Asymptotic Behavior of Periodic Wave Solution to the Hirota–Satsuma Equation

The one- and two-periodic wave solutions for the Hirota–Satsuma (HS) equation are presented by using the Hirota derivative and Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given such that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.