迪姆方程

迪姆方程是以色列数学家哈利·迪姆创建的三阶非线性偏微分方程:

${\displaystyle u_{t}=u^{3}u_{xxx}.\,}$

Bäcklund变换解

Dym eq Backlund transform solution animation

通过Bäcklund变换可得迪姆方程的分析解^[1]。

${\displaystyle u(t,x)=(-3\alpha (x+4\alpha ^{2}t)^{2/3}.}$

近似解

Harry Dym nlpde 3d animation

迪姆方程有解析解^[2]。

${\displaystyle u(x,t)=\tanh({\frac {{\sqrt {(}}V)}{2}}*(x-x[0]+V*t+ef(x,t)))^{2}}$

其中

${\displaystyle ef(x,t)={\frac {2}{{\sqrt {(}}V)}}*tanh(({\frac {{\sqrt {(}}V)}{2}}*(x-x[0]+V*t+ef(x,t)))}$

五次迭代后可得近似解：

${\displaystyle tanh((1/2)*x*{\sqrt {(}}V)-(1/2)*x[0]*{\sqrt {(}}V)+(1/2)*V^{(}3/2)*t+tanh((1/2)*x*{\sqrt {(}}V)-(1/2)*x[0]*{\sqrt {(}}V)+(1/2)*V^{(}3/2)*t+tanh((1/2)*x*{\sqrt {(}}V)-(1/2)*x[0]*{\sqrt {(}}V)+(1/2)*V^{(}3/2)*t+tanh((1/2)*x*{\sqrt {(}}V)-(1/2)*x[0]*{\sqrt {(}}V)+(1/2)*V^{(}3/2)*t+tanh((1/2)*{\sqrt {(}}V)*(x-x[0]+V*t))))))}$

阿多米安近似法

用阿多米安分解法可求得迪姆方程的柯西问题近似解^[3]。

初始条件：u（0）=cos(x)

pa := (.53823*sin(10.*x)+0.16931e-1*sin(4.*x)+.72240*sin(8.*x)+.24408*sin(6.*x)+0.57870e-4*sin(2.*x))*t^9+(0.59326e-2*cos(3.*x)+0.13563e-5*cos(x)+.55850*cos(7.*x)+.46338*cos(9.*x)+.15138*cos(5.*x))*t^8+(-0.13889e-2*sin(2.*x)-.43393*sin(6.*x)-0.88889e-1*sin(4.*x)-.40635*sin(8.*x))*t^7+(-.33908*cos(5.*x)-0.47461e-1*cos(3.*x)-0.10851e-3*cos(x)-.36474*cos(7.*x))*t^6+(.26667*sin(4.*x)+0.20833e-1*sin(2.*x)+.33750*sin(6.*x))*t^5+(.21094*cos(3.*x)+.32552*cos(5.*x)+0.52083e-2*cos(x))*t^4+(-.16667*sin(2.*x)-.33333*sin(4.*x))*t^3+(-.12500*cos(x)-.37500*cos(3.*x))*t^2+.50000*t*sin(2.*x)+cos(x)

初始条件 u(0)=cosh(x)

pa := (-.5382*sinh(10.*x)-.7224*sinh(8.*x)-.2441*sinh(6.*x)-0.5787e-4*sinh(2.*x)-0.1693e-1*sinh(4.*x))*t^9+(.4634*cosh(9.*x)+0.5933e-2*cosh(3.*x)+.5585*cosh(7.*x)+.1514*cosh(5.*x)+0.1356e-5*cosh(x))*t^8+(-.4063*sinh(8.*x)-0.8889e-1*sinh(4.*x)-.4339*sinh(6.*x)-0.1389e-2*sinh(2.*x))*t^7+(0.1085e-3*cosh(x)+0.4746e-1*cosh(3.*x)+.3647*cosh(7.*x)+.3391*cosh(5.*x))*t^6+(-0.2083e-1*sinh(2.*x)-.2667*sinh(4.*x)-.3375*sinh(6.*x))*t^5+(.3255*cosh(5.*x)+0.5208e-2*cosh(x)+.2109*cosh(3.*x))*t^4+(-.3333*sinh(4.*x)-.1667*sinh(2.*x))*t^3+(.3750*cosh(3.*x)+.1250*cosh(x))*t^2-.5000*t*sinh(2.*x)+cosh(x)

参考文献

1. ^ Fritz Gesztesy and Karl Unterkofler, Isospectral deformations for Sturm–Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys. 31 (1992), 113–137.
2. W HeremanP P, BanerjeeSS and M R Chatterjee, Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg-de Vries equation;J. Phys. A: Math. Gen. 22 (1989) 241-255.
3. Inna Shingareve Carlos Lizarraga Celaya，Solving Nonlinear Partial Differential Equations with Maple and Mathematica p230-236, Springer

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
6. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
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
12. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
13. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759