The sine-Gordon equation in the finite line. (English) Zbl 1026.65071

From the text: A rather cornplets numerical study of the perturbed and nonperturbed sine-Gordon equation (sGE) in one-dimensional Cartesian coordinates and finite domains is presented. The numerical study is based on five, three-point, linearly implicit finite difference methods of different spatial accuracy. For the unperturbed sGE, the accuracy of these five methods is found to be very sensitive to the implicitness parameter, less sensitive to the spatial order of accuracy, and almost no sensitive to the time step for second-order accurate, temporal discretizations. The largest errors of y the five methods were found to occur at the front of the soliton solution of the unperturbed sGE, but the techniques were very accurate for long-time computations of the unperturbed sGE in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries, despite the fact that they do not preserve a discrete energy.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI


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