双曲型刘维方程(Hyperbolic Liouville equation)是一个非线性偏微分方程:[1]
作变换:
得:
求得 v(x,t) 的行波解,作反代换得回 u(x,t)。
解析解
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01949858041dc8614eb403b0fb6ea34b1a37c907)
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d4f114c013889b8d478168629e7db404a74ce7)
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1da19b0a2bbfb31b19d02ba1392fdbb0c0f22f7)
![{\displaystyle {u(x,t)=ln((2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}))*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac670f8b1659f322521b3b163e6dca2c8ccd8cfc)
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}3^{2}-_{C}4^{2}))*csc(_{C}2+_{C}3*x+_{C}4*t)^{2}/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a08c3090933d08de398fe4d3522b2dd3efb260)
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}3^{2}-_{C}4^{2}))*sec(_{C}2+_{C}3*x+_{C}4*t)^{2}/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/266e4f21e9c28ec4e9882efe92690bfbb9332d41)
![{\displaystyle {u(x,t)=ln((2*(\alpha ^{2}*_{C}3^{2}-_{C}4^{2}))*sech(_{C}2+_{C}3*x+_{C}4*t)^{2}/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc426180fa2071345e89785b4936e409fe792aff)
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}4^{2}-_{C}5^{2}))*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,0,0)/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d272a37727c027dce45c1cfa8ebf40918ccc95fb)
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}+cot(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-cot(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5210c362e427ca1c7f04569dad59471b7ec37b26)
![{\displaystyle {u(x,t)=ln(-(2*(\alpha ^{2}*_{C}2^{2}-_{C}3^{2}+tan(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-tan(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2009f19808d3322c8a963263192e6d2f9c2e2e3c)
![{\displaystyle {u(x,t)=ln(-(2*(-\alpha ^{2}*_{C}2^{2}+_{C}3^{2}+coth(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-coth(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b07ac0bea91b463b83aa8cd05c857b5aa3ac8abc)
![{\displaystyle {u(x,t)=ln(-(2*(-\alpha ^{2}*_{C}2^{2}+_{C}3^{2}+tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}*\alpha ^{2}*_{C}2^{2}-tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}*_{C}3^{2}))/(\gamma *\beta ))/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/250f0925db2ded432384ad8796d3f7f5bf1121a3)
行波图
双曲型刘维方程行波图
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双曲型刘维方程行波图
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双曲型刘维方程行波图
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双曲型刘维方程行波图
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双曲型刘维方程行波图
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双曲型刘维方程行波图
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双曲型刘维方程行波图
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双曲型刘维方程行波图
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参考文献
- ^ Graham W.Griffiths Chapter 15 p275-292 Academic Press 2012
- *谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社
- *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
- 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
- 王东明著 《消去法及其应用》 科学出版社 2002
- *何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445
- Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
- Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
- Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
- Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
- Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
- Dongming Wang, Elimination Practice,Imperial College Press 2004
- David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
- George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759