The sine-Gordon equation in the finite line

A rather complete numerical study of the perturbed and nonperturbed sine-Go rdon equation in one-dimensional Cartesian coordinates and finite domains i s presented. The numerical study is based on five, three-point, linearly im plicit finite difference methods of different spatial accuracy. For the unp erturbed sGE, the accuracy of these five methods is found to be very sensit ive to the implicitness parameter, less sensitive to the spatial order of a ccuracy, and almost no sensitive to the time step for second-order accurate , temporal discretizations. The largest errors of the five methods were fou nd to occur at the front of the soliton solution of the unperturbed sGE, bu t the techniques were very accurate for long-time computations of the unper turbed sGE in both infinite domains and in finite domains upon many collisi ons of the kinks and antikinks with the boundaries, despite the fact that t hey do not preserve a discrete energy. It is shown that the largest values of the kinetic and total energies, and the smallest values of the strain an d potential energies occur when solitons and antisolitons strike on the bou ndaries subject to homogeneous Neumann conditions, and the four energies re main constant between collisions with the boundaries. For the Malomed's per turbation, one of the fronts of the soliton-soliton doublet and soliton-ant isoliton doublet is trapped at one boundary and the other one undergoes a c hange in amplitude and its front describes a curved trajectory; the soliton -antisoliton breather oscillates in a damped manner and slowly drifts towar ds one boundary. The soliton, antisoliton, soliton-soliton doublet and soli ton-antisoliton doublet of the unperturbed sGE are robust under Kivshar and Malomed's perturbations, whereas the soliton antisoliton breather loses it s integrity, becomes a kink antikink pair and then yields a chaotic behavio ur in both space and time. Small amplifications proportional to the first-o rder temporal derivative of the amplitude do not alter substantially the in itial dynamics of the soliton, antisoliton, soliton-soliton doublet and sol iton-antisoliton doublet, but their amplitudes exhibit oscillatory behaviou r of increasing amplitude at large times; however, small amplifications cau se first a transition from the soliton-antisoliton breather to the kink-ant ikink solution. A linearized stability analysis of the spatially homogeneou s solutions of the unperturbed sGE is performed and analytical solutions ar e obtained by means of the method of separation of variables for both homog eneous Dirichlet and Neumann boundary conditions, and the results show that the solution corresponding to a zero amplitude is oscillatory in both spac e and time, whereas that corresponding to an amplitude equal to pi may exhi bit exponential or linear growth depending on the length of the domain. If either of these growths is present, it is shown that the evolution of small , initial perturbations may result in homoclinic crossings and chaotic beha viour in both space and time in the absence of perturbations in the sGE and the existence of an attractor when perturbations of the Malomed's type are present. In either case, the solution may exhibit certain spatial symmetri es or anti-symmetries at large times. Finally, the results presented in thi s paper indicate that the accurate computation of homoclinic orbits of the sGE is not only controlled by the spatial accuracy of the numerical method, but also by its energy-preserving characteristics, and, perhaps, by the nu mber of grid points used in the numerical technique. (C) 2001 Elsevier Scie nce Inc. All rights reserved.