On the convergence of the nonlocal nonlinear model to the classical elasticity equation

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional propagation nonlinear waves in continuous medium. In the limit vanishing nonlocality we study behavior solutions to Cauchy problem. prove that, as kernel functions convolution integral approach Dirac delta function, converge strongly corresponding classical elasticity equation. An energy estimate with no loss derivative plays critical role proving convergence result. As typical example, discrete lattice dynamic model (the Fermi-Pasta-Ulam-Tsingou model) and show spacing approaches zero, equation