# 非线性偏微分方程列表

## 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。

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

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

${\displaystyle \xi =k*(x-c*t)}$

${\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]

## Lax 可积系统

Equation 中文 方程
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
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
4. 王东明著 《消去法及其应用》 科学出版社 2002
5. *何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445
6. Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION CRC PRESS
7. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
8. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
9. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
10. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
11. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
12. Dongming Wang, Elimination Practice,Imperial College Press 2004
13. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
14. T.Roubicek: Nonlinear Partial Differential Equations with Applications, 2nd ed., Birkhäuser, Basel, 2013, ISBN 978-3-0348-0512-4.
15. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759