Slope modulation of ring waves governed by two-dimensional sine–Gordon equation

• A theory of slope modulation of waves governed by the 2-D sine–Gordon equation is proposed.
• A large time asymptotic solution describing the slope modulation of trains of axi-symmetric kinks is obtained.
• The comparison with the numerical solution of 2-D sine–Gordon equation shows excellent agreement.

A theory of slope modulation of waves governed by the two-dimensional sine–Gordon equation is proposed. A large time asymptotic solution describing the slope modulation of trains of ring kinks is obtained. The comparison with the numerical solution of two-dimensional sine–Gordon equation shows excellent agreement.