Large number of hyperbolic and periodic solutions of sine-Gordon equation with nontrivial time dependence

By using suitable ansatz which reduces the problem of obtaining exact solutions of the sine-Gordon equation with nontrivial time dependence to obtaining exact solutions of two first order symmetric $ϕ^4$-like coupled equations in $x$ and $t$, we obtain a large number of hyperbolic and periodic solutions of the sine-Gordon equation. In particular, we not only obtain the well-known two-soliton, soliton-antisoliton and the breather solutions but also obtain periodic two-soliton, periodic soliton-antisoliton as well as periodic breather solutions. Besides, we also obtain superposed real and complex (parity-time reversal or) $\mathcal{PT}$-invariant solutions related to the two-soliton, soliton-antisoliton and the breather solutions of the sine-Gordon equation.