## The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation.(English)Zbl 1169.76401

Summary: This paper presents the dual reciprocity boundary element method (DRBEM) for solving two-dimensional sine-Gordon (SG) equation. The integral equation formulation employs the fundamental solution of the Laplace equation, and hence a domain integral arises in the boundary integral equation. Furthermore, the time derivatives are approximated by the time-stepping method, and the domain integral also appears from these approximations. The domain integral is transformed into boundary integral by using the dual reciprocity method (DRM). The linear radial basis function (RBF) is employed for DRM. The dynamics of line solitons and ring solitons of circular and elliptic shapes are studied. Numerical results are presented for some problems involving line and ring solitons to demonstrate the usefulness and accuracy of this approach.

### MSC:

 76M15 Boundary element methods applied to problems in fluid mechanics 76B25 Solitary waves for incompressible inviscid fluids 35Q53 KdV equations (Korteweg-de Vries equations)

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 [1] Ang, W.T., The two-dimensional reaction-diffusion Brusselator system: a dual-reciprocity boundary element solution, Engrg. anal. bound. elem., 27, 897-903, (2003) · Zbl 1060.76599 [2] Ang, W.T.; Ang, K.C., A dual reciprocity boundary element solution of a generalized nonlinear Schrödinger equation, Numer. methods partial differ. eq., 20, 843-854, (2004) · Zbl 1062.65109 [3] Argyris, J.; Haase, M.; Heinrich, J.C., Finite element approximation to two-dimensional sine-Gordon solitons, Comput. methods appl. mech. engrg., 86, 1-26, (1991) · Zbl 0762.65073 [4] Behbahani-nia, A.; Kowsary, F., A dual reciprocity BE-based sequential function specification solution method for inverse heat conduction problems, Int. J. heat mass transfer, 47, 1247-1255, (2004) · Zbl 1053.80012 [5] Bogolyubskii˘, I.L.; Makhankov, V.G., Lifetime of pulsating solitons in certain classical models, JETP lett., 24, 1, 12-14, (1976) [6] Bogolyubskii˘, I.L., Oscillating particle-like solutions of the nonlinear klein – gordon equation, JETP lett., 24, 10, 535-538, (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] Bratsos, A.G., A modified predictor – corrector scheme for the two-dimensional sine-Gordon equation, Numer. algor., 43, 295-308, (2006) · Zbl 1112.65077 [9] A.G. Bratsos, The solution of the two-dimensional sine-Gordon equation using the method of lines, J. Comput. Appl. Math. in press. · Zbl 1117.65126 [10] A.G. Bratsos, A third order numerical scheme for the two-dimensional sine-Gordon equation, Mathematics and Computers in Simulation, in press. · Zbl 1135.65358 [11] Brebbia, C.A.; Nardini, D., Dynamic analysis in solid mechanics by an alternative boundary element procedure, Int. J. soil dyn. earthquake engrg., 2, 228-233, (1983) [12] Brebbia, C.A.; Walker, S., Boundary element technique in engineering, (1980), Newnes-Butterworths London · Zbl 0444.73065 [13] Christiansen, P.L.; Olsen, O.H., Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation, Phys. lett. A, 68, 2, 185-188, (1978) [14] Christiansen, P.L.; Lomdahl, P.S., Numerical solutions of 2+1 dimensional sine-Gordon solitons, Phys. D: nonlinear phenom., 2, 3, 482-494, (1981) · Zbl 1194.65122 [15] Christiansen, P.L.; Grønbech-Jensen, N.; Lomdahl, P.S.; Malomed, B.A., Oscillations of eccentric pulsons, Phys. scripta, 55, 131-134, (1997) [16] Djidjeli, K.; Price, W.G.; Twizell, E.H., Numerical solutions of a damped sine-Gordon equation in two space variables, J. engrg. math., 29, 347-369, (1995) · Zbl 0841.65083 [17] 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 [18] Hirota, R., Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. phys. soc. Japan, 35, 1566, (1973) [19] Kaliappan, P.; Lakshmanan, M., Kadomtsev – petviashvili and two-dimensional sine-Gordon equations: reduction to painlevè transcendents, J. phys. A: math. gen., 12, L249-L252, (1979) · Zbl 0425.35080 [20] Karur, S.R.; Ramachandran, P.A., Radial