查看“︁非线性偏微分方程列表”︁的源代码

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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==
可以藉代数来将方程式重新编排，让方程式的一部分只含有一个变量，而剩余部分则跟此变量无关。这样，隔离出的两个部分的值，都分别等于常数，而两个部分的值的代数和等于零。

==反散射法==

==达布变换==

==混合指数法==

==齐次平衡法==

==Tanh 函数展开法==
{{main|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：

<math>\psi(u,u_{t},u_{x},u_{tt},u_{xx},u_{tx})=0</math>

作变数代换：

<math>u(x,t)</math>->  <math>U(\xi)</math>

<math>\xi=k*(x-c*t)</math>

得到：

<math>\psi(U(\psi),-kc*\frac{\partial U}{\partial \psi},k*\frac{\partial U}{\partial \psi},</math><math>
k^2*c^2*\frac{\partial^2 U}{\partial \psi^2}</math><math>,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</math>

1992年数学家 Malfliet 首先应用 tanh 展开法<ref>W. Malfliet, Solitary Wave Solution of Nonlinear wave equation, Am J.of Physics  60(7) 1992,650-654</ref>

==对称分析==

==Lax 可积系统==


{| class="wikitable"
|-
! Equation !! 中文 !! 方程 !! 图
|-
|BBM  ||[[班傑明-小野方程]] || :<math>u_t+u_x+uu_x-u_{xxt}=0.\,</math>
||[[File:BBM animation.gif|80px]]
|-
| Belousov-Zhabotinsky || [[别洛乌索夫-扎伯廷斯基方程]] || <math> u_{t}=d*u_{xx}+u*(1-r*u-u)</math>,
<math>v_{t}=v_{xx}-s*u*v</math>
 ||[[File:Belousov-Zhabotinsky equation traveling wave plot 11.gif|80px]]
|-
| (Benjamin-Ono equation ||[[本杰明-小野方程]] ||<math>u_{tt}+\alpha*(u^2)_{xx}-\beta*u_{xxxx}=0</math>
 || [[File:Benjamin-Ono equation traveling wave plot 8.gif|80px]]
|-
|Bogoyavlenski-Konoplechenko||[[波格雅夫连斯基-科譳普勒琛科方程]] ||<math>u_{xt}+\alpha*u_{xxxx}+\beta*u_{xxxy}</math><math>+6*\alpha*u_{xx}*u_{x}+4*\beta*u_{xy}*u_{x}+\beta*u_{xx}*u_{y}=0</math>
 ||[[File:Bogoyavlenski-Konoplechenko equation traveling wave plot 4.gif|80px]]
|-
| Born-Infeld ||[[玻恩-英费尔德方程]]||<math>\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0</math> ||[[File:Born Infeld equation animation3.gif|80px]]
|-
| Boussinesq ||[[博欣内斯克方程]]||<math>\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</math>
||[[File:Boussinesq pde Maple animation2.gif|80px]]
|-
|Boussinesa type ||[[博欣内斯克型方程]]||<math> u_{tt}-u_{xx}-2*\alpha*(u*u_{x})_{x}-\beta*u_{xxtt}=0   </math>
 ||[[File:Boussinesq type equation traveling wave plot 4.gif|80px]]
|-
|Unnormalized Boussinesq ||[[非规范博欣内斯克方程]]||<math>u_{tt}-\alpha*(u*u_{x})_{x}-\beta*u_{xxxx}=0    </math>||[[File:Unnormalized Boussinesq equation traveling wave plot 3.gif|80px]]
|-
|Broer-Kaup||[[布罗尔-库普方程组]]||<math>u_{y,t}+(2*u*u_{x})_{x}+2*v_{xx}-u_{xxy}=0</math>

