Emergence and Decay of $\pi$-Kinks in the Sine-Gordon Model with High-Frequency Pumping

In this paper, we consider the sine-Gordon equation with a high-frequency parametrical pumping and a weak dissipative force. We examine the class of $\pi$-kink-type solutions that are soliton solutions of the nonperturbed sine-Gordon equation. In contrast to stable $2\pi$-kinks, these these solutions are unstable. We prove that the time of decaying of $\pi$-kinks due to small perturbations is proportional to the cube of the inverse period of fast oscillations of the parametrical pumping. We derive a two-time asymptotic expansion of a solution of the boundary-value problem and analyze evolution of a wave packet whose leading term has the form of a $\pi$-kink. Numerical simulations of solutions confirm a good qualitative agreement with asymptotic expansions.