Study of Two-Dimensional Axisymmetric Breathers Using Padé Approximants

E. Emaci, A. F. Vakakis, I. V. Andrianov, Yu Mikhlin

Research output: Contribution to journalArticlepeer-review


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 Fade 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.

Original languageEnglish (US)
Pages (from-to)327-338
Number of pages12
JournalNonlinear Dynamics
Issue number4
StatePublished - 1997


  • Nonlinear differential equations
  • Nonlinear localization
  • Padé approximations

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering


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