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

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 k =2 π/3. 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.