DSW 方程

DSW 方程(Drinfeld-Solokov-Wilson equation)是一组非线性偏微分方程：

${\frac {\partial u}{\partial t}}+3*v*{\frac {\partial v}{\partial x}}=0$

${\frac {\partial v}{\partial t}}-2*{\frac {\partial ^{3}v}{\partial x^{3}}}+{\frac {\partial u}{\partial x}}*v+2u*{\frac {\partial v}{\partial x}}$

解析解

DSW方程有多种解析解

对称消减法

DSW Equation symmetry reduction method animation

用减对称法可求DSW方程的解析解

$u(x,t):=-sqrt(2*m)*c2*tanh(c1+c2*(m*t-x))$

参数：c2 = .5, c1 = 1.4, m = 2.3

$u(x,t)=-.758*sqrt(2)*tanh(1.4+1.15*t-.5*x)$

变分法

DSW Equation variational method animation

$u(x,t)=f:=(3*c*(1/2))*sech({\sqrt {(1/2)*c}}*(x-c*t))^{2}$。

达布变换法

利用达布变换可求DSW方程的解析解

$v:=-(1/2)*k^{2}*(-1+{\sqrt {(}}3)*tanh(k^{3}*t+(1/2)*k*y)*cot((1/2)*{\sqrt {(}}3)*k*y))$

DSW breather with Dabourx transform

同伦法

Drinfeld-Sokolov-Wilson Equation Homotopy method animation

利用函数的同伦，可求DSW方程的解析解。

$u(x,t)=-360*t^{2}*sec(x)^{2}*tan(x)^{4}-396*t^{2}*sec(x)^{2}*tan(x)$ $-60*t^{2}*sec(x)^{2}+360*t^{2}*sec(x)^{4}*tan(x)^{2}+72*t^{2}*sec(x)^{4}$

行波分析法

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参考文献

1. *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
2. *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
3. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
4. 王东明著 《消去法及其应用》 科学出版社 2002
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6. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
7. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
8. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
9. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
11. Dongming Wang, Elimination Practice,Imperial College Press 2004
12. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
13. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759