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Numerical homoclinic instabilities in the sine-Gordon equation. (English) Zbl 0785.65086
This paper is concerned with the instabilities which arise when discrete numerical methods are applied to the integration of initial value problems where the initial conditions are close to a homoclinic orbit.
In the first part of the paper the well known pendulum equation is considered. For this planar Hamiltonian system, the authors recall some results on the integrability and the chaos induced by symplectic methods around the homoclinic orbit. In particular, they present the numerical results obtained by means of an explicit symplectic method which confirm numerically the exponential stepsize dependence of the width of the stochastic layer surrounding the homoclinic orbit.
In the second part of this paper the authors proceed to extend, to higher dimensional problems, aspects of the above study by considering the sine- Gordon equation. Three different spatial discretizations are considered: Finite differences, a pseudospectral method and a special integrable discretization in the sense of Liouville. Although all of them retain the Hamiltonian character of the original problem, the numerical experiments carried out by the authors show that the numerical results for the three discretizations are completely different. Thus, for the finite difference discretization several numerical methods (symplectic and non symplectic) are not able to represent the qualitative behavior of the exact solutions. However, in the case of the pseudospectral discretization the low-order symplectic methods considered by the authors give good qualitative results and of course better than the ones for non-symplectic methods. In conclusion, it is shown that spatial finite difference discretizations of the sine-Gordon equation are not adequate to preserve the homoclinic structures and in any case symplectic methods should be applied in the vicinity of these orbits.
Reviewer: M.Calvo (Zaragoza)

65L07 Numerical investigation of stability of solutions to ordinary differential equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
35Q53 KdV equations (Korteweg-de Vries equations)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI
[1] Sanz-Serna J. M., integrators for Hamiltonian problems: An overview (1991) · Zbl 0762.65043
[2] Kang Feng, J. Comput. Math. 4 pp 279– (1986)
[3] Kang Feng, Lecture Notes in Mathematics # 1297, in: In Numerical methods for Partial Differential Equations (1987)
[4] DOI: 10.1007/BF01954907 · Zbl 0655.70013
[5] DOI: 10.1088/0951-7715/3/2/001 · Zbl 0704.65052
[6] Forest E., D 43 pp 105– (1990)
[7] Yoshida H., Phys. A. 150 pp 262– (1990)
[8] Yoshida H., Conserved quantities of symplectic integrators for Hamiltonian systems (1990)
[9] DOI: 10.1007/BF00048986 · Zbl 0724.70019
[10] DOI: 10.1063/1.165823 · Zbl 0899.58016
[11] Lazutkin V. F., Math. Dokl. 42 pp 5– (1991)
[12] Herbst B. M., J. Comput. Phys. (1992)
[13] Herbst B. M., Numerical evidence of exponentially small splitting distances in symplectic discretizations of planar Hamiltonian systems (1991) · Zbl 0941.37504
[14] Ablowitz M. J., Solitons and the Inverse Scattering Transform (1981) · Zbl 0472.35002
[15] DOI: 10.1017/CBO9780511623998
[16] DOI: 10.1016/0167-2789(90)90142-C · Zbl 0705.58026
[17] Ablowitz M. J., ’Hamiltonian Systems, Transformation Groups and Spectral Transform Methods’ (1990)
[18] DOI: 10.1137/0150021 · Zbl 0707.35141
[19] Faddeev L. D., methods in the theory of solitons (1987) · Zbl 0632.58004
[20] Arnold V. I., Mathematical methods of classical mechanics (1978) · Zbl 0386.70001
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