“…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 )…”