非规范KdV方程(Unnormalized KdV equation)是一个非线性偏微分方程:[1]
解析解[编辑]
![{\displaystyle u(x,t)=-_{C}5/(\beta *_{C}4)-12*\alpha *_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,_{C}2,_{C}1)/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bf2a641be75935090f6eccea33083133c8b7c1)
![{\displaystyle u(x,t)=(4*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d933a5a10eebfa1e269b18e011db9cee14c5d6)
![{\displaystyle u(x,t)=(4*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/025a765f416dd53118187e40db8e9bad1ae9294b)
![{\displaystyle u(x,t)=-(4*\alpha *_{C}2^{3}+_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2363d8381a585397c8cad99167ac588616118fb7)
![{\displaystyle u(x,t)=-(4*\alpha *_{C}2^{3}+_{C}3)/(\beta *_{C}2)+12*\alpha *_{C}2^{2}*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aebdb1b917ec10fb7089570a67a0c2a0942528b)
![{\displaystyle u(x,t)=(8*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c74b56e5f4f1b93a48b2656c82dc33f5eb0f23ce)
![{\displaystyle u(x,t)=(8*\alpha *_{C}2^{3}-_{C}3)/(\beta *_{C}2)-12*\alpha *_{C}2^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc0297c4a6219e2da3952c88fbf84bb7a18ce63)
![{\displaystyle u(x,t)=(-8*\alpha *_{C}3^{3}+4*\alpha *_{C}3^{3}*_{C}1^{2}-_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/02b592134eaac981c7d4e3f230f8b087789dea0b)
![{\displaystyle u(x,t)=(-8*\alpha *_{C}3^{3}+4*\alpha *_{C}3^{3}*_{C}1^{2}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*(-1+_{C}1^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f3ddf0e37b404e11dbe0a6df6eb221d34811a6)
![{\displaystyle u(x,t)=(4*\alpha *_{C}3^{3}*_{C}1^{2}+4*\alpha *_{C}3^{3}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59eedc40862ed42af2b8a7156003977929e01c9e)
![{\displaystyle u(x,t)=(4*\alpha *_{C}3^{3}*_{C}1^{2}+4*\alpha *_{C}3^{3}-_{C}4)/(\beta *_{C}3)-12*\alpha *_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a05e88a1de417561d1ddaea21e751984d9d3f47d)
![{\displaystyle u(x,t)=-(8*\alpha *_{C}3^{3}*_{C}1^{2}-4*\alpha *_{C}3^{3}+_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0ff32c0484be9875675409e48bb64849d1425f)
![{\displaystyle u(x,t)=-(8*\alpha *_{C}3^{3}*_{C}1^{2}-4*\alpha *_{C}3^{3}+_{C}4)/(\beta *_{C}3)+12*\alpha *_{C}3^{2}*(-1+_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6944cdaaf61fb7e41a38b5dcef36b4b074b561b1)
行波图[编辑]
Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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Unnormalized KdV equation traveling wave plot
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参考文献[编辑]
- ^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS
- *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