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

Abstract

The sharp pulse method is applied to Fermi-Pasta-Ulam (FPU) and Lennard-Jones\n(LJ) anharmonic lattices. Numerical simulations reveal the presence of high\nenergy strongly localized ``discrete'' kink-solitons (DK), which move with\nsupersonic velocities that are proportional to kink amplitudes. For small\namplitudes, the DK's of the FPU lattice reduce to the well-known ``continuous''\nkink-soliton solutions of the modified Korteweg-de Vries equation. For high\namplitudes, we obtain a consistent description of these DK's in terms of\napproximate solutions of the lattice equations that are obtained by restricting\nto a bounded support in space exact solutions with sinusoidal pattern\ncharacterized by the ``magic'' wavenumber $k=2\\pi/3$. Relative displacement\npatterns, velocity versus amplitude, dispersion relation and exponential tails\nfound in numerical simulations are shown to agree very well with analytical\npredictions, for both FPU and LJ lattices.\n