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].
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}))}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c0ff0cbb64a3f47e4a089571b2a1a86ad34dd9b7)
![{\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))}}](//wikimedia.org/api/rest_v1/media/math/render/svg/88cfda45da55e0c130a31991881a64916ff95cf8)
![{\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}))}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a19a77c3e27b585480a9aa9c7d201269ab305012)
![{\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))}}](//wikimedia.org/api/rest_v1/media/math/render/svg/486c1cc85cc6b147f46e072d3b8a4a7812a08a50)
![{\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c796a9164c4ea734bdf42c6005c6c3db0c3c6ca3)
![{\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}2+_{C}3*x-(2*b+a)*t/_{C}3)/a}}](//wikimedia.org/api/rest_v1/media/math/render/svg/04527237f20c8da27e408653714686aa26d4f8b5)
![{\displaystyle {v={\sqrt {(}}a*(2*b+a))*sec(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b04bd2e5b38d02bc143f7a51fc721295b71becf5)
![{\displaystyle {v={\sqrt {(}}a*(2*b+a))*sech(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}](//wikimedia.org/api/rest_v1/media/math/render/svg/aa8674c3b26ec6e4634ec1ea6b34ed65475f4264)
![{\displaystyle {v={\sqrt {(}}-a*(2*b+a))*csch(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}](//wikimedia.org/api/rest_v1/media/math/render/svg/14511830e26870c5ff6c0dcac1a9d84635040cac)
![{\displaystyle {v={\sqrt {(}}(a-2*b)*a)*cosh(_{C}2+_{C}3*x-(a-2*b)*t/_{C}3)/(a-2*b)}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f3b02d669d1a5218366c6e711c33b5558c1d9ce1)
![{\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)}}](//wikimedia.org/api/rest_v1/media/math/render/svg/660117869b72c9678193e407b87527c18cefc292)
其中
特解
![{\displaystyle u(x,t)=2arctanh(1.5*JacobiCN(1.2+1.3*x+3.2307692307692307692*t,1.0555973258234951998))}](//wikimedia.org/api/rest_v1/media/math/render/svg/bc847f9c32c307587013fec44ac9d037a389af58)
![{\displaystyle u(x,t)=2arctanh(1.5*JacobiDN(1.2+1.3*x+3.6000000000000000000*t,.94733093343134184593))}](//wikimedia.org/api/rest_v1/media/math/render/svg/a378dd9d596183a376cfc58f15b1f654638331b3)
![{\displaystyle u(x,t)=2*arctanh(1.5*JacobiNC(-1.2-1.3*x+3.2307692307692307692*t,.33806170189140663100*I))}](//wikimedia.org/api/rest_v1/media/math/render/svg/03f40b700bd2490c4ad4f79d031714c7ee4604c1)
![{\displaystyle u(x,t)=2*arctanh(1.5*JacobiND(1.2+1.3*x+.36923076923076923077*t,2.9580398915498080213*I))}](//wikimedia.org/api/rest_v1/media/math/render/svg/3126f51395b494fd0d9f60ec923166fb28c520bf)
![{\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(15.1-1.2*x+2.5000000000000000000*t))}](//wikimedia.org/api/rest_v1/media/math/render/svg/21e09490a2aac6d18933adccfb2b1ac5cafd4979)
![{\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(-1.2-1.3*x+2.3076923076923076923*t))}](//wikimedia.org/api/rest_v1/media/math/render/svg/42bc0e86c653c9ad968f04d4483bcd7e2045acd0)
![{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sec(15.1-1.2*x+2.5000000000000000000*t))}](//wikimedia.org/api/rest_v1/media/math/render/svg/6d2ae029cd10fcb659be3ba416c26b0bd5f68e6e)
![{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(-15.1+1.2*x+2.5000000000000000000*t))}](//wikimedia.org/api/rest_v1/media/math/render/svg/ceb23fc3670d6fbde680abf9ce294b779f45df3e)
![{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(1.2+1.3*x+2.3076923076923076923*t))}](//wikimedia.org/api/rest_v1/media/math/render/svg/3cb2953daa64511ee9af0f844c6b87d9c17abd8c)
![{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}-3)*csch(-15.1+1.2*x+2.5000000000000000000*t))}](//wikimedia.org/api/rest_v1/media/math/render/svg/104ac1b61380e9a3b66b4d1d4d15bcfac48230e6)
![{\displaystyle }](//wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](//wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](//wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
行波图
参考文献
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