查看“︁双曲型刘维方程”︁的源代码

''' 双曲型刘维方程'''(Hyperbolic Liouville equation)是一个非线性偏微分方程：<ref name=gg>Graham W.Griffiths Chapter 15 p275-292 Academic Press 2012</ref>

<math> u_{tt}=\alpha^2*u_{xx}+\gamma*exp(\beta*u)   </math>

作变换：

<math>u(x, t) = ln(v(x, t))/\beta</math> 得：

<math>v_{tt}*v-(v_{t})^2-\alpha^2*v_{xx}*v+\alpha^2*(v_{x})^2-\beta^2*v=0</math>

求得 v(x,t) 的行波解，作反代换得回 u(x,t)。

==解析解==
:<math> {u(x, t) = ln(-(2*(\alpha^2*_C2^2-_C3^2))*csc(_C1+_C2*x+_C3*t)^2/(\gamma*\beta))/\beta}  </math>
:<math> {u(x, t) = ln(-(2*(\alpha^2*_C2^2-_C3^2))*csch(_C1+_C2*x+_C3*t)^2/(\gamma*\beta))/\beta}  </math>
:<math> {u(x, t) = ln(-(2*(\alpha^2*_C2^2-_C3^2))*sec(_C1+_C2*x+_C3*t)^2/(\gamma*\beta))/\beta}  </math>
:<math>  {u(x, t) = ln((2*(\alpha^2*_C2^2-_C3^2))*sech(_C1+_C2*x+_C3*t)^2/(\gamma*\beta))/\beta} </math>
:<math> {u(x, t) = ln(-(2*(\alpha^2*_C3^2-_C4^2))*csc(_C2+_C3*x+_C4*t)^2/(\gamma*\beta))/\beta}  </math>
:<math>  {u(x, t) = ln(-(2*(\alpha^2*_C3^2-_C4^2))*sec(_C2+_C3*x+_C4*t)^2/(\gamma*\beta))/\beta} </math>
:<math> {u(x, t) = ln((2*(\alpha^2*_C3^2-_C4^2))*sech(_C2+_C3*x+_C4*t)^2/(\gamma*\beta))/\beta}  </math>
:<math>  {u(x, t) = ln(-(2*(\alpha^2*_C4^2-_C5^2))*WeierstrassP(_C3+_C4*x+_C5*t, 0, 0)/(\gamma*\beta))/\beta} </math>
:<math> {u(x, t) = ln(-(2*(\alpha^2*_C2^2-_C3^2+cot(_C1+_C2*x+_C3*t)^2*\alpha^2*_C2^2-cot(_C1+_C2*x+_C3*t)^2*_C3^2))/(\gamma*\beta))/\beta}  </math>
:<math> {u(x, t) = ln(-(2*(\alpha^2*_C2^2-_C3^2+tan(_C1+_C2*x+_C3*t)^2*\alpha^2*_C2^2-tan(_C1+_C2*x+_C3*t)^2*_C3^2))/(\gamma*\beta))/\beta}  </math>
:<math> {u(x, t) = ln(-(2*(-\alpha^2*_C2^2+_C3^2+coth(_C1+_C2*x+_C3*t)^2*\alpha^2*_C2^2-coth(_C1+_C2*x+_C3*t)^2*_C3^2))/(\gamma*\beta))/\beta}  </math>
:<math>  {u(x, t) = ln(-(2*(-\alpha^2*_C2^2+_C3^2+tanh(_C1+_C2*x+_C3*t)^2*\alpha^2*_C2^2-tanh(_C1+_C2*x+_C3*t)^2*_C3^2))/(\gamma*\beta))/\beta} </math>


==行波图==
{|
|[[File:Hyperbolic Liouville equation traveling wave plot 1.gif|thumb|双曲型刘维方程行波图]]
|[[File:Hyperbolic Liouville equation traveling wave plot 2.gif|thumb|双曲型刘维方程行波图]]
|[[File:Hyperbolic Liouville equation traveling wave plot 3.gif|thumb|双曲型刘维方程行波图]]
|[[File:Hyperbolic Liouville equation traveling wave plot 4.gif|thumb|双曲型刘维方程行波图]]
|}
{|
|[[File:Hyperbolic Liouville equation traveling wave plot 5.gif|thumb|双曲型刘维方程行波图]]
|[[File:Hyperbolic Liouville equation traveling wave plot 6.gif|thumb|双曲型刘维方程行波图]]
|[[File:Hyperbolic Liouville equation traveling wave plot 7.gif|thumb|双曲型刘维方程行波图]]
|[[File:Hyperbolic Liouville equation traveling wave plot 8.gif|thumb|双曲型刘维方程行波图]]
|}

==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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