查看“齐次蒙日-安培方程”的源代码

'''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。<ref>Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p775-776 CRC PRESS</ref>
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：<br />
<math>L[u] = A(u_{xx}u_{yy} - u_{xy}^2) + Bu_{xx} + 2Cu_{xy} + Du_{yy} + E = 0,</math><br />
這裡的A,B,C,D,E為一階變量x,y,u<sub>x</sub>和u<sub>y</sub>唯一的非獨立函數。

==解析解==
根據齊次蒙日-安培方程：
<math>sys = {(U[x,t])^2-U[x,x]*U[t,t]=0 };</math><br />
其對應的解析解為：
:<math> u[1] := C1*y^2+C2*x*y+C2^2*x^2/(4*C1)+C3*y+C4*x+C5  </math>
:<math> u[2] := (C2*y^2+C3*y+C3^2/(4*C2))/(x+C1)+C4*y+C5*x+C6  </math>
:<math> u[3] := (C1*x+C2*y+C3)^(k+1)/(C4*x+C5*y+C6)^k+C7*x+C8*y+C9  </math>
:<math> u[4] := (C1*x+C2*y+C3)^k+C4*x+C5*y+C6  </math>
:<math> u[5] := -(C1*x+C2*y+C3)^k+C4*x+C5*y+C6  </math>
:<math> u[6] := -(C1*x+C2*y+C3)^(k+1)/(C4*x+C5*y+C6)^k+C7*x+C8*y+C9 </math>
:<math> u[7] := \sqrt{C1*x^2+2*C1*x*a+C1*a^2+C2*x*y+C2*x*b+C2*a*y+C2*a*b+C3*y^2+2*C3*y*b+C3*b^2}+C5*x+C6*y+C7  </math>
:<math> u[8] := C1*(C1*y^2+C2*x*y+(1/4)*C2^2*x^2/C1+C3*y+C4*x+C5)*(C2*x+C3*y+C4)+C5*x+C6*y+C7  </math>
:<math> u[9] := C1*((C2*y^2+C3*y+(1/4)*C3^2/C2)/(x+C1)+C4*y+C5*x+C6)*(C2*x+C3*y+C4)+C5*x+C6*y+C7  </math>
:<math> u[10] := C1*((C1*x+C2*y+C3)^(k+1)/(C4*x+C5*y+C6)^k+C7*x+C8*y+C9)*(C2*x+C3*y+C4)+C5*x+C6*y+C7  </math>

==行波图==
{|
|[[File:Homogeneous Monge-Ampere equation plot 1.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 2.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 3.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 4.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|}
{|
|[[File:Homogeneous Monge-Ampere equation plot 5.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 6.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 7.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 8.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|}
{|
|[[File:Homogeneous Monge-Ampere equation plot 9.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 10.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 11.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|[[File:Homogeneous Monge-Ampere equation plot 12.gif|thumb|Homogeneous Monge-Ampere equation plot]]
|}

==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
#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
# George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

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[[category:非线性偏微分方程]]