The solution of the two-dimensional sine-Gordon equation using the method of lines. (English) Zbl 1117.65126
Summary: The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor-corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Keywords:
soliton; sine-Gordon equation; finite-difference method; method of lines; numerical examples; error bounds; stability; convergence; predictor-corrector schemeReferences:
| [1] | M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser. (1991) 149.; M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser. (1991) 149. · Zbl 0762.35001 |
| [2] | Argyris, J.; Haase, M.; Heinrich, J. C., Finite element approximation to two-dimensional sine-Gordon solitons, Comput. Methods Appl. Mech. Eng., 86, 1-26 (1991) · Zbl 0762.65073 |
| [3] | Benabdallah, A.; Caputo, J. G.; Scott, A. C., Exponentially tapered Josephson flux-flow oscillator, Phys. Rev. B, 54, 22, 16139-16146 (1996) |
| [4] | Bogolyubskii˘, I. L., Oscillating particle-like solutions of the nonlinear Klein-Gordon equation, JETP Lett., 24, 10, 535-538 (1976) |
| [5] | Bogolyubskii˘, I. L., Dynamics of spherically symmetrical pulsons of large amplitude, JETP Lett., 25, 2, 107-110 (1977) |
| [6] | Bogolyubskii˘, I. L.; Makhankov, V. G., Lifetime of pulsating solitons in certain classical models, JETP Lett., 24, 1, 12-14 (1976) |
| [7] | Bratsos, A. G., An explicit numerical scheme for the sine-Gordon equation in \(2 + 1\) dimensions, Appl. Numer. Anal. Comput. Math., 2, 2, 189-211 (2005) · Zbl 1075.65111 |
| [8] | Christiansen, P. L.; Grønbech-Jensen, N.; Lomdahl, P. S.; Malomed, B. A., Oscillations of eccentric pulsons, Phys. Scripta, 55, 131-134 (1997) |
| [9] | Christiansen, P. L.; Lomdahl, P. S., Numerical solutions of \(2 + 1\) dimensional sine-Gordon solitons, Physica, 2D, 482-494 (1981) · Zbl 1194.65122 |
| [10] | Christiansen, P. L.; Olsen, O. H., Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation, Phys. Lett., 68A, 2, 185-188 (1978) |
| [11] | Christiansen, P. L.; Olsen, O. H., On dynamical two-dimensional solutions to the sine-Gordon equation, Z. Angew. Math. Mech., 59, T30 (1979) · Zbl 0409.35069 |
| [12] | Christiansen, P. L.; Olsen, O. H., Ring-shaped quasi-soliton solutions to the two and three-dimensional sine-Gordon equations, Phys. Scripta, 20, 531-538 (1979) · Zbl 1063.81529 |
| [13] | Djidjeli, K.; Price, W. G.; Twizell, E. H., Numerical solutions of a damped sine-Gordon equation in two space variables, J. Eng. Math., 29, 347-369 (1995) · Zbl 0841.65083 |
| [14] | Gaididei, Yu. B.; Mingaleev, S. F.; Christiansen, P. L., Curvature-induced symmetry breaking in nonlinear Schrödinger models, Phys. Rev. E, 62, 1, R53-R56 (2000) · Zbl 1031.35133 |
| [15] | Geicke, J., On the reflection of radially symmetrical sine-Gordon kinks in the origin, Phys. Lett., 98A, 4, 147-150 (1983) |
| [16] | Geicke, J., Cylindrical pulsons in nonlinear relativistic wave equations, Phys. Scripta, 29, 431-434 (1984) |
| [17] | Gibbon, J. D.; Zambotti, G., The interaction of \(n\)-dimensional soliton wave fronts, Nuovo Cimento, 25B, 1 (1976) |
| [18] | C. Gorria, Yu.B. Gaididei, M.P. Soerensen, P.L. Christiansen, J.G. Caputo, Kink propagation and trapping in a two-dimensional curved Josephson junction, Phys. Rev. B 69 (2004) 134506 (1-10).; C. Gorria, Yu.B. Gaididei, M.P. Soerensen, P.L. Christiansen, J.G. Caputo, Kink propagation and trapping in a two-dimensional curved Josephson junction, Phys. Rev. B 69 (2004) 134506 (1-10). |
