川原方程

川原方程（Kawahara pde）是一個非線性偏微分方程：^[1]

${\frac {\partial u}{\partial t}}+u*{\frac {\partial u}{\partial x}}+a*{\frac {\partial ^{3}u}{\partial x^{3}}}=b*{\frac {\partial ^{5}u}{\partial x^{5}}}$

廣義川原方程有如下形式：^[2]

${\frac {\partial u}{\partial t}}+(1+u^{2})*{\frac {\partial u}{\partial x}}+a*{\frac {\partial ^{3}u}{\partial x^{3}}}=b*{\frac {\partial ^{5}u}{\partial x^{5}}}$

行波解

川原方程有孤立子解：^[3]

$u(x,t)={\frac {105*a^{2}}{169*b*cosh^{4}(z)}}+C1$

$z={\frac {1}{2}}*k*x-(18*b*k^{5}+C1*k)*t$

$k={\sqrt {\frac {a}{13*b}}}$

tanh 行波解

利用tanh 法，可得^[4]

$u={\frac {-c(-1+\tanh(1/2(c(-x-x0+ct))}{(a)))}}$

Maple 軟件包行波解

利用Maple的TWSolutions軟件包可得廣義川原方程的一系列行波解：^[5]

行波圖

Kawahara equation traveling wave plot

參考文獻

^ Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p197-238 Equations Academy Press
^ Inna Shingareva, Carlos Lizarraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica , Springer, p192
^ 李志斌非線性數學物理方程的行波解第137頁科學出版社
^ Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations p391-409 Academy Press
^ Inna Shingareva et al Solving Nonlinear Partial Differential Equations with Maple and Mathematica Springer, p 192

*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

[1] Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p197-238 Equations Academy Press

[2] Inna Shingareva, Carlos Lizarraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica , Springer, p192

[3] 李志斌非線性數學物理方程的行波解第137頁科學出版社

[4] Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations p391-409 Academy Press

[5] Inna Shingareva et al Solving Nonlinear Partial Differential Equations with Maple and Mathematica Springer, p 192

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