<math>v_{t}+2*(vu)_{x}+v_{xx}=0</math>
 ||[[File:Broer-Kaup equation traveling wave plot U3.gif|80px]]
|-
|Burgers||[[伯格斯方程]]||<math>\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  </math>||[[File:Burgers equation traveling wave plot 14.gif|80px]]
|-
|Burgers-Fisher||[[伯格斯-费希尔 方程]]||:<math> \frac{\partial u}{\partial t}+u^2*\frac{\partial u}{\partial x}-\frac{\partial^2 u}{\partial u^2}=u*(1-u^2) </math>
||[[File:Burgers Fisher equation tanh Adomian plot.gif|80px]]
|-
|Modified Burgers||[[变形伯格斯方程]]||<math>u_{t}+\frac{k}{t}*u+b*u*u_{x}=a*u_{xx}    </math>
||[[File:Modified Burgers equation 3D plot 5.png|80px]]
|-
|Unnormalized Burgers||[[非规范伯格斯方程]]||<math>u_{t}-\alpha*u_{xx}-\beta*u*u_{x}=0  </math>||[[File:Unnormalized Burgers equation traveling wave plot 2.gif|80px]]
|-
|Generalized Burgers||[[广义伯格斯方程]]||<math>  </math> ||  ||  ||
|-
|Burgers-Huxley||[[伯格斯-赫胥黎方程]]||<math>u[t]-nu*u[x, x]+a*u*u[x] = b*u*(1-u)*(u-c)</math>
||[[File:Burgers Huxley eq animation1.gif|80px]]
|-
|Bretherton||[[布雷瑟顿方程]]||<math>u_{tt}+u_{xx}+u_{xxxx}-\alpha*u^3=0</math>
||[[File:Bretherton equation traveling wave Jacobi function plot 6.gif|80px]]
|-
|Cahn-Hilliard ||[[卡恩-希利亚德方程]]||  || 
|-
|Cassama-Holm||[[卡马萨-霍尔姆方程]]||:<math>
u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}</math>
 ||[[File:Camassa Holm equation traveling wave sech plot5.gif|80px]]
|-
|Chaffee-Infante||[[查菲 - 堙方特方程]]||<math>u_{t}-u_{xx}+\lambda*(u^3-u)=0 </math>||[[File:Chaffee-Infante equation traveling wave plot 02.gif|80px]]
|-
|Chaplygin||[[查普里金方程]] ||<math>0.5*u_{tt}+u_{x}*u_{xt}-u_{t}*u_{xx}=0</math>||[[File:Chaplygin equation traveling wave plot 1.gif|80px]]
|-
|Davey–Stewartson||[[戴维-斯图尔森方程组]]||:<math>i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \phi_x,\,</math>

:<math>\phi_{xx} + c_3 \phi_{yy} = ( |u|^2 )_x.\,</math>
 ||[[File:Davey-Stewardson equation traveling wave plot 4.gif|80px]]
|-
|Degasperis-Procesi||[[DP 方程]]||<math>\displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx}</math>
 ||[[File:Degasperis-Procesi equation traveling wave plot 05.gif|80px]]
|-
|Drinfeld-Solokov-Wilson||DSW 方程||<math>\frac{\partial u}{\partial t}+3*v*\frac{\partial v}{\partial x}=0</math>

<math>\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}</math>
||[[File:Drinfeld-Sokolov-Wilson Equation Homotopy method animation.gif|80px]]
|-
|Dodd-Bullough-Mikhailov||[[多德-布洛-米哈伊洛夫方程]]||[[<math>u_{xt}+\alpha*e^u+\gamma*e^{-2*u} = 0</math>
||[[File:Dodd-Bullough-Mikhailov equation traveling wave plot5.gif|80px]]
|-
|Nonlinear Diffusion||[[非线性扩散方程]]||<math> u_{t}=\alpha*u_{xx}-\beta*u^3-\gamma*u^2   </math>
||[[File:Nonlinear Diffusion equation traveling wave plot 7.gif|80px]] 
|-
|Harry Dym||[[迪姆方程]]||:<math>u_t = u^3u_{xxx}.\,</math>
||[[File:Dym eq Backlund solution animation.gif|80px]]
|-
|Eckhaus||[[艾克豪斯方程]]||<math>u(x, y, t)_t+v(x, t)_x+1.0*u(x, y, t)*(u(x, y, t)_x) = 0</math>

