Localized Waves of the $\varphi^{4}$ Equation in the Model with Two-Point Impurities

In this work, the interaction of the kink in the $\varphi^{4}$ model with two point impurities is considered. A point impurity is described using the Dirac delta-function. The case of an attractive impurity is analyzed. It is shown that the interaction of the kink with the impurities leads to the excitation of long-lived small-amplitude breather-type waves localized on them. Their structure and associated dynamics have been investigated analytically and numerically. Using the collective variable method, a system of two differential equations describing the coupled dynamics of the waves localized on the impurities has been obtained. This system of equations has solutions: in the form of in-phase oscillations, if the initial amplitudes of the waves localized on the impurities are equal; and in the form of antiphase oscillations, if one of the initial amplitudes is zero. In all other cases of initial amplitudes, the system has solutions in the form of beats. Numerically, using the method of lines, coupled in-phase oscillations, antiphase oscillations, and beats of the waves localized on the impurities were also obtained. The oscillations of the waves localized on the impurities are accompanied by radiation. The existence of two possible oscillation frequencies was found, both analytically and numerically. It is shown that these frequencies do not depend on the initial kink velocity but strongly depend on the distance between the impurities. As the distance between the impurities increases, the frequencies merge into one — the frequency obtained for the case of a single impurity. The dependencies of the frequencies on the distance between the impurities, found numerically and analytically, agree well for large distances, when the interaction between the impurities is weak, and begin to differ noticeably at small distances,