川原方程

川原方程（Kawahara pde）是一个非线性偏微分方程：^[1]

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

广义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 软件包行波解

利用Maple的TWSolutions软件包可得广义川原方程的一系列行波解：^[5]

行波图

Kawahara equation traveling wave plot

参考文献

^ 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
^ 李志斌非线性数学物理方程的行波解第137页科学出版社
^ 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

[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] 李志斌非线性数学物理方程的行波解第137页科学出版社

[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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