Dynamics of kink-soliton solutions of the $(2+1)$-dimensional sine-Gordon equation

We study the dynamics of explicit solutions of the $(2+1)$-dimensional (2D) sine-Gordon equation. The Darboux transformation is applied to the associated linear eigenvalue problem to construct nontrivial solutions of the $2$D sine-Gordon equation in terms of a ratio of determinants. We obtain a generalized expression for an $N$-fold transformed dynamical variable, which enables us to calculate explicit expressions of nontrivial solutions. To explore the dynamics of kink soliton solutions, explicit expressions for one- and two-soliton solutions are derived for particular column solutions. Different profiles of kink–kink and kink–anti-kink interactions are illustrated for different parameters and arbitrary functions. We also present a first-order bound state solution.