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

Abstract

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 $\sqrt{\frac{3}{2}}$ 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