The solution of the two-dimensional sine-Gordon equation using the method of lines

M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Math. Soc. Lecture Note Ser. (1991) 149. Argyris, 1991, Finite element approximation to two-dimensional sine-Gordon solitons, Comput. Methods Appl. Mech. Eng., 86, 1, 10.1016/0045-7825(91)90136-T Benabdallah, 1996, Exponentially tapered Josephson flux-flow oscillator, Phys. Rev. B, 54, 16139, 10.1103/PhysRevB.54.16139 Bogolyubskii˘, 1976, Oscillating particle-like solutions of the nonlinear Klein–Gordon equation, JETP Lett., 24, 535 Bogolyubskii˘, 1977, Dynamics of spherically symmetrical pulsons of large amplitude, JETP Lett., 25, 107 Bogolyubskii˘, 1976, Lifetime of pulsating solitons in certain classical models, JETP Lett., 24, 12 Bratsos, 2005, An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions, Appl. Numer. Anal. Comput. Math., 2, 189, 10.1002/anac.200410035 Christiansen, 1997, Oscillations of eccentric pulsons, Phys. Scripta, 55, 131, 10.1088/0031-8949/55/2/002 Christiansen, 1981, Numerical solutions of 2+1 dimensional sine-Gordon solitons, Physica, 2D, 482 Christiansen, 1978, Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation, Phys. Lett., 68A, 185, 10.1016/0375-9601(78)90798-3 Christiansen, 1979, On dynamical two-dimensional solutions to the sine-Gordon equation, Z. Angew. Math. Mech., 59, T30 Christiansen, 1979, Ring-shaped quasi-soliton solutions to the two and three-dimensional sine-Gordon equations, Phys. Scripta, 20, 531, 10.1088/0031-8949/20/3-4/032 Djidjeli, 1995, Numerical solutions of a damped sine-Gordon equation in two space variables, J. Eng. Math., 29, 347, 10.1007/BF00042761 Gaididei, 2000, Curvature-induced symmetry breaking in nonlinear Schrödinger models, Phys. Rev. E, 62, R53, 10.1103/PhysRevE.62.R53 Geicke, 1983, On the reflection of radially symmetrical sine-Gordon kinks in the origin, Phys. Lett., 98A, 147, 10.1016/0375-9601(83)90570-4 Geicke, 1984, Cylindrical pulsons in nonlinear relativistic wave equations, Phys. Scripta, 29, 431, 10.1088/0031-8949/29/5/003 Gibbon, 1976, The interaction of n-dimensional soliton wave fronts, Nuovo Cimento, 25B, 1 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). Grella, 1978, Special solution of the sine-Gordon equation in 2+1 dimensions, Lett. Nuovo Cimento, 23, 459, 10.1007/BF02770537 Guo, 1986, Numerical solution of the sine-Gordon equation, Appl. Math. Comput., 18, 1, 10.1016/0096-3003(86)90025-1 Harayama, 1999, Nonlinear whispering gallery modes, Phys. Rev. Lett., 82, 3803, 10.1103/PhysRevLett.82.3803 Hirota, 1973, Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. Phys. Soc. Japan, 35, 1566, 10.1143/JPSJ.35.1566 Kaliappan, 1979, Kadomtsev–Petviashvili and two-dimensional sine-Gordon equations: reduction to Painlevè transcendents, J. Phys. A, 12, L249, 10.1088/0305-4470/12/10/002 Kobayashi, 1976, Exact solution of the n-dimensional sine-Gordon equation, J. Phys. Soc. Japan, 41, 1091, 10.1143/JPSJ.41.1091 Kodama, 1980, Transverse instability of breathers in resonant media, J. Math. Phys., 21, 928, 10.1063/1.524480 Lamb, 1980 Leanhardt, 2002, Propagation of Bose–Einstein condensates in a magnetic waveguide, Phys. Rev. Lett., 89, 1, 10.1103/PhysRevLett.89.040401 Leibbrandt, 1978, New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions, Phys. Rev. Lett., 41, 435, 10.1103/PhysRevLett.41.435 Malomed, 1987, Decay of shrinking solitons in multidimensional sine-Gordon equation, Phys. D, 24, 155, 10.1016/0167-2789(87)90071-6 Malomed, 1991, Dynamic of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model, Phys. D, 52, 157, 10.1016/0167-2789(91)90118-S Maslov, 1985, Dynamics of rotationally symmetric solitons in near-SG field model with applications to large-area Josephson junctions and ferromagnets, Phys. D, 15, 433, 10.1016/S0167-2789(85)80009-9 Maslov, 1988, Rotationally symmetric sG oscillator with tunable frequency, Phys. Lett. A, 131, 364, 10.1016/0375-9601(88)90789-X Mingaleev, 2000, Long-range interaction and nonlinear localized modes in photonic crystal waveguides, Phys. Rev. E, 62, 5777, 10.1103/PhysRevE.62.5777 Nakajima, 1974, Numerical analysis of vortex motion on Josephson structures, J. Appl. Phys., 45, 4095, 10.1063/1.1663917 Sheng, 2005, Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. Comput. Simulation, 68, 355, 10.1016/j.matcom.2005.02.017 Wallraff, 2000, Whispering vortices, Phys. Rev. Lett., 84, 151, 10.1103/PhysRevLett.84.151 Xin, 2000, Modeling light bullets with the two-dimensional sine-Gordon equation, Phys. D, 135, 345, 10.1016/S0167-2789(99)00128-1 Zagrodzinsky, 1979, Particular solutions of the sine-Gordon equation in 2+1 dimensions, Phys. Lett., 72A, 284, 10.1016/0375-9601(79)90469-9