紐厄爾-懷特海德方程

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：

$u(x, t) = 1/\sqrt(\beta)$,

$u(x, t) = -1/\sqrt(\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)-(1/2)*cot(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)-(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -(1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*cot(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*cot(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1-(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)-(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x-(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = (1/2*I)/\sqrt(-\beta)+(1/2)*tan(_C1+(1/4*I)*\sqrt(2)*x+(3/4*I)*t)/\sqrt(-\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*coth(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*coth(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*coth(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*coth(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))+(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = -1/(2*\sqrt(\beta))-(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*coth(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*coth(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*coth(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*coth(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$,

$u(x, t) = 1/(2*\sqrt(\beta))-(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)/\sqrt(\beta)$,

$$u(x, t) = 1/(2*\sqrt(\beta))+(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)/\sqrt(\beta)$$

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