From Newton's cradle to the discrete p-Schrödinger equation

Brigitte Bidégaray-Fesquet (1) , Eric Dumas (2) , Guillaume James (3, 4)

1 EDP - Equations aux Dérivées Partielles
2 IF - Institut Fourier
3 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
4 CASYS - Calculs Algébriques et Systèmes Dynamiques

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.

Mots clés: breather; discrete p-Schrödinger equation; Hertzian interactions; multiple-scale analysis; Newton's cradle; nonlinear lattice

Domaines: Systèmes dynamiques [math.DS]

Brigitte Bidégaray-Fesquet, Eric Dumas, Guillaume James. From Newton's cradle to the discrete p-Schrödinger equation. SIAM Journal on Mathematical Analysis, 2013, 45 (6), pp.3404-3430. ⟨10.1137/130924196⟩. ⟨hal-00831868⟩