雙重sinh-Gordon方程

雙重sinh-Gordon方程（Double sinh-Gordon equation）是一個非線性偏微分方程。^{[1][2][3][4][5]}.

${\displaystyle u_{xt}=asinh(u)+bsinh(2u)}$

行波解

• ${\displaystyle {v=_{C}5*JacobiCN(_{C}2+_{C}3*x-(a*_{C}5^{2}-2*b*_{C}5^{2}-2*b-a)*t/(_{C}3*(_{C}5^{2}-1)),{\sqrt {(}}(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b*_{C}5^{2}-a))*_{C}5/(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4}))}}$
• ${\displaystyle {v=_{C}5*JacobiDN(_{C}2+_{C}3*x-_{C}5^{2}*(a*_{C}5^{2}-2*b*_{C}5^{2}-a)*t/(_{C}3*(-2*_{C}5^{2}+1+_{C}5^{4})),{\sqrt {(}}(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b*_{C}5^{2}-a))/((a*_{C}5^{2}-2*b*_{C}5^{2}-a)*_{C}5))}}$
• ${\displaystyle {v=_{C}5*JacobiNC(_{C}2+_{C}3*x+(a*_{C}5^{2}-2*b*_{C}5^{2}-2*b-a)*t/(_{C}3*(_{C}5^{2}-1)),{\sqrt {(}}-(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b-a))/(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4}))}}$
• ${\displaystyle {v=_{C}5*JacobiND(_{C}2+_{C}3*x-(a*_{C}5^{2}-2*b-a)*t/(_{C}3*(-2*_{C}5^{2}+1+_{C}5^{4})),{\sqrt {(}}-(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b-a))/(a*_{C}5^{2}-2*b-a))}}$
• ${\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}$
• ${\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}2+_{C}3*x-(2*b+a)*t/_{C}3)/a}}$
• ${\displaystyle {v={\sqrt {(}}a*(2*b+a))*sec(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}$
• ${\displaystyle {v={\sqrt {(}}a*(2*b+a))*sech(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}$
• ${\displaystyle {v={\sqrt {(}}-a*(2*b+a))*csch(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}$
• ${\displaystyle {v={\sqrt {(}}(a-2*b)*a)*cosh(_{C}2+_{C}3*x-(a-2*b)*t/_{C}3)/(a-2*b)}}$
• ${\displaystyle {v={\sqrt {(}}(a-2*b)*(2*b+a))*tanh(_{C}1+_{C}2*x+(1/8)*(a^{2}-4*b^{2})*t/(_{C}2*b))/(a-2*b)}}$

其中 ${\displaystyle v=tanh((1/2)*u)}$

特解

• ${\displaystyle u(x,t)=2arctanh(1.5*JacobiCN(1.2+1.3*x+3.2307692307692307692*t,1.0555973258234951998))}$
• ${\displaystyle u(x,t)=2arctanh(1.5*JacobiDN(1.2+1.3*x+3.6000000000000000000*t,.94733093343134184593))}$
• ${\displaystyle u(x,t)=2*arctanh(1.5*JacobiNC(-1.2-1.3*x+3.2307692307692307692*t,.33806170189140663100*I))}$
• ${\displaystyle u(x,t)=2*arctanh(1.5*JacobiND(1.2+1.3*x+.36923076923076923077*t,2.9580398915498080213*I))}$
• ${\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(15.1-1.2*x+2.5000000000000000000*t))}$
• ${\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(-1.2-1.3*x+2.3076923076923076923*t))}$
• ${\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sec(15.1-1.2*x+2.5000000000000000000*t))}$
• ${\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(-15.1+1.2*x+2.5000000000000000000*t))}$
• ${\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(1.2+1.3*x+2.3076923076923076923*t))}$
• ${\displaystyle u(x,t)=2*arctanh({\sqrt {(}}-3)*csch(-15.1+1.2*x+2.5000000000000000000*t))}$
• ${\displaystyle }$
• ${\displaystyle }$
• ${\displaystyle }$

行波圖

參考文獻

1. Andrei D. Polyanin, Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS SECOND EDITION CRC Press, A Chapman & Hall Book ISBN 9781420087239
2. Zeitschrift Für Naturforschung: A journal of physical sciences 2004 p933-937
3. A. M. WAZWAZ Exact solutions to the double sinh-gordon equation by the tanh method and a variable separated ODE method,Computers & Mathematics with Applications,Volume 50, Issues 10–12, November–December 2005, Pages 1685–1696
4. Issues in Logic, Operations, and Computational Mathematics and Geometry 2013 p484
5. Mathematical Reviews - Page 3708 2007