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.

我们考虑对非线性波在连续介质中的双向传播进行建模的一类通用卷积型非局部波动方程。在消失非定域性的极限中，我们研究柯西问题的解的行为。我们证明，当卷积积分的核函数逼近 Dirac delta 函数时，解强烈收敛到经典弹性方程的对应解。没有导数损失的能量估计在证明收敛结果方面起着关键作用。作为一个典型的例子，我们考虑离散晶格动力学模型（Fermi-Pasta-Ulam-Tsingou 模型）的连续极限，并表明，随着晶格间距接近零，离散晶格方程的解收敛到相应的解经典弹性方程。