Short-Time Existence for Scale-Invariant Hamiltonian Waves

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh wave

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

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 H\"older-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When $\alpha \rightarrow 1^+$, 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

Elastic wave dispersion in microstructured membranes

The effect of microstructural properties on the wave dispersion in linear elastic membranes is addressed in this paper. The periodic spring-mass lattice at the lower level of observation is continualised and a gradient-enriched membrane model is obtained to account for the characteristic microstructural length scale of the material. In the first part of this study, analytical investigations show that the proposed model is able to capture correctly the physical phenomena of wave dispersion in microstructured membrane which is overlooked by classical continuum theories. In the second part, a finite element discretisation of microstructured membrane is formulated by introducing the pertinent inertia and stiffness terms. Importantly, the proposed modifications do not increase the size of the problem with respect to the classical elasticity. Numerical simulations are included for validation purposes. The results confirm that the structural characteristics of the material can have a huge impact on the vibrational properties, particularly in the high-frequency range