纽厄尔-怀特海德方程

**紐厄爾 - 懷特海德方程**（Newell-Whitehead equation) 是一個非線性偏微分方程：

$$
u_{t}-u_{xx}-\alpha *u+beta*u^{3}=0
$$

## 行波解

紐厄爾 - 懷特海德方程有行波解：

　
: $u(x,t)=1/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*cot(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1-(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )-(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x-(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=(1/2*I)/{\sqrt {(}}-\beta )+(1/2)*tan(_{C}1+(1/4*I)*{\sqrt {(}}2)*x+(3/4*I)*t)/{\sqrt {(}}-\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=-1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*coth(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1-(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))-(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x-(3/4)*t)/{\sqrt {(}}\beta )$ ,

　
: $u(x,t)=1/(2*{\sqrt {(}}\beta ))+(1/2)*tanh(_{C}1+(1/4)*{\sqrt {(}}2)*x+(3/4)*t)/{\sqrt {(}}\beta )$ </math>

## 行波圖

## 參考文獻

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4. 王東明著 《消去法及其應用》 科學出版社 2002
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