\frac {\partial u}{\partial t}}=D{\frac {\partial ^{2}u}{\partial ^{2}x^{2}}}-u(1-u)*(a-u)$
Fisher	費雪方程	$u_{t}=u_{xx}+au(1-u)$
Fisher-Kolmogorov	費雪-科摩哥洛夫方程	:: ${\frac {\partial u}{\partial t}}={\frac {\alpha }{k}}*u(1-u^{q})+{\frac {\partial ^{2}u}{\partial x^{2}}}.\,$
Fujita-Storm	藤田-斯托姆方程	$u_{t}=a(u^{-2}u_{x})_{x}$
Gardner	加德納方程	${\frac {\partial u}{\partial t}}+(2au-3bu^{2})*{\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0$
Gibbons-Tsarev	吉本斯-查理夫方程	$u_{t}u_{xt}-u_{x}u_{tt}+u_{xx}+1=0$
Ginzburg-Landau	金茲堡－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	廣田-薩摩方程組	: $u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0$ $v_{t}+v_{xxx}-3uv_{x}=0$ $w_{t}+w_{xxx}-3uw_{x}=0$
Hunt-Saxton	亨特 - 薩克斯頓方程	: $(u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}$
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	變形KdV-Burgers方程	$u_{t}+u_{xxx}-\alpha u^{2}u_{x}-\beta *u_{xx}=0$
Fifth order KdV	五階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	五階色散KdV方程	$u_{t}+\alpha uu_{x}+\beta *u_{xxx}+u_{xxxxx}=0$
Seventh order KdV	七階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	九階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	非規範KdV方程	$u_{t}+\alpha u_{xxx}+\beta u*u_{x}=0$
Generalized Burgers-KdV	廣義伯格斯-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	非規範變形KdV方程	$u_{t}+u_{xxx}+\alpha u^{2}u_{x}=0$
von Karman	馮·卡門方程	$\Delta \Delta (u)=a((w_{xy})^{2}-w_{xx}w_{yy})$ $\Delta \Delta (w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c$

參考文獻

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

*谷超豪《孤立子理論中的達布轉換及其幾何應用》上海科學技術出版社
*閻振亞著《複雜非線性波的構造性理論及其應用》科學出版社 2007年
李志斌編著《非線性數學物理方程的行波解》科學出版社
王東明著《消去法及其應用》科學出版社 2002
*何青王麗芬編著《Maple 教程》科學出版社 2010 ISBN 9787030177445
Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION CRC PRESS
Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
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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T.Roubicek: Nonlinear Partial Differential Equations with Applications, 2nd ed., Birkhäuser, Basel, 2013, ISBN 978-3-0348-0512-4.
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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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