齐次蒙日-安培方程

非线性偏微分方程

齐次蒙日-安培方程(Homogeneous Monge-Ampère equation)是一个常见于黎曼几何的非线性偏微分方程,同时也是卡拉比-丘流形证明时曾用的工具。[1] 广义而言,定义两个独立变量x,y,以及一个非独立变量u,蒙日-安培方程可以表述为:

这里的A,B,C,D,E为一阶变量x,y,ux和uy唯一的非独立函数。

解析解

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根据齐次蒙日-安培方程:  
其对应的解析解为:

 
 
 
 
 
 
 
 
 
 

行波图

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Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot
 
Homogeneous Monge-Ampere equation plot

参考文献

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  1. ^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p775-776 CRC PRESS
  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. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
  7. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
  8. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
  9. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
  10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
  11. Dongming Wang, Elimination Practice,Imperial College Press 2004
  12. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
  13. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759