劉維方程(Liouville equation)是一個非線性偏微分方程:[1]
![{\displaystyle u*u_{xt}-u*u_{x}+\alpha *u^{3}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09048142a5fdd64a02023d1cda7c1931904bf987)
解析解
![{\displaystyle pu[2]:=ln(-3.4101973587048390836*csc(1.32+1.4934776966447732662*x^{1}.2+1.6440433652163821333*t^{1}.2)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0287a43d7d1f47f02d660e543b67b9f284efe9f)
![{\displaystyle pu[3]:=ln(-3.4101973587048390836*csch(1.32+1.4934776966447732662*x^{1}.2+1.6440433652163821333*t^{1}.2)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de212ee57ec02662e99ea45eff9caa927fb8b264)
![{\displaystyle pu[4]:=ln(-3.4101973587048390836*sec(1.32+1.4934776966447732662*x^{1}.2+1.6440433652163821333*t^{1}.2)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af3baba212f81db49930b5781c1b8addf0533fb)
![{\displaystyle pu[5]:=ln(3.4101973587048390836*sech(1.32+1.4934776966447732662*x^{1}.2+1.6440433652163821333*t^{1}.2)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1228d3931a527861b1a4c703df23f5c8ba0710f)
![{\displaystyle pu[9]:=ln(-3.4101973587048390836-3.4101973587048390836*cot(1.32+1.4934776966447732662*x^{1}.2+1.6440433652163821333*t^{1}.2)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53199f3758db4fca4225b8375b3e06ec65bb9c45)
![{\displaystyle pu[12]:=ln(3.4101973587048390836-3.4101973587048390836*tanh(1.32+1.4934776966447732662*x^{1}.2+1.6440433652163821333*t^{1}.2)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a553745ecffa5dde5c0f0393077c763670736bc)
![{\displaystyle pu[13]:=ln(4.1031336303472692620-4.1031336303472692620*coth(1.44+1.6440433652163821333*x^{1}.2+1.7969454312181156991*t^{1}.2)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04256ce138ace26e212b6cbeb083eee93b432393)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
![{\displaystyle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68)
行波圖
Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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Liouville equation traveling wave plot
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參考文獻
- ^ 李志斌編著 《非線性數學物理方程的行波解》105頁 科學出版社 2008
- *谷超豪 《孤立子理論中的達布變換及其幾何應用》 上海科學技術出版社
- *閻振亞著 《複雜非線性波的構造性理論及其應用》 科學出版社 2007年
- 李志斌編著 《非線性數學物理方程的行波解》 科學出版社
- 王東明著 《消去法及其應用》 科學出版社 2002
- *何青 王麗芬編著 《Maple 教程》 科學出版社 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