A numerical scheme and some theoretical aspects for the cylindrically and spherically symmetric sine-Gordon equations

• A predictor-corrector technique is proposed to numerically integrate the sine-Gordon equation for the ring-shaped solution.
• A pseudo-analytical expression of the returning time is obtained.
• The modulation solutions corresponding to the ring-shaped waves of kink type are obtained.

A finite difference formula based on the predictor–corrector technique is presented to integrate the cylindrically and spherically symmetric sine-Gordon equations numerically. Based on various numerical observations, one property of the waves of kink type is conjectured and used to explain their returning effect. Several numerical experiments are carried out and they are in excellent agreement with the existing results. In addition, the corresponding modulation solution for the two-dimensional ring-shaped kink is extended to that in three-dimension. Both numerical and theoretical aspects are utilized to verify the reliability of the proposed numerical scheme and thus the analytical modulation solutions.