Quasi-periodic waves and an asymptotic property for the asymmetrical Nizhnik–Novikov–Veselov equation

Based on a multi-dimensional Riemann theta function, the Hirota bilinear method is extended to explicitly construct multi-periodic (quasi-periodic) wave solutions for the asymmetrical Nizhnik–Novikov–Veselov equation. Among these periodic waves, two-periodic waves are a direct generalization of well-known cnoidal waves; their surface pattern is two dimensional. The main physical result is the description of the behavior of nonlinear waves in shallow water. A limiting procedure is presented to analyze asymptotic properties of the two-periodic waves in details. Relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a 'small amplitude' limit.