纽厄尔-怀特海德方程

**纽厄尔-怀特海德方程**（Newell-Whitehead equation)是一个非线性偏微分方程：

u_{t}-u_{xx}-\alpha*u+beta*u^3=0
## 行波解
纽厄尔-怀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(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = -(1/2*I)/\sqrt(-\beta)-(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = -(1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = -(1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*cot(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta), 
:u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta),
: u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*coth(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta), 
:u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*coth(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta), 
:u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*coth(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta), 
:u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*coth(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta),
: u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta),
: u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta), 
:u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta), 
:u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta), 
:u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*coth(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta), 
:u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*coth(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta), 
:u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*coth(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta),
: u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*coth(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta), 
:u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta), 
:u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta), 
:u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta), 
:u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)

## 行波图

## 参考文献

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