Solitary waves and kinks in FPU lattices with soft-hard-soft trilinear interactions

TL;DR

This paper studies solitary waves and kinks in a modified FPU lattice with soft-hard-soft interactions, revealing new localized and delocalized wave structures, explicit solutions, and stability properties through numerical and analytical methods.

Contribution

It introduces a non-symmetric soft-hard-soft trilinear force model in FPU lattices, providing explicit solutions and analyzing wave stability and dynamics.

Findings

Existence of supersonic kinks and flat-top solitary waves.

Explicit solutions for slow solitary waves in special cases.

Numerical evidence of Whitham shocks and wave localization.

Abstract

We consider a version of the classical Hamiltonian Fermi-Pasta-Ulam (FPU) problem with a trilinear force-strain relation of soft-hard-soft type that is in general non-symmetric. In addition to the classical spatially localized solitary waves, such hardening-softening model also exhibits supersonic kinks and finite-amplitude, spatially delocalized flat-top solitary waves that acquire the structure of a kink-antikink bundle when their velocity approaches the kink limit. Exploiting the fact that traveling waves are periodic modulo shift by a lattice spacing, we compute these solutions as fixed points of the corresponding nonlinear map and investigate how their properties depend on the parameter measuring the asymmetry of the problem. In a particularly interesting case when one of the soft regimes has zero elastic modulus, we obtain explicit solutions for sufficiently slow solitary waves. In contrast to conventional delocalization in the sonic limit, these compact structures mounted on a constant background become localized at the lattice scale as their velocity tends to zero. Numerical simulations of Riemann-type initial value problem in this degenerate model show the emergence of Whitham shocks that involve periodic trains of solitary waves. We investigate stability of the obtained solutions using direct numerical simulations and Floquet analysis. We also obtain explicit solutions for a quasicontinuum model that captures some important features of the discrete problem.