卡馬薩-霍爾姆方程

卡馬薩-霍爾姆方程（Camassa Holm equation）是流體力學中的一個非線性偏微分方程

$$u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}.$$

1993年卡馬薩和霍爾姆以此偏微分方程模擬淺水波，其中$\kappa$是大於0的參數。

## 行波解

卡馬薩-霍爾姆方程有行波解：

$$u_2 := \frac{3c(c - 2\kappa)}{2(c + \kappa)\cosh^2\left(\frac{1}{2}\sqrt{\frac{c - 2\kappa}{c}}(ct -x - x_0)\right)}.$$

將參數 $c = 1, x_0 = 1, \kappa = 0.3$ 代入得：

$$u(x, t) \approx \frac{0.46154}{\cosh^2[0.31623(-x + t - 1)]}.$$

### Maple TWSolution

Maple軟體包TWSolution可提供多種行波解，有以下形式：

$\textbf{sech}$ 展開$$\begin{aligned} g[2] := u(x, t) &= -\frac{1}{2}\kappa + C_2 \operatorname{sech}^2\left(C_1 + \frac{\sqrt{3}}{2}x - \frac{\sqrt{3}\kappa}{4}t\right), \\ g[3] := u(x, t) &= -\frac{1}{2}\kappa + C_2 \operatorname{sech}^2\left(C_1 - \frac{\sqrt{3}}{2}x + \frac{\sqrt{3}\kappa}{4}t\right), \\ g[4] := u(x, t) &= C_2 - \left(\frac{3}{2}C_2 + \frac{3}{4}\kappa\right)\operatorname{sech}^2\left(C_1 + \frac{\sqrt{3}}{2}ix - \frac{\sqrt{3}\kappa}{4}it\right), \\ g[5] := u(x, t) &= C_2 - \left(\frac{3}{2}C_2 + \frac{3}{4}\kappa\right)\operatorname{sech}^2\left(C_1 - \frac{\sqrt{3}}{2}ix + \frac{\sqrt{3}\kappa}{4}it\right), \\ g[6] := u(x, t) &= -\frac{C_3 - 4C_3 C_2^2 + 2\kappa C_2}{C_2(3 + 4C_2^2)} + \frac{24C_2(2C_3 + \kappa C_2)}{16C_2^4 - 9}\operatorname{sech}^2(C_1 + C_2 x + C_3 t). \end{aligned}$$

$\textbf{exp}$ 展開$$\begin{aligned} g[2] := u(x, t) &= -\frac{\sqrt{3}}{9}C_1 - \frac{1}{3}\kappa + C_2 \exp(-\sqrt{3}x + C_1 t), \\ g[3] := u(x, t) &= \frac{\sqrt{3}}{9}C_1 - \frac{1}{3}\kappa + C_2 \exp(\sqrt{3}x + C_1 t), \\ g[5] := u(x, t) &= -\frac{\sqrt{3}}{3}C_1 - \frac{1}{3}\kappa + C_2 \exp(-\sqrt{3}x + 3C_1 t). \end{aligned}$$

$\textbf{csch}$ 展開$$\begin{aligned} g[2] := u(x, t) &= -\frac{1}{2}\kappa + C_2 \operatorname{csch}^2\left(C_1 + \frac{\sqrt{3}}{2}x - \frac{\sqrt{3}\kappa}{4}t\right), \\ g[4] := u(x, t) &= C_2 + \left(\frac{3}{2}C_2 + \frac{3}{4}\kappa\right)\operatorname{csch}^2\left(C_1 + \frac{\sqrt{3}}{2}ix - \frac{\sqrt{3}\kappa}{4}it\right), \\ g[6] := u(x, t) &= -\frac{C_3 - 4C_3 C_2^2 + 2\kappa C_2}{C_2(4C_2^2 + 3)} - \frac{24C_2(2C_3 + \kappa C_2)}{16C_2^4 - 9}\operatorname{csch}^2(C_1 + C_2 x + C_3 t). \end{aligned}$$

$\textbf{sec}$ 展開$$\begin{aligned} g[3] := u(x, t) &= -\frac{1}{2}\kappa + C_2 \sec^2\left(C_1 - \frac{\sqrt{3}}{2}ix + \frac{\sqrt{3}\kappa}{4}it\right), \\ g[5] := u(x, t) &= C_2 - \left(\frac{3}{2}C_2 + \frac{3}{4}\kappa\right)\sec^2\left(C_1 - \frac{\sqrt{3}}{2}x + \frac{\sqrt{3}\kappa}{4}t\right), \\ g[6] := u(x, t) &= \frac{C_3 + 4C_3 C_2^2 + 2\kappa C_2}{C_2(4C_2^2 - 3)} - \frac{24C_2(2C_3 + \kappa C_2)}{16C_2^4 - 9}\sec^2(C_1 + C_2 x + C_3 t). \end{aligned}$$

$\textbf{JacobiSN}$ 展開$$\begin{aligned} g[3] := u(x, t) &= \frac{\sqrt{3}}{9}i C_1 - \frac{1}{3}\kappa + C_2 \sin\left(C_3 - \sqrt{3}ix + C_1 t\right), \\ g[4] := u(x, t) &= -\frac{2\sqrt{3}}{9}i C_1 - \frac{1}{2}C_2 - \frac{1}{3}\kappa + C_2 \sin^2\left(C_3 + \frac{\sqrt{3}}{2}ix + C_1 t\right). \end{aligned}$$

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