Abstract:
The problem of nonlinear dynamics of kinks for the $\varphi^4$ equation in a model with two point impurities is considered using numerical methods. The $\varphi^4$ equation is used in many areas of physics, from cosmology and particle physics to biophysics and condensed matter theory. Topological defects, or kinks, in this theory describe stable, corpuscular excitations. In practice, these excitations necessarily interact with impurities or imperfections of the background potential as they propagate. For numerical calculations, we used the line method. A point impurity is described using the delta function . The case of an attractive impurity is analyzed. The kink was launched in the direction of the impurities with different initial velocities. In the paper, all possible scenarios of the kink dynamics are determined and described, taking into account resonance effects. It is established that among the found scenarios of kink dynamics there are scenarios of resonance dynamics of the kink obtained for the case of one point impurity, for example, quasi-tunneling and repulsion from the attractive potential. It is shown that the dynamics of the kink with two impurities also contains new scenarios of its dynamics compared to the case of one impurity. Critical and resonance velocities of the kink motion are found depending on the parameters of the impurity and the distance between them. A diagram of possible scenarios of the kink dynamics is constructed depending on its initial velocity and the distance between the impurities.
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