Abstract
Hirota bilinear method is proposed to directly construct periodic wave solutions in terms of Riemann theta functions for $(2+1)$ -dimensional Toda lattice equations. The asymptotic properties of the periodic waves are analyzed in detail, including one-periodic and two-periodic solutions. Furthermore, the curves of the solutions are plotted to analyze the solutions. It is shown that well-known soliton solutions can be reduced from the periodic wave solutions.
Highlights
It is well known that there are many successful methods to construct explicit solutions for differential equations, such as the scattering transform [ ], the Darboux transformation [ ], Hirota direct method [ – ], algebra-geometrical approach [ – ], etc
Once nonlinear equations are written in bilinear forms by a dependent variable transformations, multisoliton solutions and rational solutions can be obtained
Nakamura [, ] in and presented oneperiodic wave solutions and two-periodic wave solutions based on the Hirota method with the help of the Riemann theta function, where the periodic solutions of the KdV and Boussinesq equations were derived
Summary
It is well known that there are many successful methods to construct explicit solutions for differential equations, such as the scattering transform [ ], the Darboux transformation [ ], Hirota direct method [ – ], algebra-geometrical approach [ – ], etc. The procedures introduced in Dai et al [ ] are adopted by other authors to study a number of soliton equations for constructing quasi-periodic solutions (see [ – ]). Ting et al Advances in Difference Equations (2016) 2016:55 tained the relation between the periodic and open Toda lattice.
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