A Study on the Semi-Discrete KP Equation: Bilinear Bäcklund Transformation, Lax Pair and Periodic Wave Solutions

In this paper, we investigate the integrability and solution structure of the semi-discrete KP equation. A bilinear Bäcklund transformation, Lax pair, and nonlinear superposition formula are systematically derived, establishing the integrability of the system. Furthermore, one- and two-periodic wave solutions are constructed using Hirota’s method combined with Riemann theta functions. By means of a rigorous limiting procedure, the asymptotic behavior of the periodic wave solutions is analyzed, and the connection between periodic wave and soliton solutions is established. These results not only enrich the solution structure of the semi-discrete KP equation but also provide new perspectives on periodic phenomena in discrete integrable systems.