Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials

Abstract

We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order $\alpha >1$. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when $\alpha$ is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV equation, and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When $\alpha \to 1^{+}$, we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed, and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.

References in corpus (4)

• Solitary wave trains in granular chains: Experiments, theory and simulations
• Compactons and Chaos in Strongly Nonlinear Lattices
• A Hamiltonian-Krein (instability) index theory for KdV-like eigenvalue problems
• Breather solutions of the discrete p-Schrödinger equation

Cited by in corpus (11)

• Nonlinear Coherent Structures in Granular Crystals
• Shock and Rarefaction Waves in Generalized Hertzian Contact Models
• On the orbital stability of Gaussian solitary waves in the log-KdV equation
• Stability and interaction of compactons in the sublinear KdV equation
• High-energy waves in superpolynomial FPU-type chains
• Solitary waves in atomic chains and peridynamical media
• Waves in Strongly Nonlinear Gardner-like Equations on a Lattice
• Long wavelength solitary waves in Hertzian chains and their properties in different nonlinearity regimes
• Asymptotic stability of viscous shocks in the modular Burgers equation
• Breathers in lattices with alternating strain-hardening and strain-softening interactions
• Fronts in dissipative Fermi-Pasta-Ulam-Tsingou chains