双重sinh-Gordon方程 （Double sinh-Gordon equation）是一个非线性偏微分方程 。[1] [2] [3] [4] [5]
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{\displaystyle u_{xt}=asinh(u)+bsinh(2u)}
行波解 [ 编辑 ]
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{\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}))}}
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{\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))}}
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{\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}))}}
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{\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))}}
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{\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}
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{\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}2+_{C}3*x-(2*b+a)*t/_{C}3)/a}}
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{\displaystyle {v={\sqrt {(}}a*(2*b+a))*sec(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}
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{\displaystyle {v={\sqrt {(}}a*(2*b+a))*sech(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}
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{\displaystyle {v={\sqrt {(}}-a*(2*b+a))*csch(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}
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{\displaystyle {v={\sqrt {(}}(a-2*b)*a)*cosh(_{C}2+_{C}3*x-(a-2*b)*t/_{C}3)/(a-2*b)}}
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{\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
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{\displaystyle v=tanh((1/2)*u)}
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1.0555973258234951998
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{\displaystyle u(x,t)=2arctanh(1.5*JacobiCN(1.2+1.3*x+3.2307692307692307692*t,1.0555973258234951998))}
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1.5
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{\displaystyle u(x,t)=2arctanh(1.5*JacobiDN(1.2+1.3*x+3.6000000000000000000*t,.94733093343134184593))}
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{\displaystyle u(x,t)=2*arctanh(1.5*JacobiNC(-1.2-1.3*x+3.2307692307692307692*t,.33806170189140663100*I))}
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2.9580398915498080213
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{\displaystyle u(x,t)=2*arctanh(1.5*JacobiND(1.2+1.3*x+.36923076923076923077*t,2.9580398915498080213*I))}
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{\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(15.1-1.2*x+2.5000000000000000000*t))}
u
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2.3076923076923076923
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{\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(-1.2-1.3*x+2.3076923076923076923*t))}
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{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sec(15.1-1.2*x+2.5000000000000000000*t))}
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{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(-15.1+1.2*x+2.5000000000000000000*t))}
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{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(1.2+1.3*x+2.3076923076923076923*t))}
u
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{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}-3)*csch(-15.1+1.2*x+2.5000000000000000000*t))}
{\displaystyle }
{\displaystyle }
{\displaystyle }
行波图 [ 编辑 ] 参考文献 [ 编辑 ]
^ Andrei D. Polyanin, Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS SECOND EDITION CRC Press, A Chapman & Hall Book ISBN 9781420087239
^ Zeitschrift Für Naturforschung: A journal of physical sciences 2004 p933-937
^ 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
^ Issues in Logic, Operations, and Computational Mathematics and Geometry 2013 p484
^ Mathematical Reviews - Page 3708 2007
