Riemann theta functions periodic wave solutions and rational characteristics for the (1+1)-dimensional and (2+1)-dimensional Ito equation

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward way is presented to explicitly construct multiperiodic Riemann theta functions periodic waves solutions for nonlinear differential equation such as the (1+1)-dimensional and (2+1)-dimensional Ito equations. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two dimensional that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze asymptotic behavior of the multiperiodic periodic waves in details and the relations between the periodic wave solutions and soliton solutions are rigorously established.