On the convergence of the nonlocal nonlinear model to the classical elasticity equation (Q2077606)

In this paper a general class of convolution-type nonlocal wave equations is studied which models the bidirectional propagation of nonlinear waves in a continuous medium. The authors consider the particular case where one of the kernels is the Dirac delta function corresponding to the zero-dispersion case. An energy estimate for strong solutions of the corresponding linearized system is obtained. The dependence of the existence time of the solutions on the initial data and the size of the nonlinear term is shown, which makes it possible to extend the local well-posedness result to long time intervals. A rigorous justification of the convergence of the Fermi-Pasta-Ulam-Tsingou discrete lattice dynamic equation corresponding to the triangular kernel to the classical elasticity equation is provided in the case when the lattice spacing approaches zero.