| [19] | Grella, G.; Marinaro, M., Special solution of the sine-Gordon equation in \(2 + 1\) dimensions, Lett. Nuovo Cimento, 23, 459 (1978) |
| [20] | Guo, B.-Y.; Pascual, P. J.; Rodriguez, M. J.; Vázquez, L., Numerical solution of the sine-Gordon equation, Appl. Math. Comput., 18, 1-14 (1986) · Zbl 0622.65131 |
| [21] | Harayama, T.; Davis, P.; Ikeda, K. S., Nonlinear whispering gallery modes, Phys. Rev. Lett., 82, 19, 3803-3806 (1999) |
| [22] | Hirota, R., Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. Phys. Soc. Japan, 35, 1566 (1973) |
| [23] | Kaliappan, P.; Lakshmanan, M., Kadomtsev-Petviashvili and two-dimensional sine-Gordon equations: reduction to Painlevè transcendents, J. Phys. A, 12, L249 (1979) · Zbl 0425.35080 |
| [24] | Kobayashi, K. K.; Izutsu, M., Exact solution of the \(n\)-dimensional sine-Gordon equation, J. Phys. Soc. Japan, 41, 1091 (1976) · Zbl 1334.35284 |
| [25] | Kodama, Y.; Ablowitz, M. J., Transverse instability of breathers in resonant media, J. Math. Phys., 21, 4, 928-931 (1980) |
| [26] | Lamb, G. L., Elements of Soliton Theory (1980), Wiley: Wiley New York · Zbl 0445.35001 |
| [27] | Leanhardt, A. E.; Chikkatur, A. P.; Kielpinski, D.; Shin, Y.; Gustavson, T. L.; Ketterle, W.; Pritchrad, D. E., Propagation of Bose-Einstein condensates in a magnetic waveguide, Phys. Rev. Lett., 89, 4, 1-4/040401 (2002) |
| [28] | Leibbrandt, G., New exact solutions of the classical sine-Gordon equation in \(2 + 1\) and \(3 + 1\) dimensions, Phys. Rev. Lett., 41, 435-438 (1978) |
| [29] | Malomed, B. A., Decay of shrinking solitons in multidimensional sine-Gordon equation, Phys. D, 24, 155-171 (1987) · Zbl 0634.35077 |
| [30] | Malomed, B. A., Dynamic of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model, Phys. D, 52, 157-170 (1991) · Zbl 0742.35060 |
| [31] | Maslov, E. M., Dynamics of rotationally symmetric solitons in near-SG field model with applications to large-area Josephson junctions and ferromagnets, Phys. D, 15, 433-443 (1985) · Zbl 0583.35088 |
| [32] | Maslov, E. M., Rotationally symmetric sG oscillator with tunable frequency, Phys. Lett. A, 131, 6, 364-367 (1988) |
| [33] | Mingaleev, S. F.; Kivshar, Y. S.; Sammut, R. A., Long-range interaction and nonlinear localized modes in photonic crystal waveguides, Phys. Rev. E, 62, 5777-5782 (2000) |
| [34] | Nakajima, K.; Onodera, Y.; Nakamura, T.; Sato, R., Numerical analysis of vortex motion on Josephson structures, J. Appl. Phys., 45, 9, 4095-4099 (1974) |
| [35] | Sheng, Q.; Khaliq, A. Q.M.; Voss, D. A., Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. Comput. Simulation, 68, 355-373 (2005) · Zbl 1073.65095 |
| [36] | Wallraff, A.; Ustinov, A. V.; Kurin, V. V.; Shereshevsky, I. A.; Vdovicheva, N. K., Whispering vortices, Phys. Rev. Lett., 84, 1, 151-154 (2000) |
| [37] | Xin, J. X., Modeling light bullets with the two-dimensional sine-Gordon equation, Phys. D, 135, 345-368 (2000) · Zbl 0936.78006 |
| [38] | Zagrodzinsky, J., Particular solutions of the sine-Gordon equation in \(2 + 1\) dimensions, Phys. Lett., 72A, 284-286 (1979) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
