# Tanh 函数展开法

Tanh 函数展开法是目前求解非线性偏微分方程行波解的最强劲的和行之有效的方法。1992年数学家 Malfliet 首先应用 tanh 展开法[1] 运用这个方法要进行的大量繁杂的运算，必须借助MapleMathematicaMatlab等计算机代数系统。

${\displaystyle \psi (u,u_{t},u_{x},u_{tt},u_{xx},u_{tx})=0}$

${\displaystyle u(x,t)}$${\displaystyle U(\xi )}$

${\displaystyle \xi =k*(x-c*t)}$

${\displaystyle \psi (U(\psi ),-kc*{\frac {dU}{d\psi }},k*{\frac {dU}{d\psi }},}$${\displaystyle k^{2}*c^{2}*{\frac {d^{2}U}{d\psi ^{2}}}}$${\displaystyle ,k^{2}*{\frac {d^{2}U}{d\psi ^{2}}},-k^{3}*c^{3}*{\frac {d^{3}U}{d\psi ^{3}}},k^{3}*{\frac {d^{3}U}{d\psi ^{3}}})=0}$

${\displaystyle Y=tanh(\xi )}$

${\displaystyle {\frac {dY}{d\xi }}=1-tanh^{2}(\xi )=1-Y^{2}}$

${\displaystyle {\frac {dF(Y)}{d\xi }}={\frac {dF(Y)}{dY}}*{\frac {dY}{d\xi }}={\frac {dF(Y)}{dY}}*(1-Y^{2})}$

${\displaystyle {\frac {d}{d\xi }}=L=(1-Y^{2})*{\frac {d}{dY}}}$

${\displaystyle {\frac {d^{2}}{d\xi ^{2}}}=L^{2}=-2*(1-Y^{2})*Y*{\frac {d}{dY}}+(1-Y^{2})^{2}*{\frac {d^{2}}{dY^{2}}}}$

## 实例

${\displaystyle \partial _{t}\phi +6\phi \partial _{x}\phi +\partial _{x}^{3}\phi =0}$

${\displaystyle tr1:={t=tau,u=U(\xi ),x=\xi /k+c*tau}}$  得常微分方程：

${\displaystyle -ck{\frac {dU(\xi )}{d\xi }}+6kU(\xi ){\frac {dU(\xi )}{d\xi }}+k^{3}{\frac {d^{3}U(\xi )}{d\xi ^{3}}}=0}$

${\displaystyle -ckU(\xi )+3kU(\xi )^{2}+k^{3}{\frac {d(U(\xi )}{d^{2}\xi ^{2}}}=0}$ ${\displaystyle U(\xi )=F(Y)}$ 得：

${\displaystyle -ckF(Y)+3kF(Y)^{2}+k^{3}(1-Y^{2})(-2Y{\frac {d(F(Y)}{dY}}+(1-Y^{2}){\frac {d^{2}(F(Y)}{d^{2}Y}})=0}$

${\displaystyle U(\xi )=F(Y)=a[0]+a[1]Y+a[2]Y^{2}+a[3]Y^{3}+a[4]Y^{4}+a[5]Y^{5}+\cdot \cdot \cdot \cdot \cdot +a[M]Y^{M}+}$ …… 得： ${\displaystyle -ck(a[0]+a[1]Y+a[2]Y^{2}+a[3]Y^{3}+a[4]Y^{4}+a[5]Y^{5}+\cdot \cdot \cdot \cdot \cdot +a[M]Y^{M})+3k(a[0]+a[1]Y+a[2]Y^{2}+a[3]Y^{3}+a[4]Y^{4}+a[5]Y^{5}+\cdot \cdot \cdot \cdot \cdot +a[M]Y^{M})^{2}+k^{3}(1-Y^{2})(-2Y(a[1]+2a[2]Y+3a[3]Y^{2}+4a[4]Y^{3}+5a[5]Y^{4}+\cdot \cdot \cdot \cdot \cdot +Ma[M]Y^{M-1})+(1-Y^{2})(2a[2]+6a[3]Y+12a[4]Y^{2}+20a[5]Y^{3}+30a[6]Y^{4}))}$

${\displaystyle (3ka[2]^{2}+6k^{3}a[2])Y^{4}+(2k^{3}a[1]+6ka[1]a[2])Y^{3}+}$ ${\displaystyle (-cka[2]+3ka[1]^{2}+6ka[0]a[2]-8k^{3}a[2])Y^{2}+}$ ${\displaystyle (-cka[1]+6ka[0]a[1]-2k^{3}a[1])Y-cka[0]+3ka[0]^{2}+2k^{3}a[2]=0}$

${\displaystyle -cka[0]+3ka[0]^{2}+2k^{3}a[2]=0}$

${\displaystyle -cka[1]+6ka[0]a[1]-2k^{3}a[1]=0}$

${\displaystyle >-cka[2]+3ka[1]^{2}+6ka[0]a[2]-8k^{3}a[2]=0}$

${\displaystyle 2k^{3}a[1]+6ka[1]a[2]=0}$

${\displaystyle 3ka[2]^{2}+6k^{3}a[2]=0}$

a[0]=2k^2,a[1]=0,a[2]=-2k^2,c=4k^2;

a[0]=(2/3)k^2,a[1]=0,a[2]=-2k^2,c=-4k^2;

${\displaystyle u(x,t)=2k^{2}(1-tanh^{2}(k(x-4k^{2}t))=2k^{2}sech^{2}(k(x-4k^{2}t))}$

${\displaystyle u(x,t)={\frac {2}{3}}k^{2}(1-3tanh^{2}(k(+4k^{2}t)))}$

## 推广

Malfliet 的tanh 函数展开法被后人推广到 三角函数、雅可比橢圓函數、魏爾斯特拉斯橢圓函數。

JacobiCN, JacobiDN, JacobiNC, JacobiND, JacobiNS, JacobiSN, WeierstrassP, arcsinh, cos, cosh, cot, coth, csc, csch, exp, ln, sec, sech, sin, sinh, tan, tanh 等。

## 软件包

Maple商业计算机代数系统内包括一个求解偏微分方程的软件包，可用于多种非线性性偏微分方程，求得显式解析解。这个软件包称为为TWSolutions,功能丰富，可求多数非线性偏微分方程的行波解，但仍非万能，对有些非线性偏微分方程无解或只有平凡解[3]

tws:={TWSolutions(pdes,functions = [arcsinh, cos, cosh, cot, coth, csc, csch, exp, identity, ln, sec, sech, sin, sinh, tan, tanh, JacobiCN, JacobiDN, JacobiNC, JacobiND, JacobiNS, JacobiSN])};

## 参考文献

1. ^ W. Malfliet, Solitary Wave Solution of Nonlinear wave equation, Am J.of Physics 60(7) 1992,650-654
2. ^ Graham W Griffiths, William E.Schiesser, Traveling Wave Analysis of Partial Differential Equations p393-396 Academic Press 2012
3. Graham Griffiths, p436-437 Maple Built-in Procedure TWSolutions
4. ^ 李志斌 《非线性数学物理方程的行波解》第119-130
5. ^ RATH 下载