A Natural Full-Discretization of the Korteweg-de-Vries Equation

In this paper, we propose an integrable full-discretization of the Korteweg-de-Vries (KdV) equation. Our method is based on the compatibility between the integrable equation and its Bäcklund transformation. By using this approach, we derive a discrete equation that is a natural discretization of the KdV equation. Specifically, in the natural limit, the discrete system approaches the continuous KdV equation. We demonstrate that the integrability of the discrete system is confirmed by a Lax pair and a Bäcklund transformation.