An approach to radially symmetrical solutions of the sine-Gordon equation

Abstract

The solution φ(r, t) of the radially symmetric sine-Gordon equation is considered in three and two spatial dimensions for initial curves, analogous to a 2 π- kink, in the expanding and in the shrinking phase, for R( t) j⩾ R(0). It is shown that the parameterization φ( r, t) = 4 arcian exp[ γ( r− R(0)] + x( r, t), where R( t) describes the exact propagation of the maximum of φ,( r, t), is suitable. Using an appoximate differential equation, recently given for the propagation of the solitary ring wave, a rough analytic approximation for the correction function x( r = R( t), t) is found and tested numerically. A relationship between the fluctuations in x( r = R( t), t) and those in R ̈ (t), t) and R(t) explains why the solitary wave is almost stable. From x( r = R( t), t) and the supposition x(1, t) ≈ x(∞, t) ≈ 0 an assymetry in φ r ( r, t) with respect to r = R( t) is predicted. It also exhibits fluctuations corresponding to those in x( r = R( t), t). The condition for validity of this approximation apparently is also a limit for the stability of the solitary ring wave.