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