布雷瑟顿方程(Bretherton equation)是一个非线性偏微分方程:[1]
解析解
利用Maple软件包TWSolution可得布雷瑟顿方程两个WeirstrassP函数行波解和六个Jacobi椭圆函数行波解。
![{\displaystyle {u(x,t)=-(1/30)*(_{C}5^{2}+_{C}4^{2})*{\sqrt {(}}30)/({\sqrt {(}}\alpha )*_{C}4^{2})-2*{\sqrt {(}}30)*_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,-(1/180)*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{8},(1/5400)*(_{C}5^{2}+_{C}4^{2})*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{1}2)/{\sqrt {(}}\alpha )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1e58d8fbc0aa1c609ac3cb205e0f41b81701f5)
![{\displaystyle {u(x,t)=(1/30)*(_{C}5^{2}+_{C}4^{2})*{\sqrt {(}}30)/({\sqrt {(}}\alpha )*_{C}4^{2})+2*{\sqrt {(}}30)*_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,-(1/180)*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{8},(1/5400)*(_{C}5^{2}+_{C}4^{2})*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{1}2)/{\sqrt {(}}\alpha )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8977c8296ffca95684d9376eef52595831a012e)
![{\displaystyle p[3]:=-4.4229081351691113421-0.93307517430300270455e-1*I+(18.208764436791314548+0.*I)*JacobiDN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/060e5d3d7a90c393b1fb721a640679dc6153b113)
![{\displaystyle p[4]:=-5.1919544739096094160+5.3436314406870407268*I+(18.208764436791314548+0.*I)*JacobiNS(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a54e15315b6e125be6cf17e66c2f92ef30cad62c)
![{\displaystyle p[5]:=-2.5743416547838020433-3.2877514938561477464*I+(2.3878725879366620662-40.119130323671954306*I)*JacobiND(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/701b65578609cb8d196cb850419fd451860dd1ba)
![{\displaystyle p[6]:=-5.5945307529851545322+1.6067031040792677714*I+(2.3878725879366620662-40.119130323671954306*I)*JacobiNC(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/116ea832ccb5c9bafe34bed8a155c3d1c97e370a)
![{\displaystyle p[7]:=-5.1805015655073574818+2.2007257608074972052*I+(21.217747598613725681-27.288660748456732356*I)*JacobiSN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb7e68fdcbfbfa60a3e72f0d28a797b59f860eca)
![{\displaystyle p[8]:=-5.7120768471542134149-.77489365858670457610*I+(21.217747598613725681-27.288660748456732356*I)*JacobiCN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71a458d91f7a24385a5695c4a403334a5c1de68)
![{\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)
行波图
Bretherton equation traveling wave WeierstrassP plot 1
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Bretherton equation traveling wave WeierstrassP plot 1
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Bretherton equation traveling wave Jacobi function plot
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Bretherton equation traveling wave Jacobi function plot
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Bretherton equation traveling wave Jacobi function plot
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Bretherton equation traveling wave Jacobi function plot
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Bretherton equation traveling wave Jacobi function plot
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Bretherton equation traveling wave Jacobi function plot
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参考文献
- ^ 李志斌编著 《非线性数学物理方程的行波解》 152页 科学出版社 2008
- *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