Elastic waves localised near the edge of a semi-infinite plate reinforced by a strip plate are considered within the framework of the 2D classical theory for plate bending. The boundary value problem for the strip plate is subject to asymptotic analysis, assuming that a typical wavelength is much greater than the strip thickness. As a result, effective conditions along the interface corresponding to a plate reinforced by a beam with a narrow rectangular cross-section are established. They support an approximate dispersion relation perturbing that for a homogeneous plate with a free edge. The accuracy of the approximate dispersion relation is tested by comparison with the numerical data obtained from the ‘exact’ matrix relation for a composite plate. The effect of the problem parameters on the localisation rate is studied.

This paper is concerned with the existence of solutions for a fully coupled Riemann–Stieltjes, integro-multipoint, boundary value problem of Caputo-type sequential fractional differential equations. The given system is studied with the aid of the Leray–Schauder alternative and contraction mapping principle. A numerical example illustrating the abstract results is also presented.

We consider a two-dimensional square lattice in which each node is restricted to the plane of the lattice, but is permitted to move in both directions of the lattice. We assume nodes are connected to nearest neighbours along the lattice directions with nonlinear springs, and to diagonal neighbours with linear springs. We consider a generalised Klein-Gordon system, that is, where there is an onsite potential at each node in addition to the (nonlinear) nearest-neighbour interactions. We derive the equations of motion for the displacements from the Hamiltonian. We use asymptotic techniques to derive the form of small amplitude breather solutions, and find necessary conditions required for their existence. We find two types of mode, which we term 'optical' and 'acoustic', based on the analysis of other lattices which support dispersion relations with multiple branches. In addition to the usual inequality on the sign of the nonlinearity in order for the NLS to be of the focusing type, we obtain an additional ellipticity constraint, that is a restriction in the two-dimensional wavenumber space, required for the spatial differential operator to be elliptic. Highlights• we consider a 2D square lattice with in-plane motion of nodes • we use a weakly nonlinear asymptotic expansion to derive envelope equation • we find breather solutions of an associated 2D NLS