Study of Two-Dimensional Axisymmetric Breathers Using Padé Approximants |
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Authors: | Emaci E Vakakis A F Andrianov I V Mikhlin Yu |
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Institution: | (1) Department of Mechanical and Industrial Engineering, University of Illinois at Urbana – Champaign, Urbana, IL, 61801, U.S.A;(2) Department of Mathematics, Prydneprovie State Academy of Civil Engineering and Architecture, Dnepropetrovsk, 320000, Ukraine;(3) Department of Applied Mathematics, Kharkov Polytechnic University, Kharkov, 310002, Ukraine |
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Abstract: | 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. |
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Keywords: | Nonlinear localization nonlinear differential equations Padé approximations |
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