广田-萨摩方程(Hirota Satsuma equation)是一个三元非线性偏微分方程组:[1]
![{\displaystyle u_{t}-0.5*u_{xxx}+3uu_{x}-3(vw)_{x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbbc309c70eac83e195c941f0e5528c353a8f65)
![{\displaystyle v_{t}+v_{xxx}-3uv_{x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea87e6e5035b25f4500022094484d4b5ebf0495)
![{\displaystyle w_{t}+w_{xxx}-3uw_{x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdb8092a50fdf6633ce4e0b48c0f0f987103634)
解析解[编辑]
![{\displaystyle u(x,t)=-(1/3)*(-_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*cos(_{C}1+_{C}2*x+_{C}3*t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55da2eff337916d2257281f29077b3cfeb3ffafa)
![{\displaystyle u(x,t)=-(1/3)*(-_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*sin(_{C}1+_{C}2*x+_{C}3*t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94a46787c948750f4fbdf6c7dc17440ddcde59ad)
![{\displaystyle u(x,t)=(1/3)*(_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*cosh(_{C}1+_{C}2*x+_{C}3*t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0387bcc8649373c5c3e8c6f5e738ea6da114957d)
![{\displaystyle u(x,t)=(1/3)*(_{C}3+_{C}2^{3})/_{C}2,v(x,t)=0,w(x,t)=_{C}7+_{C}8*sinh(_{C}1+_{C}2*x+_{C}3*t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b65a1417019d8c7902f22b9238b5ea198adfbeb6)
![{\displaystyle u(x,t)=-(1/3)*(9*_{C}3^{3}-_{C}4)/_{C}3,v(x,t)=_{C}6-(3/4)*_{C}9*cos(_{C}2+_{C}3*x+_{C}4*t)+_{C}9*cos(_{C}2+_{C}3*x+_{C}4*t)^{3},w(x,t)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65df623a73bd0c2ee9fb1785a98c3cf9a10f6197)
![{\displaystyle u(x,t)=-_{C}2^{2}+2*_{C}2^{2}*coth(_{C}1+_{C}2*x-_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*coth(_{C}1+_{C}2*x-_{C}2^{3}*t),w(x,t)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9c0375fa3ac4e5fba6cea9a018f43c31886cb3)
![{\displaystyle {u(x,t)=-(1/2)*_{C}2^{2}+2*_{C}2^{2}*csc(_{C}1+_{C}2*x-(1/2)*_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*csc(_{C}1+_{C}2*x-(1/2)*_{C}2^{3}*t),w(x,t)=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2cdcfd3b9d76cdbb363676ef982edd607d1a96c)
![{\displaystyle {u(x,t)=(1/2)*_{C}2^{2}+2*_{C}2^{2}*csch(_{C}1+_{C}2*x+(1/2)*_{C}2^{3}*t)^{2},v(x,t)=_{C}5+_{C}6*csch(_{C}1+_{C}2*x+(1/2)*_{C}2^{3}*t),w(x,t)=0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d30fc274da7fc2b15fd0eafb6640fcd3685521f)
![{\displaystyle u(x,t)=-(1/3)*(-_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})/_{C}3-2*_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=(2/3)*_{C}3*(2*_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*_{C}8/_{C}9^{2}-(2/3)*_{C}3*(2*_{C}4-2*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)/_{C}9,w(x,t)=_{C}8+_{C}9*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e51779d3999e50ca81ad23ea6f481478061a1590)
![{\displaystyle u(x,t)=-(1/3)*(-_{C}4+_{C}3^{3}*_{C}1^{2}+_{C}3^{3})/_{C}3+2*_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=-(2/3)*_{C}1^{2}*_{C}3*(_{C}3^{3}*_{C}1^{2}+_{C}3^{3}+2*_{C}4)*_{C}10/_{C}11^{2}+(2/3)*_{C}3*_{C}1^{2}*(_{C}3^{3}*_{C}1^{2}+_{C}3^{3}+2*_{C}4)*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)/_{C}11,w(x,t)=_{C}10+_{C}11*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb4fd44a43d45f21a7197c81c931c7e84c813ac)
![{\displaystyle u(x,t)=(1/3)*(-2*_{C}3^{3}+4*_{C}3^{3}*_{C}1^{2}-_{C}4)/_{C}3+(2*_{C}3^{2}-2*_{C}3^{2}*_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=_{C}6,w(x,t)=_{C}10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5cc8457239b9cb42c007e74a5da99512e02d2a8)
![{\displaystyle u(x,t)=-(1/2)*_{C}3^{2}+_{C}3^{2}*_{C}1^{2}-2*_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+(-(1/2)*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1)^{2},v(x,t)=_{C}6+_{C}7*JacobiCN(_{C}2+_{C}3*x+(-(1/2)*_{C}3^{3}+_{C}3^{3}*_{C}1^{2})*t,_{C}1),w(x,t)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177914e0c3ec1ca7dc052d581e6e8e7508ae877d)
![{\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[编辑]
Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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Hirota Satsuma equations traveling wave plot
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参考文献[编辑]
- ^ 李志斌编著 《非线性数学物理方程的行波解》 140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