Supersonic Discrete Kink-Solitons and Sinusoidal Patterns with "Magic" wavenumber in Anharmonic Lattices

TL;DR

This paper uncovers high-energy, supersonic localized kink-solitons in anharmonic lattices, revealing a 'magic' wavenumber pattern and demonstrating strong agreement between numerical simulations and analytical models.

Contribution

It introduces the concept of discrete kink-solitons with supersonic speeds and a specific sinusoidal pattern at a 'magic' wavenumber in anharmonic lattices, supported by both simulations and analytical solutions.

Findings

Discovery of high-energy, localized kink-solitons moving supersonically.

Identification of a sinusoidal pattern with 'magic' wavenumber in these solitons.

Excellent agreement between numerical results and analytical predictions.

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 DK's 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 DK's 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'' wavenumber $k=2\pi/3$. Relative displacement patterns, velocity versus 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.