Geodesic
Emergence and Decay of $\pi$-Kinks in the Sine-Gordon Model with High-Frequency Pumping
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 53-66

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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.
Mots-clés : sine-Gordon equation
Keywords: $\pi$-kink, Kapitsa pendulum, averaging method, asymptotic expansion, stability of solitons. 
@article{INTO_2018_152_a5,
     author = {O. M. Kiselev and V. Yu. Novokshenov},
     title = {Emergence and {Decay} of $\pi${-Kinks} in the {Sine-Gordon} {Model} with {High-Frequency} {Pumping}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {53--66},
     publisher = {mathdoc},
     volume = {152},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_152_a5/}
}
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O. M. Kiselev; V. Yu. Novokshenov. Emergence and Decay of $\pi$-Kinks in the Sine-Gordon Model with High-Frequency Pumping. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 53-66. http://geodesic.mathdoc.fr/item/INTO_2018_152_a5/