<math>v(x, t)_{xt}+u(x, y, t)_{xx}*v(x, t)+</math><math>2*u(x, y, t)_x*v(x, t)_x+u(x, y, t)*(v(x, t)_{xx}+</math><math>
u(x, y, t)_{xx}+u(x, y, t)_{xxxx}+u(x, y, t)_{yy} = 0</math>
||[[File:Eckhaus dispersion equation traveling wave plotV2.gif|80px]]
|-
|Eikonal||[[程函方程]]||<math>sys := (u(x, t)_t))^2+(u(x, t)_x)^2-4 = 0</math>
||[[File:Eikonal equation traveling wave plot 1.gif|80px]]
|-
|Estevez-Mansfield-Clarkson ||[[埃斯特韦斯-曼斯菲尔德-克拉克森方程]]||<math> u_{tyyy}+\beta*u_{y}*u_{yt}+\beta*u_{yy}*u_{t}+u_{tt}=0</math>
 ||[[File:Estevez-Mansfield-Clarkson equation traveling wave plot 5.gif|80px]]
|-
|Fitzhugh-Nagumo ||[[菲茨休 - 南云方程]]||<math>\frac{\partial u}{\partial t}=D*\frac{\partial^2 u}{\partial^2 x^2}-u*(1-u)*(a-u)</math>

||[[File:1-Fitzhugh Nagumo plot 5.jpg|80px]]
|-
|Fisher||[[费希尔方程]]||<math> u_{t}=u_{xx}+a*u*(1-u)   </math>

||[[File:Fisher equation traveling wave plot 10.gif|80px]]
|-
|Fisher-Kolmogorov||[[费希尔-柯尔莫哥洛夫方程]]||::<math> \frac{\partial u}{\partial t}=\frac{\alpha}{k}*u(1-u^q)+\frac{\partial^2 u}{\partial x^2}.\, </math>
||[[File:Fisher Kolmogorov equation traveling wave plot11.gif|80px]]
|-
|Fujita-Storm||[[藤田-斯托姆方程]]||<math> u_{t}=a*(u^{-2}*u_{x})_{x}   </math>||[[File:Fujita-Storm equation plot 9.gif|80px]]
|-
| Gardner||[[加德纳方程]]||<math>\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</math>

||[[File:Gardner equation traveling wave plot4.gif|80px]]
|-
|Gibbons-Tsarev||[[吉本斯-查理夫方程]]||<math>u_{t}*u_{xt}-u_{x}*u_{tt}+u_{xx}+1=0</math>
||[[File:Gibsons-Tsarev equation traveling wave plot 3.gif|80px]]
|-
|Ginzburg-Landau||[[金兹堡－朗道方程]]||<math>\frac{\partial u}{\partial t}-a*u*\frac{\partial^2 u }{\partial x^2}-b*u+c*|u|^2*u=0</math>

 ||[[File:Ginzburg Landau equation animation1.gif|80px]]
|-
|Hirota Satsuma||[[广田-萨摩方程组]] ||:<math>u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0</math>
:<math>v_{t}+v_{xxx}-3uv_{x}=0</math>
:<math>w_{t}+w_{xxx}-3uw_{x}=0</math>
||[[File:Hirota Satsuma equations traveling wave plot 2.gif|80px]]
|-
|Hunt-Saxton||[[亨特 - 萨克斯顿方程]]||:<math>
(u_t + u u_x)_x = \frac{1}{2} \, u_x^2
</math>
||[[File:Hunter Saxton eq traveling wave plot 17.gif|80px]]
|-
|Ito||[[伊藤方程]]||<math> U_{t}+((6*U^5+10*\alpha*(U^2*U_{xx}+U*U_{x}^2)+U_{xxxx})_{x}=0 </math>

