Quasi-periodic wave solution and asymptotic behavior for the (2Â +Â 1)-dimensional Toda lattice equation

One- and two-periodic wave solutions for the (2 + 1)-dimensional Toda lattice equation are presented based on the Hirota bilinear method and the Riemann theta function. The asymptotic behaviors of these two solutions are considered and the rigorous proof is given that the periodic wave solutions tend to the soliton solutions in an appropriate limiting procedure.