Study of Two-Dimensional Axisymmetric Breathers Using Padé Approximants

We analyze axisymmetric, spatially localized standing wave solutions with periodic time dependence (breathers) of a nonlinear partial differential equation. This equation is derived in the 'continuum approximation' of the equations of motion governing the anti-phase vibrations of a two-dimensional array of weakly coupled nonlinear oscillators. Following an asymptotic analysis, the leading order approximation of the spatial distribution of the breather is shown to be governed by a two-dimensional nonlinear Schrödinger (NLS) equation with cubic nonlinearities. The homoclinic orbit of the NLS equation is analytically approximated by constructing [2N × 2N] Padé approximants, expressing the Padé coefficients in terms of an initial amplitude condition, and imposing a necessary and sufficient condition to ensure decay of the Padé approximations as the independent variable (radius) tends to infinity. In addition, a convergence study is performed to eliminate 'spurious' solutions of the problem. Computation of this homoclinic orbit enables the analytic approximation of the breather solution.