Supersonic discrete kink-solitons and sinusoidal patterns with “magic” wave number in anharmonic lattices

¹ Instituto de Investigación en Comunicación Optica Universidad Autonóma de San Luis Potosí, Alvaro Obregon 64 78000 San Luis Potosí, SLP, Mexico
² Department of Physics, Tbilisi State University - 3 Chavchavadze avenue Tbilisi 380028, Republic of Georgia
³ Dipartimento di Energetica “Sergio Stecco” and CSDC Università di Firenze Via S. Marta, 3, I-50139 Firenze, Italy
⁴ INFM and INFN - Firenze, Italy

Received: 24 March 2003
Accepted: 3 February 2004

Abstract

The sharp-pulse method is applied to Fermi-Pasta-Ulam (FPU) and Lennard-Jones (LJ) anharmonic lattices. Numerical simulations reveal the presence of high-energy strongly localized “discrete” kink-solitons (DK), which move with supersonic velocities that are proportional to kink amplitudes. For small amplitudes, the DKs of the FPU lattice reduce to the well-known “continuous” kink-soliton solutions of the modified Korteweg-de Vries equation. For high amplitudes, we obtain a consistent description of these DKs in terms of approximate solutions of the lattice equations that are obtained by restricting to a bounded support in space exact solutions with sinusoidal pattern characterized by the “magic” wave number . Relative displacement patterns, velocity vs. amplitude, dispersion relation and exponential tails found in numerical simulations are shown to agree very well with analytical predictions, for both FPU and LJ lattices.