伯格斯方程

伯格斯方程（Burgers equation）是一個模擬衝擊波的傳播和反射的非線性偏微分方程，給定函數 $u(x,t)$ ，與擴散系數 $\nu$ ，則一維的伯格斯方程可以寫成以下的形式:

${\frac {\partial u(x,t)}{\partial t}}+u(x,t){\frac {\partial (u(x,t))}{\partial x}}=\nu {\frac {\partial ^{2}(u(x,t))}{\partial x^{2}}}$

Backlund 變換

伯格斯方程通過下列Backlund變換^[1]。

$w[x]={\frac {1}{2}}wu$

$w[t]={\frac {1}{2}}w[x]u+{\frac {1}{2}}wu[x]$

則化為熱方程：

$HEq:=w[t]=w[x,x]$

伯格斯方程有鐘行孤波和扭波兩種形式的行波解^[2]

Burgers equation traveling wave plot

^ Inna Shingareva, Carlos Lizarraga Celaya, Solving Nonlinear Differential Equations with Maple and Mathematica, p47,Springer
^ Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple p57-58Springer

*谷超豪《孤立子理論中的達布變換及其幾何應用》上海科學技術出版社
*閻振亞著《複雜非線性波的構造性理論及其應用》科學出版社 2007年
李志斌編著《非線性數學物理方程的行波解》科學出版社
王東明著《消去法及其應用》科學出版社 2002
*何青王麗芬編著《Maple 教程》科學出版社 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