https://doi.org/10.1140/epjp/s13360-020-00587-2
Regular Article
On the generation and propagation of solitary waves in integrable and nonintegrable nonlinear lattices
1
Department of Physics, State University of New York, 14260, Buffalo, NY, USA
2
Department of Mathematics and Statistics, Macquarie University, 12 Wally’s Walk L6, Sydney, Australia
3
Department of Mathematics, State University of New York, 14260, Buffalo, NY, USA
4
Department of Mathematics and Statistics, University of Massachusetts Amherst, 01003, Amherst, MA, USA
Received:
18
February
2020
Accepted:
3
July
2020
Published online:
24
July
2020
We investigate the generation and propagation of solitary waves in the context of the Hertz chain and Toda lattice, with the aim to highlight the similarities, as well as differences between these systems. We begin by discussing the kinetic and potential energy of a solitary wave in these systems and show that under certain circumstances the kinetic and potential energy profiles in these systems (i.e., their spatial distribution) look reasonably close to each other. While this and other features, such as the connection between the amplitude and the total energy of the wave, bear similarities between the two models, there are also notable differences, such as the width of the wave. We then study the dynamical behavior of these systems in response to an initial velocity impulse. For the Toda lattice, we do so by employing the inverse scattering transform, and we obtain analytically the ratio between the energy of the resulting solitary wave and the energy of the impulse, as a function of the impulse velocity; we then compare the dynamics of the Toda system to that of the Hertz system, for which the corresponding quantities are obtained through numerical simulations. In the latter system, we obtain a universality in the fraction of the energy stored in the resulting solitary traveling wave irrespectively of the size of the impulse. This fraction turns out to only depend on the nonlinear exponent. Finally, we investigate the relation between the velocity of the resulting solitary wave and the velocity of the impulse. In particular, we provide an alternative proof for the numerical scaling rule of Hertz-type systems.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020