Attractive Impurity as a Generator of Wobbling Kinks and Breathers in the $\varphi^4$ Model

The $\varphi^4$ theory is widely used in many areas of physics, from cosmology and elementary particle physics to biophysics and condensed matter theory. However, in the $\varphi^4$ model, there are no spatially localized solutions in the form of breathers. Topological defects, or kinks, in this theory describe stable, solitary wave excitations. In practice, these excitations, as they propagate, necessarily interact with impurities or imperfections in the on-site potential. In this work, with the help of numerical calculations using the method of lines, the interaction of the kink in the $\varphi^4$ model with extended impurities is considered. The case of an attractive rectangular impurity is analyzed. It is found that after the kink-impurity interaction, an internal mode with frequency $\sqrt{\frac32}$ is excited on the kink and it becomes a wobbling kink. It is shown that with the help of kink-impurity interaction, an extended rectangular attracting impurity, as well as a point impurity, can be used as a generator for excitation of long-lived high-amplitude localized breather waves. The structure of the excited wobbling breather (or wobbler), which consists of a compact core and an extended tail, is described. It is shown that the wobbler tail has the form of a spatially unbounded quasi-sinusoidal function with a classical frequency $\sqrt{2}$. To determine the lifetime of the wobbler, the dependence of the amplitude of the impurity mode on time is found. For the case of small impurities, it turned out that it practically does not change for a long time. For the case of large impurities, the wobbler amplitude begins to noticeably decrease with time. The frequency of wobbler oscillations does not depend on the initial velocity of the kink. The dependence of the impurity mode oscillation amplitude on the initial kink velocity has minima and maxima. By changing the impurity parameters, one can also control the dynamic parameters of the wobbler. A linear approximation is considered that allows an analytical solution of the problem for localized breather waves, and the limits of its applicability for this model are found.