||[[File:Ito equation traveling wave plot 2.gif|80px]]
|-
|KdV||[[KdV方程]]||:<math>\partial_t\phi+6\phi\partial_x\phi+\partial^3_x\phi=0</math>
 ||[[File:KdV equation traveling wave plot 8.gif|80px]]
|-
|Modified KdV||[[MKdV方程]]||<math>u_{t}+\alpha*u^2*u_{x}+u_{xxx}=0</math>
 ||[[File:MKdV equation traveling wave plot 3.gif|80px]]
|-
|KdV-mKdV ||[[KdV-mKdV方程]]||<math>u_{t}+6*\alpha*u*u_{x}+6*\beta*u^2*u_{}+\gamma*U_{xxx}=0</math>

||[[File:Kdv-mKdv equation traveling wave plot 5.gif|80px]]
|-
|KdV-Burgers||[[KdV-Burgers方程]]||<math>u_{t}+u*u_{x}-\alpha*u_{xx}-\beta*u_{xxx}=0</math>

||[[File:KdV-Burgers equation traveling wave plot 6.gif|80px]]
|-
|Modified KdV-Burgers||[[变形KdV-Burgers方程]]||<math>u_{t}+u_{xxx}-\alpha*u^2*u_{x}-\beta*u_{xx}=0 </math>
 ||[[File:Modified KdV-Burgers equation traveling wave plot 7.gif|80px]]
|-
|Fifth order KdV||[[五阶KdV方程]]||<math>u_{t}+\alpha*u^2*u_{x}+\beta*u_{x}*u_{xx}+\gamma*u*u_{xxx}+\delta*u_{xxxxx}=0</math>
||[[File:General Fifth order KdV equation traveling wave plot 16.gif|80px]]
|-
|Fifth order dispersion KdV||[[五阶色散KdV方程]]||<math> u_{t}+\alpha*u*u_{x}+\beta*u_{xxx}+u_{xxxxx}=0   </math>
||[[File:Fifth order dispersion KdV equation traveling wave plot 2.gif|80px]]
|-
|Seventh order KdV||[[七阶KdV方程]]||<math>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</math>
||[[File:Seventh order KdV equation traveling wave plot 2.gif|80px]]
|-
|Nineth order KdV ||[[九阶KdV方程]]||<math>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</math>

||[[File:Nineth order KdV equation traveling wave plot 1.gif|80px]]
|-
|Unnormalized KdV equation||[[非规范KdV方程]]||<math>u_{t}+\alpha*u_{xxx}+\beta*u*u_{x}=0</math>
 ||[[File:Unnormalized KdV equation traveling wave plot 6.gif|80px]]
|-
|Generalized Burgers-KdV ||[[广义伯格斯-KdV方程]]||<math>U[t]-\alpha*\frac{\partial^n u(x,t)}{\partial x^n}-\beta*u(x,t)*\frac{\partial u(x,t)}{\partial x}=0    </math>
||[[File:Generalized 7th order Burgers-KdV equation plot 16.gif|80px]]
|-
|Unnormalized modified  KdV  ||[[非规范变形KdV方程]]||<math>u_{t}+u_{xxx}+\alpha*u^2*u_{x}=0    </math>
||[[File:Unnormalized modifed KdV equation traveling wave plot 15.gif|80px]]
|-
| von Karman || [[冯·卡门方程]] || <math>\Delta\Delta(u)=a((w_{xy})^2-w_{xx}w_{yy})</math>

<math>\Delta\Delta(w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c</math>
 || [[File:Von Karman equation U Maple plot.png|80px]]
|}

==参考文献==
<references/>
# *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
#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
#Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
#Dongming Wang, Elimination Practice,Imperial College Press 2004
# David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
#T.Roubicek: Nonlinear Partial Differential Equations with Applications, 2nd ed., Birkhäuser, Basel, 2013, ISBN 978-3-0348-0512-4.
# George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

==外部链接==
*[http://www.maths.ox.ac.uk/groups/oxpde 牛津大学非线性偏微分方程研究中心] {{Wayback|url=http://www.maths.ox.ac.uk/groups/oxpde |date=20210413231613 }}

[[Category:非线性偏微分方程| ]]