Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota–Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey–Dodd–Gibbon–Sawada–Kotera equation and (2+1)-dimensional breaking soliton equation. 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 so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota–Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.