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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.

MSC:

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)
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[1] Ramos, J.I., Linearization methods for reaction – diffusion equations: 1-D problems, Appl. math. comput., 88, 199-224, (1997) · Zbl 0904.65088
[2] Ramos, J.I., Linearization methods for reaction – diffusion equations: multidimensional problems, Appl. math. comput., 88, 225-254, (1997) · Zbl 0904.65089
[3] Ramos, J.I., Implicit, compact, linearized θ-methods with factorization for multidimensional reaction – diffusion equations, Appl. math. comput., 94, 17-43, (1999) · Zbl 0943.65098
[4] Kivshar, Y.S.; Malomed, B.A., Dynamics of solitons in nearly integrable systems, Rev. modern phys., 61, 763-915, (1989)
[5] Kivshar, Y.S.; Malomed, B.A., Many-particle effects in nearly integrable systems, Physica D, 24, 125-154, (1987) · Zbl 0634.35076
[6] Bishop, A.R.; Krumhansl, J.A.; Trullinger, S.E., Solitons in condensed matter physics: a paradigm, Physica D, 1, 1-44, (1980)
[7] Ustinov, A.V., Solitons in Josephson junctions, Physica D, 123, 315-329, (1998)
[8] Christiansen, P.L.; Savin, A.V.; Zolotaryuk, A.V., Soliton analysis in complex molecular systems: a zig-zag chain, J .comput. phys., 134, 108-121, (1997) · Zbl 0884.65126
[9] Gorshkov, A.S.; Lyakhov, G.A.; Voliak, K.I.; Yarovoi, L.A., Parametric generation in anomalously dispersive media, Physica D, 122, 161-177, (1997)
[10] Argyris, J.; Haase, M., An Engineer’s guide to soliton phenomena: application of the finite element method, Comput. meth. appl. mech. engrg., 61, 71-122, (1987) · Zbl 0624.76020
[11] Tourigny, Y., Product approximation for nonlinear klein – gordon equations, IMA J. numer. anal., 9, 449-462, (1990) · Zbl 0707.65088
[12] Guo, B.-Y.; Li, X.; Vázquez, L., A Legendre spectral method for solving the nonlinear klein – gordon equation, Comp. appl. math., 15, 19-36, (1996) · Zbl 0856.65117
[13] Jiménez, S.; Vázquez, L., Analysis of four numerical schemes for a nonlinear klein – gordon equation, Appl. math. comput., 35, 61-94, (1990) · Zbl 0697.65090
[14] Strauss, W.A.; Vázquez, L., Numerical solution of a nonlinear klein – gordon equation, J. comput. phys., 28, 271-278, (1978) · Zbl 0387.65076
[15] Ablowitz, M.J.; Kruskal, M.D.; Ladik, J.F., Solitary wave collisions, SIAM J. appl. math., 36, 428-437, (1979) · Zbl 0408.65075
[16] Guo, B.-Y.; Pascual, P.J.; Rodrı́guez, M.J.; Vázquez, L., Numerical solution of the sine-Gordon equation, Appl. math. comput., 18, 1-14, (1986) · Zbl 0622.65131
[17] Fei, Z.; Vázquez, L., Two energy conserving numerical schemes for the sine-Gordon equation, Appl. math. comput., 45, 17-30, (1991) · Zbl 0732.65107
[18] Li, S.; Vu-Quoc, L., Finite difference calculus invariant structure of a class of algorithms for the nonlinear klein – gordon equation, SIAM J. numer. anal., 32, 1839-1875, (1995) · Zbl 0847.65062
[19] Wong, Y.S.; Chang, Q.; Gong, L., An initial-boundary value problem of a nonlinear klien – gordon equation, Appl. math. comput., 84, 77-93, (1997)
[20] Jiménez, S., Derivation of the discrete conservation laws for a family of finite difference schemes, Appl. math. comput., 64, 13-45, (1994) · Zbl 0806.65081
[21] Ablowitz, M.J.; Herbst, B.M.; Schober, C.M., Numerical simulation of quasi-periodic solutions of the sine-Gordon equation, Physica D, 87, 37-47, (1995) · Zbl 1194.65121
[22] Ablowitz, M.J.; Herbst, B.M.; Schober, C.M., On the numerical solution of the sine-Gordon equation: I. integrable discretizations and homoclinic manifolds, J. comput. phys., 126, 299-314, (1996) · Zbl 0866.65064
[23] Ablowitz, M.J.; Herbst, B.M.; Schober, C.M., On the numerical solution of the sine-Gordon equation: II. performance of numerical schemes, J. comput. phys., 131, 354-367, (1997) · Zbl 0874.65076
[24] Kouri, D.J.; Zhang, D.S.; Wei, G.W.; Konshak, T.; Hoffman, D.K., Numerical solution of nonlinear wave equations, Phys. rev. E, 59, 1274-1277, (1999)
[25] DeLeonardis, R.M.; Trullinger, S.E.; Wallis, R.F., Theory of boundary effects on sine-Gordon solitons, J. appl. phys., 51, 1211-1226, (1980)
[26] Cao, W.-M.; Guo, B.-Y., Fourier collocation method for solving nonlinear klein – gordon equation, J. comput. phys., 108, 296-305, (1993) · Zbl 0791.65095
[27] Hanna, S.N., Stability of periodic plane wave solutions of wave equations, J. comput. appl. math., 34, 41-46, (1991) · Zbl 0725.35091
[28] Adomian, G., Non-perturbative solution of the klein – gordon – zakharov equation, Appl. math. comput., 81, 89-92, (1997) · Zbl 0869.65063
[29] Deera, E.Y.; Khuri, S.A., A decomposition method for solving the nonlinear klein – gordon equation, J. comput. phys., 124, 442-448, (1996) · Zbl 0849.65073
[30] Ferguson, C.D.; Willis, C.R., One- and two-collective variable descriptions of two interacting sine-Gordon kinks, Physica D, 119, 283-300, (1998)
[31] Zhang, F., Breather scattering by impurities in the sine-Gordon model, Phys. rev. E, 58, 2558-2563, (1998)
[32] Sánchez, A.; Bishop, A.R., Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations, SIAM rev., 40, 579-615, (1998) · Zbl 0926.35135
[33] Friedland, L., Autoresonance of coupled nonlinear waves, Phys. rev. E, 57, 3494-3501, (1998)
[34] Bass, F.G.; Kivshar, Y.S.; Knotop, V.V.; Sinitsyn, Y.A., Dynamics of solitons under random perturbations, Phys. reports, 157, 63-181, (1988)
[35] Abdullaev, F.Kh., Dynamical chaos of solitons and nonlinear periodic waves, Phys. reports, 179, 1-78, (1989)
[36] Ghidaglia, J.M.; Marzocchi, A., Longtime behaviour of strongly damped wave equations: global attractors and their dimension, SIAM J. math. anal., 22, 879-895, (1991) · Zbl 0735.35015
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