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,同時也是卡拉比-丘流形證明時曾用的工具。[1] 廣義而言,定義兩個獨立變量x,y,以及一個非獨立變量u,蒙日-安培方程可以表述為:

這裡的A,B,C,D,E為一階變量x,y,ux和uy唯一的非獨立函數。

解析解

根據齊次蒙日-安培方程:  
其對應的解析解為:

 
 
 
 
 
 
 
 
 
 

行波图

 
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

  1. ^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p775-776 CRC PRESS
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