耦合KdV方程组

耦合KdV方程组(Coupled KdV equation)是一个二元非线性偏微分方程组：^[1]

${\displaystyle u_{t}+6*\alpha *u*u_{x}-6*v*v_{x}+\alpha *u_{xxx}=0}$

${\displaystyle v_{t}+3*\alpha *u*v_{x}+\alpha *v_{xxx}=0}$

解析解

${\displaystyle u(x,t)=(1/3)*(-_{C}3+\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*csc(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}2^{4}+6*_{C}2*_{C}3)*csc(_{C}1+_{C}2*x+_{C}3*t)}$

${\displaystyle u(x,t)=(1/3)*(-_{C}3+\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*sec(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}2^{4}+6*_{C}2*_{C}3)*sec(_{C}1+_{C}2*x+_{C}3*t)}$

${\displaystyle u(x,t)=(1/3)*(-_{C}3+2*\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}6*_{C}2*_{C}3+24*\alpha *_{C}2^{4})*coth(_{C}1+_{C}2*x+_{C}3*t)}$

${\displaystyle u(x,t)=(1/3)*(-_{C}3+2*\alpha *_{C}2^{3})/(alpha*_{C}2)-2*_{C}2^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}6*_{C}2*_{C}3+24*alpha*_{C}2^{4})*tanh(_{C}1+_{C}2*x+_{C}3*t)}$

${\displaystyle u(x,t)=-(1/3)*(_{C}3+\alpha *_{C}2^{3})/(\alpha *_{C}2)-2*_{C}2^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/3)*{\sqrt {(}}6*_{C}2*_{C}3-12*alpha*_{C}2^{4})*csch(_{C}1+_{C}2*x+_{C}3*t)}$

${\displaystyle u(x,t)=-(1/3)*(_{C}3+8*\alpha *_{C}2^{3})/(\alpha *_{C}2)-4*_{C}2^{2}*tan(_{C}1+_{C}2*x+_{C}3*t)^{2},v(x,t)=-(1/6)*(_{C}3+8*\alpha *_{C}2^{3})*{\sqrt {(}}2)/(_{C}2*{\sqrt {(}}\alpha ))-2*{\sqrt {(}}2)*{\sqrt {(}}\alpha )*_{C}2^{2}*tan(_{C}1+_{C}2*x+_{C}3*t)^{2}}$

${\displaystyle u(x,t)=(1/3)*(-_{C}4+\alpha *_{C}3^{3}+\alpha *_{C}3^{3}*_{C}1^{2})/(\alpha *_{C}3)-2*_{C}3^{2}*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}3^{4}*_{C}1^{2}+12*\alpha *_{C}3^{4}+6*_{C}3*_{C}4)*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)}$

${\displaystyle u(x,t)=(1/3)*(-_{C}4+\alpha *_{C}3^{3}+\alpha *_{C}3^{3}*_{C}1^{2})/(\alpha *_{C}3)-2*_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2},v(x,t)=-(1/3)*{\sqrt {(}}12*\alpha *_{C}3^{4}*_{C}1^{2}+12*\alpha *_{C}3^{4}+6*_{C}3*_{C}4)*_{C}1*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)}$

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行波图

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

Coupled KdV equation traveling wave plot

参考文献

1. 李志斌编著 《非线性数学物理方程的行波解》 38页 科学出版社 2008

1. *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
2. *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
3. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
4. 王东明著 《消去法及其应用》 科学出版社 2002
5. *何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445
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7. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
8. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
9. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
10. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
11. Dongming Wang, Elimination Practice,Imperial College Press 2004
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