RT Journal Article
T1 Propagation of solitons in a two-dimensional nonlinear square lattice
A1 Zaera, Ramón
A1 Vila Morán, Javier
A1 Fernández-Sáez, José
A1 Ruzzene, Massimo
AB We investigate the existence of solitary waves in a nonlinear square spring-mass lattice. In the lattice, the masses interact with their neighbors through linear springs, and are connected to the ground by a nonlinear spring whose force is expressed as a polynomial function of the masses out-of-plane displacement. The low-order Taylor series expansions of the discrete equations lead to a continuum representation that holds in the long wavelength limit. Under this assumption, solitary wave solutions are sought within the long wavelength approximation, and the subsequent application of multiple scales to the resulting nonlinear continuum equations. The study focuses on weak nonlinearities of the ground stiffness and reveals the existence of 3 types of solitons, namely a bright, a dark, and a vortex soliton. These solitons result from the balance of dispersive and nonlinear effects in the lattice, setting aside other relevant phenomena in 2D waves such as diffraction that may lead to a field that does not change during propagation in nonlinear media. For equal constants of the in-plane springs, the governing equation reduces to the Klein-Gordon type, for which bright and dark solitons replicate solutions for one-dimensional lattices. However, unequal constants of the in-plane springs aligned with the two principal lattice directions lead to conditions in which the soliton propagation direction, defined by the group velocity, differs from the wave vector direction, which is unique to two-dimensional assemblies. Furthermore, vortex solitons are obtained for isotropic lattices, which shows similarities with results previously found in optics, thermal media and quantum plasmas. The paper describes the main parameters defining the existence of these solitary waves, and verifies the analytical predictions through numerical simulations. Results show the validity of obtained solutions and illustrate the main characteristics of the solitary waves found in the considered n
PB Elsevier
SN 0020-7462
SN 1878-5638 (online)
YR 2018
FD 2018-11-01
LK https://hdl.handle.net/10016/36233
UL https://hdl.handle.net/10016/36233
LA eng
NO M. Ruzzene acknowledges the support of the UC3M-Santander Chairs of Excellence Program during academic year 2016-17. J. Vila Moran acknowledges the support from the US National Science Foundation (Grant number 1332862). The authors from the Universidad Carlos III de Madrid are indebted to the Ministerio de Ciencia e Innovación de España (Project DPI-2014-57989-P) for the financial support.
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RD 30 jun. 2024