Finite element approximation to two-dimensional sine-Gordon solitons

Meshless Singular Boundary Method for Nonlinear Sine-Gordon Equation

“…We can see that the soliton is moving in straightness when = 18. These graphical results match those obtained by the FEM [7], the FDM [8], the BEM [9,10], the differential quadrature method (DQM) [11], and the RBF [21].…”

“…When = 12.6, the ring soliton appears to be again in its shrinking phase. These graphical results match those obtained by the FEM [7], the BEM [9,10], the DQM [11], the RBF [21], and the mesh-free reproducing kernel particle Ritz method [23]. the sine-Gordon equation for ( 1 , 2 ) = 1 with initial conditions [6,7,10,21] ( 1 , 2 )…”

“…As in [6,7,10,21], the solution is computed by the SBM over −10 ≤ 1 ≤ 10 and −7 ≤ 2 ≤ 7 and then is expanded at the lines 1 = −10 and 2 = −7 by symmetry relations.…”

“…The circular ring soliton can be simulated by solving the sine-Gordon equation for ( 1 , 2 ) = 1 and = 0, with initial conditions [7,[9][10][11]21, 23]…”

“…In the past thirty years, a variety of numerical methods [6][7][8][9][10][11][12][13][14][15], such as the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM), have been used to numerical simulation of the sine-Gordon equation. The success of these methods relies on the ability to construct and retain meshes of sufficient quality in the process of computation.…”