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、带电密度波（英语：Charge density wave）、液晶研究、氦自旋波（英语：spin wave）、超短光脉冲（英语：ultrashort pulse）等工程物理系领域有广泛的運用。双重sine-Gordon方程形式如下： u x t = a s i n ( u ) + b s i n ( 2 u ) {\displaystyle u_{xt}=asin(u)+bsin(2u)} 作变换 u = 2 ∗ a r c t a n ( v ) {\displaystyle u=2*arctan(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 ) {\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})} 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 ) ) , ( ( 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*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}))}} 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 ) ) , ( ( 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*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))}} 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 ) ) , ( ( 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*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}))}} 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 ) ) , ( ( 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*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))}} 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 ) ) , ( − ( 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 ) ) {\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 v = ( ( a − 2 ∗ b ) ∗ a ) ∗ s i n h ( C 1 + C 2 ∗ x − ( a − 2 ∗ b ) ∗ t / C 2 ) / ( a − 2 ∗ b ) {\displaystyle {v={\sqrt {(}}(a-2*b)*a)*sinh(_{C}1+_{C}2*x-(a-2*b)*t/_{C}2)/(a-2*b)}} v = ( − ( a − 2 ∗ b ) ∗ a ) ∗ c o s h ( 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)}} v = ( − ( 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 ) {\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)}} v = − ( a ∗ ( a + 2 ∗ b ) ) ∗ c s c h ( C 1 + C 2 ∗ x + ( a + 2 ∗ b ) ∗ t / C 2 ) / a {\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 v = ( ( 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 ) {\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)}} v = ( ( 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 ) {\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)}} v = ( − ( a − 2 ∗ b ) ∗ a ) ∗ c o s ( C 1 + C 2 ∗ x + ( a − 2 ∗ b ) ∗ t / C 2 ) / ( 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)}} v = ( − ( a − 2 ∗ b ) ∗ a ) ∗ s i n ( 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)}} …… 作反变换 u = t a n ( v / 2 ) {\displaystyle u=tan(v/2)} 即得原来sine-Gordon方程的行波解： ∗ 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 ) ) {\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}}} 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）