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

TL;DR

This paper proves that nonlocal nonlinear wave models converge to classical elasticity equations as the nonlocality diminishes, using energy estimates, with applications to the Fermi-Pasta-Ulam-Tsingou lattice model.

Contribution

It establishes the strong convergence of solutions from nonlocal wave equations to classical elasticity equations as the kernel approaches a delta function, including discrete-to-continuum limits.

Findings

Solutions to nonlocal wave equations converge to classical elasticity solutions.

Energy estimates ensure convergence without loss of derivatives.

Discrete lattice models converge to continuum elasticity models.

Abstract

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