查看“︁特里忒蔡卡方程”︁的源代码

[[File:Gheorghe Titeica (timbre roumain).jpg|thumb|right|200px|罗马尼亚数学家George Tritzeica]]

'''特里忒蔡卡方程'''(Tritzeica equation)是一个最早由罗马尼亚数学家George Tritzeica在1907年在微分几何领域研究的非线性偏微分方程<ref>G. Tzitz´eica, “Geometric infinitesimale-sur une nouvelle classes
de surfaces,”Comptes Rendus de l’Acad´emie des Sciences, vol. 144,pp. 1257–1259, 1907.</ref>常见于微分几何学和物理学的非线性偏微分方程：<ref>Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS</ref>

<math>u_{xy}=exp(u_{x,y})-exp(-2*u_{x,y})</math>

作变换<math>w(x, y) = exp(u(x, y))</math>

得

<math> w(x, y)_{y, x}*w(x, y)-w(x, y)_{x}*w(x, y)_{y}-w(x, y)^3+1 = 0</math>

求得行波解，再用反代换 <math>u(x,y)=ln(w(x,y))</math>即得 原方程的行波解。

==解析解==
:<math>u(x, y)=ln( -1/2-(1/2*I)*\sqrt(3)+(3/4+(3/4*I)*\sqrt(3))*csc(_C1+_C2*x+(3/4)*(1/2+(1/2*I)*\sqrt(3))*y/_C2)^2)</math>
:<math>u(x, y) =ln( -1/2-(1/2*I)*\sqrt(3)+(3/4+(3/4*I)*\sqrt(3))*sec(_C1+_C2*x+(3/4)*(1/2+(1/2*I)*\sqrt(3))*y/_C2)^2)</math>
:<math>u(x, y) =ln( -1/2+(1/2*I)*\sqrt(3)+(3/4-(3/4*I)*\sqrt(3))*csc(_C1+_C2*x+(3/4)*(1/2-(1/2*I*\sqrt(3))*y/_C2)^2)</math>
:<math>u(x, y) =ln(1/4-(1/4*I)*\sqrt(3)+(-3/4+(3/4*I)*\sqrt(3))*coth(_C1+_C2*x+(3/4)*(-1/2+(1/2*I)*sqrt(3))*y/_C2)^2)</math>
:<math>u(x, y) = ln(1/4-(1/4*I)*\sqrt(3)+(-3/4+(3/4*I)*\sqrt(3))*tanh(_C1+_C2*x+(3/4)*(-1/2+(1/2*I)*sqrt(3))*y/_C2)^2)</math>
:<math>u(x, y) =ln( 1/4-(1/4*I)*\sqrt(3)+(3/4-(3/4*I)*\sqrt(3))*cot(_C1+_C2*x+(3/4)*(1/2-(1/2*I)*\sqrt(3))*y/_C2)^2)</math>
:<math>u(x, y) =ln( 1/4-(1/4*I)*\sqrt(3)+(3/4-(3/4*I)*\sqrt(3))*tan(_C1+_C2*x+(3/4)*(1/2-(1/2*I)*\sqrt(3))*y/_C2)^2)</math>
:<math>w(x, y) = (8/3)*_C4^2-(1/3)*RootOf(64*_C4^6+27-24*_C4^4*_Z-6*_C4^2*_Z^2+_Z^3)-4*_C4^2*JacobiDN(_C2+(1/2)*_C4*x+_C4*t, (1/2)*RootOf(-RootOf(64*_C4^6+27-24*_C4^4*_Z-6*_C4^2*_Z^2+_Z^3)+_Z^2)/_C4)^2</math>

==行波图==
{|
|[[File:Tritzeica equation traveling wave plot 1.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 2.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 3.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 4.gif|thumb|Tritzeica equation traveling wave plot]]
|}
{|
|[[File:Tritzeica equation traveling wave plot 5.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 6.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 7.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 8.gif|thumb|Tritzeica equation traveling wave plot]]
|}
{|
|[[File:Tritzeica equation traveling wave plot 9.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 10.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 11.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 12.gif|thumb|Tritzeica equation traveling wave plot]]
|}
{|
|[[File:Tritzeica equation traveling wave plot 13.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 14.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 29.gif|thumb|Tritzeica equation traveling wave plot]]
|[[File:Tritzeica equation traveling wave plot 16.gif|thumb|Tritzeica equation traveling wave 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

{{非线性偏微分方程}}
[[category:非线性偏微分方程]]