**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、带电密度波（英语：）、液晶研究、氦自旋波（英语：）、超短光脉冲（英语：）等工程物理系领域有广泛的運用。双重sine-Gordon方程形式如下：

{\displaystyle u\_{xt}=asin(u)+bsin(2u)}

变换
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作变换

{\displaystyle u=2\*arctan(v)}

经简化后，上列方程化为

{\displaystyle 2\*v\_{xt}+2\*v\_{xt}\*v^{2}-4\*v\_{t}\*v\*v\_{x}=2\*v\*(a+a\*v^{2}+2\*b-2\*b\*v^{2})}

中间解
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此v方程有雅可比椭圆函数解：

{\displaystyle {v=\_{C}5\*JacobiCN(\_{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}))}}

{\displaystyle {v=\_{C}5\*JacobiDN(\_{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))}}

{\displaystyle {v=\_{C}5\*JacobiNC(\_{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}))}}

{\displaystyle {v=\_{C}5\*JacobiND(\_{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))}}

{\displaystyle {v=\_{C}5\*JacobiNS(\_{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

{\displaystyle {v={\sqrt {(}}(a-2\*b)\*a)\*sinh(\_{C}1+\_{C}2\*x-(a-2\*b)\*t/\_{C}2)/(a-2\*b)}}

{\displaystyle {v={\sqrt {(}}-(a-2\*b)\*a)\*cosh(\_{C}1+\_{C}2\*x-(a-2\*b)\*t/\_{C}2)/(a-2\*b)}}

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

{\displaystyle {v=-{\sqrt {(}}a\*(a+2\*b))\*csch(\_{C}1+\_{C}2\*x+(a+2\*b)\*t/\_{C}2)/a}}

  

………………

和三角函数解

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

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

{\displaystyle {v={\sqrt {(}}-(a-2\*b)\*a)\*cos(\_{C}1+\_{C}2\*x+(a-2\*b)\*t/\_{C}2)/(a-2\*b)}}

{\displaystyle {v={\sqrt {(}}-(a-2\*b)\*a)\*sin(\_{C}1+\_{C}2\*x+(a-2\*b)\*t/\_{C}2)/(a-2\*b)}}

……

行波解
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作反变换

{\displaystyle u=tan(v/2)}即得原来sine-Gordon方程的行波解：

  

{\displaystyle {\begin{aligned}\*2arctan(1.5JacobiCN(-1.2-1.3x+0.17751479289940828402t,1.0741723110591493207I))\\\*2arctan(1.5JacobiDN(1.2+1.3x+0.20482476103777878925t,0.93094933625126274463I))\\\*2arctan(1.5JacobiNC(1.2+1.3x+0.17751479289940828402t,1.4675987714106856141))\\2arctan(1.5JacobiND(1.2+1.3x+0.38233955393718707328t,0.68138514386924689225))\\\*-2arctan(1.5JacobiNS(-1.2-1.3x+0.20482476103777878925t,1.3662601021279464511))\\\*-2arctan(1.5JacobiSN(-1.2-1.3x+0.38233955393718707328t,0.73192505471139988450))\\\*-(2\*I)\*arctanh(sinh(-15.1+1.2\*x+.83333333333333333333\*t))\\\*2\*arctan(sin(15.1-1.2\*x+.83333333333333333333\*t))\\\*2\*arctan(sqrt(3)\*coth(15.1-1.2\*x+.31250000000000000000\*t))\\\end{aligned}}}

行波图
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参考文献
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1. ↑  Juntao Fu,Shikuo Liu and Shida Liu,Exact Jacobian Elliptic Function Solutions to the Double Sine-Gordon Equation,Z.Naturforsch. 60a,301-312 2005（英文）