双重sine-Gordon方程

双重sine-Gordon方程是一个非线性偏微分方程，它在诸如铁磁物质研究、Template:Tsl、液晶研究、氦 Template:Tsl、Template:Tsl等工程物理系领域有广泛的運用。双重sine-Gordon方程形式如下：^[1]

$u_{x t} = a s i n (u) + b s i n (2 u)$

变换

作变换

$u = 2 * a r c t a n (v)$

经简化后，上列方程化为

$2 * v_{x t} + 2 * v_{x t} * v^{2} - 4 * v_{t} * v * v_{x} = 2 * v * (a + a * v^{2} + 2 * b - 2 * b * v^{2})$

中间解

此v方程有雅可比椭圆函数解：

$v =_{C} 5 * J a c o b i C N (_{C} 2 +_{C} 3 * x - (a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a + 2 * b) * t / (_{C} 3 * (_{C} 5^{2} + 1)), \sqrt{(} (a *_{C} 5^{4} + 2 * a *_{C} 5^{2} + a + 2 * b - 2 * b *_{C} 5^{4}) * (a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a)) *_{C} 5 / (a *_{C} 5^{4} + 2 * a *_{C} 5^{2} + a + 2 * b - 2 * b *_{C} 5^{4}))$

$v =_{C} 5 * J a c o b i D N (_{C} 2 +_{C} 3 * x -_{C} 5^{2} * (a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a) * t / (_{C} 3 * (_{C} 5^{4} + 2 *_{C} 5^{2} + 1)), \sqrt{(} (a *_{C} 5^{4} + 2 * a *_{C} 5^{2} + a + 2 * b - 2 * b *_{C} 5^{4}) * (a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a)) / ((a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a) *_{C} 5))$

$v =_{C} 5 * J a c o b i N C (_{C} 2 +_{C} 3 * x + (a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a + 2 * b) * t / (_{C} 3 * (_{C} 5^{2} + 1)), \sqrt{(} (a *_{C} 5^{4} + 2 * a *_{C} 5^{2} + a + 2 * b - 2 * b *_{C} 5^{4}) * (a *_{C} 5^{2} + a + 2 * b)) / (a *_{C} 5^{4} + 2 * a *_{C} 5^{2} + a + 2 * b - 2 * b *_{C} 5^{4}))$

$v =_{C} 5 * J a c o b i N D (_{C} 2 +_{C} 3 * x + (a *_{C} 5^{2} + a + 2 * b) * t / (_{C} 3 * (_{C} 5^{4} + 2 *_{C} 5^{2} + 1)), \sqrt{(} (a *_{C} 5^{4} + 2 * a *_{C} 5^{2} + a + 2 * b - 2 * b *_{C} 5^{4}) * (a *_{C} 5^{2} + a + 2 * b)) / (a *_{C} 5^{2} + a + 2 * b))$

$v =_{C} 5 * J a c o b i N S (_{C} 2 +_{C} 3 * x +_{C} 5^{2} * (a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a) * t / (_{C} 3 * (_{C} 5^{4} + 2 *_{C} 5^{2} + 1)), \sqrt{(} - (a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a) * (a *_{C} 5^{2} + a + 2 * b)) / ((a *_{C} 5^{2} - 2 * b *_{C} 5^{2} + a) *_{C} 5))$

…………………………

以及双曲函数解

$v = \sqrt{(} (a - 2 * b) * a) * s i n h (_{C} 1 +_{C} 2 * x - (a - 2 * b) * t /_{C} 2) / (a - 2 * b)$

$v = \sqrt{(} - (a - 2 * b) * a) * c o s h (_{C} 1 +_{C} 2 * x - (a - 2 * b) * t /_{C} 2) / (a - 2 * b)$

$v = \sqrt{(} - (a - 2 * b) * (a + 2 * b)) * t a n h (_{C} 1 +_{C} 2 * x + (1 / 8) * (a^{2} - 4 * b^{2}) * t / (_{C} 2 * b)) / (a - 2 * b)$

$v = - \sqrt{(} a * (a + 2 * b)) * c s c h (_{C} 1 +_{C} 2 * x + (a + 2 * b) * t /_{C} 2) / a$

………………

和三角函数解

$v = \sqrt{(} (a - 2 * b) * (a + 2 * b)) * c o t (_{C} 1 +_{C} 2 * x - (1 / 8) * (a^{2} - 4 * b^{2}) * t / (_{C} 2 * b)) / (a - 2 * b)$

$v = \sqrt{(} (a - 2 * b) * (a + 2 * b)) * t a n (_{C} 1 +_{C} 2 * x - (1 / 8) * (a^{2} - 4 * b^{2}) * t / (_{C} 2 * b)) / (a - 2 * b)$

$v = \sqrt{(} - (a - 2 * b) * a) * c o s (_{C} 1 +_{C} 2 * x + (a - 2 * b) * t /_{C} 2) / (a - 2 * b)$

$v = \sqrt{(} - (a - 2 * b) * a) * s i n (_{C} 1 +_{C} 2 * x + (a - 2 * b) * t /_{C} 2) / (a - 2 * b)$

……

行波解

作反变换

$u = t a n (v / 2)$ 即得原来sine-Gordon方程的行波解：

$\begin{matrix} * 2 a r c t a n (1.5 J a c o b i C N (- 1.2 - 1.3 x + 0.17751479289940828402 t, 1.0741723110591493207 I)) \\ * 2 a r c t a n (1.5 J a c o b i D N (1.2 + 1.3 x + 0.20482476103777878925 t, 0.93094933625126274463 I)) \\ * 2 a r c t a n (1.5 J a c o b i N C (1.2 + 1.3 x + 0.17751479289940828402 t, 1.4675987714106856141)) \\ 2 a r c t a n (1.5 J a c o b i N D (1.2 + 1.3 x + 0.38233955393718707328 t, 0.68138514386924689225)) \\ * - 2 a r c t a n (1.5 J a c o b i N S (- 1.2 - 1.3 x + 0.20482476103777878925 t, 1.3662601021279464511)) \\ * - 2 a r c t a n (1.5 J a c o b i S N (- 1.2 - 1.3 x + 0.38233955393718707328 t, 0.73192505471139988450)) \\ * - (2 * I) * a r c t a n h (s i n h (- 15.1 + 1.2 * x + . 83333333333333333333 * t)) \\ * 2 * a r c t a n (s i n (15.1 - 1.2 * x + . 83333333333333333333 * t)) \\ * 2 * a r c t a n (s q r t (3) * c o t h (15.1 - 1.2 * x + . 31250000000000000000 * t)) \end{matrix}$

行波图

参考文献

↑ Juntao Fu,Shikuo Liu and Shida Liu,Exact Jacobian Elliptic Function Solutions to the Double Sine-Gordon Equation,Z.Naturforsch. 60a,301-312 2005Template:En

[1] Juntao Fu,Shikuo Liu and Shida Liu,Exact Jacobian Elliptic Function Solutions to the Double Sine-Gordon Equation,Z.Naturforsch. 60a,301-312 2005Template:En

[1]

双重sine-Gordon方程

目录

变换

中间解

行波解

行波图

参考文献

导航菜单

双重sine-Gordon方程

变换

中间解

行波解

行波图

参考文献

导航菜单

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