DSW 方程

DSW 方程(Drinfeld-Solokov-Wilson equation)是一組非線性偏微分方程^[1]：

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

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

解析解

DSW方程有多種解析解^[2]

對稱消減法

DSW Equation symmetry reduction method animation

用減對稱法可求DSW方程的解析解^[3]

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

參數：c2 = .5, c1 = 1.4, m = 2.3

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

變分法

DSW Equation variational method animation

${\displaystyle u(x,t)=f:=(3*c*(1/2))*sech({\sqrt {(1/2)*c}}*(x-c*t))^{2}}$^[4]。

達布變換法

利用達布變換可求DSW方程的解析解^[5]

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

DSW breather with Dabourx transform

同倫法

Drinfeld-Sokolov-Wilson Equation Homotopy method animation

利用函數的同倫，可求DSW方程的解析解^[6]。

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

行波分析法

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參考文獻

1. Esmaeil Alibeiki and Ahmad Neyrameh Application of Homotopy Perturbation Method to Nonlinear Drinfeld-Sokolov-Wilson Equation
2. Esmaeil Alibeiki and Ahmad Neyrameh Application of Homotopy Perturbation Method to Nonlinear Drinfeld-Sokolov-Wilson Equation
3. B.Kauer, Symmetry Reduction Mothod for Exact Solution of some Nonlinear Systems p27
4. Wei-Min Zhang Solitary Solutions and Singular Periodic Solutions of the Drinfeld-Sokolov-Wilson Equation by Variational Approach,Applied Mathematical Sciences, Vol. 5, 2011, no. 38, 1887 - 1894
5. 耿獻國、吳麗華《Darboux Transformation and Explicit Solutions for Drinfeld–Sokolov–Wilson Equation
6. Esmaeil Alibeiki and Ahmad Neyrameh,Application of Homotopy Perturbation Method to Nonlinear Drinfeld-Sokolov-Wilson Equation

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