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From Newton’s cradle to the discrete \(p\)-Schrödinger equation. (English) Zbl 1292.34008

Authors’ abstract: We investigate the dynamics of a chain of oscillators coupled by fully nonlinear interaction potentials. This class of models includes Newton’s cradle with Hertzian contact interactions between neighbors. By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times. The envelope equation leading to approximate solutions is a discrete \(p\)-Schrödinger equation. Our results include the existence of long-lived breather solutions to the original model. For a large class of localized initial conditions, we also estimate the maximal decay of small amplitude solutions over long times.

MSC:

34A33 Ordinary lattice differential equations
39A12 Discrete version of topics in analysis
34E13 Multiple scale methods for ordinary differential equations
70F45 The dynamics of infinite particle systems
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
70K75 Nonlinear modes
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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