Evolution of two-dimensional standing and travelling breather solutions for the Sine–Gordon equation

In the present work the problem of the evolution of standing and travelling breather-type waves for the two-space-dimensional Sine–Gordon equation is studied asymptotically and numerically. This work was motivated by the work of Xin [Physica D 135 (2000) 345] on modulated travelling wave solutions and the work of Piette and Zakrzewski [Nonlinearity 11 (1998) 1103] on radially symmetric, periodic standing wave solutions of the two-dimensional Sine–Gordon equation. In the present work it is shown that the dispersive radiation shed by a pulse as it evolves ultimately stabilises it and that the internal breathing motion of the pulse is intimately involved in this process. It is further shown that the linear momentum shed in radiation by a travelling pulse ultimately stabilises it. In [Nonlinearity 11 (1998) 1103] it was shown numerically that radially symmetric, periodic solutions exist for very long times. The present results describe approximately the dynamical approach to these periodic solutions, starting from distorted initial conditions. These distorted initial conditions were not considered in [Nonlinearity 11 (1998) 1103]. Moreover in the present work the stability of periodic solutions evolving from non-radially symmetric initial conditions is studied. Solutions of the approximate equations are found to be in remarkable agreement with full numerical solutions. This agreement confirms the crucial role that radiation plays in the evolution of the breathers.