摘要The soliton calculation method put forward by Zabusky and Kruskal has played an important role in the development of soliton theory, however numerous numerical results show that even though the parameters satisfy the linear stability condition, nonlinear instability will also occur. We notice an exception in the numerical calculation of soliton, gain the linear stability condition of the second order Leap-frog scheme constructed by Zabusky and Kruskal, and then draw the perturbed equation with the finite difference method. Also, we solve the symmetry group of the KdV equation with the knowledge of the invariance of Lie symmetry group and then discuss whether the perturbed equation and the conservation law keep the corresponding symmetry. The conservation law of KdV equation satisfies the scaling transformation, while the perturbed equation does not satisfy the Galilean invariance condition and the scaling invariance condition. It is demonstrated that the numerical simulation destroy some physical characteristics of the original KdV equation. The nonlinear instability in the calculation of solitons is related to the breaking of symmetry
Abstract：The soliton calculation method put forward by Zabusky and Kruskal has played an important role in the development of soliton theory, however numerous numerical results show that even though the parameters satisfy the linear stability condition, nonlinear instability will also occur. We notice an exception in the numerical calculation of soliton, gain the linear stability condition of the second order Leap-frog scheme constructed by Zabusky and Kruskal, and then draw the perturbed equation with the finite difference method. Also, we solve the symmetry group of the KdV equation with the knowledge of the invariance of Lie symmetry group and then discuss whether the perturbed equation and the conservation law keep the corresponding symmetry. The conservation law of KdV equation satisfies the scaling transformation, while the perturbed equation does not satisfy the Galilean invariance condition and the scaling invariance condition. It is demonstrated that the numerical simulation destroy some physical characteristics of the original KdV equation. The nonlinear instability in the calculation of solitons is related to the breaking of symmetry
[1] Russell J S 1844 Fourteenth Meeting of the BritishAssociation for the Advancement of Science 26 September--2 October 1844(London: John Murray) p 311 [2] Ablowitz M J and Segur H 1981 Solitons and theInverse Scattering Transformations (Philadelphia: SIAM) [3] Gu C H et al 1990 Theory and Application of Solitons(Hangzhou: Zhejiang Science and Technology Press) (in Chinese) [4] Fermi A, Pasta J and Ulam S 1940 Los Alamos Sci.Lab. Report Fermi A, Pasta J and Ulam S 1955 LosAlamos National Laboratory [5] Perring J K, Skyrime and T H R 1962 Nucl. Phys. 31 550 [6] Zabusky N J and Kruskal M D 1965 Phys. Rev. Lett. 15 240 [7] Guo B Y 1990 Theory and Application of Solitons (inChinese) (Hangzhou: Zhejiang Science and Technology Press) [8] Aoyagi A 1989 J. Comput. Phys. 83 447 [9] Herman R L 1993 J. Comput. Phys. 104 50 [10] Olver P J 1991 Applications of Lie Groups toDifferential Equations 2nd edn (New York: Springer) [11] Ran Z 2005 Chin. Q. Mech. 26 650 (in Chinese) [12] Ran Z 2007 SIAM J. Numer. Anal. 46 325 [13] Ran Z 2007 Chin. Phys. Lett. 24 3332 Ran Z 2008 Chin. Phys. Lett. 25 3867 [14] Chen J B et al 2002 SIAM J. Numer. Anal. 392164 [15] Chen J B et al 2008 Chin. Phys. Lett. 4 1168 [16] Ran Z 2008 Lie Symmetries Preservation and theSoliton Calculation (Report of SIAM).
2009, Vol. 26 