basis function approximation in dual reciprocity method, Math. comput. modell., 20, 7, 59-70, (1994) · Zbl 0812.65101 [21] Krishna, K.M.; Tanaka, M., Dual reciprocity boundary element analysis of nonlinear diffusion: temporal discretization, Engrg. anal. bound. elem., 23, 419-433, (1999) · Zbl 0955.74074 [22] Krishna, K.M.; Tanaka, M., Dual reciprocity boundary element analysis of inverse heat conduction problems, Comput. methods appl. mech. engrg., 190, 5283-5295, (2001) · Zbl 0988.80005 [23] 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) [24] Malomed, B.A., Decay of shrinking solitons in multidimensional sine-Gordon equation, Physica D, 24, 155-171, (1987) · Zbl 0634.35077 [25] Malomed, B.A., Dynamic of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model, Physica D, 52, 157-170, (1991) · Zbl 0742.35060 [26] Maslov, E.M., Dynamics of rotationally symmetric solitons in near-SG field model with applications to large-area Josephson junctions and ferromagnets, Physica D, 15, 433-443, (1985) · Zbl 0583.35088 [27] Maslov, E.M., Rotationally symmetric SG oscillator with tunable frequency, Phys. lett. A, 131, 6, 364-367, (1988) [28] Minzoni, A.A.; Smyth, N.F.; Worthy, A.L., Pulse evolution for a two-dimensional sine-Gordon equation, Physica D, 159, 101-123, (2001) · Zbl 0980.35144 [29] Paris, F.; Canas, J., Boundary element method: fundamental and applications, (1997), Oxford University Press Oxford [30] Partridge, P.W.; Brebbia, C.A., The dual reciprocity boundary element method for the Helmholtz equation, (), 543-555 · Zbl 0712.73094 [31] Partridge, P.W., Towards criteria for selecting approximation functions in the dual reciprocity method, Engrg. anal. bound. elem., 24, 519-529, (2000) · Zbl 0968.65097 [32] Partridge, P.W.; Sensale, B., The method of fundamental solutions with dual reciprocity for diffusion and diffusion-convection using subdomains, Engrg. anal. bound. elem., 24, 633-641, (2000) · Zbl 1005.76064 [33] Ramachandran, P.A., Boundary element methods in transport phenomena, (1994), Computational Mechanics Publication Southampton, Boston · Zbl 0865.65081 [34] Sheng, Q.; Khaliq, A.Q.M.; Voss, D.A., Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. comput. simulat., 68, 355-373, (2005) · Zbl 1073.65095 [35] Tanaka, M.; Matsumoto, T.; Takakuwa, S., Dual reciprocity BEM for time-stepping approach to the transient heat conduction problem in nonlinear materials, Comput. methods appl. mech. engrg., 195, 4953-4961, (2006) · Zbl 1126.80008 [36] Vu-Quoc, L.; Li, S., Invariant-conserving finite difference algorithms for the nonlinear klein – gordon equation, Comput. methods appl. mech. engrg., 107, 341-391, (1993) · Zbl 0790.65101 [37] Wrobel, L.C.; Brebbia, C.A., The dual reciprocity boundary element formulation for nonlinear diffusion problems, Comput. methods appl. mech. engrg., 65, 147-164, (1987) · Zbl 0612.76094 [38] Wrobel, L.C., The boundary element method, Applications in thermo-fluids and acoustics, vol. 1, (2001), John Wiley Press · Zbl 0772.76042 [39] Xin, J.X., Modeling light bullets with the two-dimensional sine-Gordon equation, Physica D, 135, 345-368, (2000) · Zbl 0936.78006 [40] Zagrodzinsky, J., Particular solutions of the sine-Gordon equation in 2+1 dimensions, Phys. lett. A, 72, 284-286, (1979) [41] Zhang, F.; Vazquez, L., Two energy conservating numerical schemes for the sine-Gordon equation, Appl. math. comput., 45, 17-30, (1991) · Zbl 0732.65107 [42] Dehghan, M., A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. methods partial differ. equations, 22, 220-257, (2006) · Zbl 1084.65099 [43] M. Dehghan, A. Shokri, A numerical method for one-dimensional nonlinear sine-Gordon equation using collocation and radial basis functions, Numer. Methods Partial Differ. Equations, in press. · Zbl 1135.65380 [44] M. Dehghan, D. Mirzaei, The boundary integral equation approach for numerical solution of the one-dimensional sine-Gordon equation, Numer. Methods Partial Differ. Equations, to appear. · Zbl 1153.65099
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