Kirill Samsonov

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.

The $\varphi^4$ theory is widely used in many areas of physics, from cosmology and elementary particle physics to biophysics and condensed matter theory. 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, we focus on the effect of the length and strength of a rectangular impurity on the kink dynamics. It is found that the interaction of a kink with an extended impurity is qualitatively similar to the interaction with a well-studied point impurity described by the delta function, but significant quantitative differences are observed. The interaction of kinks with an extended impurity described by a rectangular function is studied numerically. All possible scenarios of kink dynamics are determined and described, taking into account resonance effects. The inelastic interaction of the kink with the repulsive impurity arises only at high initial kink velocities. The dependencies of the critical and resonant velocities of the kink on the impurity parameters are found. It is shown that the critical velocity of the repulsive impurity passage is proportional to the square root of the barrier area, as in the case of the sine-Gordon equation with an impurity. It is shown that the resonant interaction in the $\varphi^4$ model with an attracting extended impurity, as well as for the case of a point impurity, in contrast to the case of the sine-Gordon equation, is due to the fact that the kink interacts not only with the impurity mode, but also with the kink’s internal mode. It is found that the dependence of the kink final velocity on the initial one has a large number of resonant windows.

Possibility of changing the dynamic parameters of localized breather and soliton waves for the sine-Gordon equation in the model with extended impurity, variable external force and dissipation was investigated using the autoresonance method. The model of ferromagnetic structure consisting of two wide identical layers separated by a thin layer with modified values of magnetic anisotropy parameter was taken as a basis. Frequency of external field is a linear function of time. The sine-Gordon equation (SGE) was solved numerically using the finite differences method with explicit scheme of integration. For certain values of the extended impurity parameters a magnetic inhomogeneity in the form of magnetic breather is formed when domain wall passes through it with constant velocity. The numerical simulation showed that using special variable force and small amplitude it is possible to resonantly increase the amplitude of breather. For each case of the impurity parameters values, there is a threshold value of the magnetic field amplitude leading to resonance. Geometric parameters of thin layer also have influence on the resonance effect — for decreasing layer width the breather amplitude grows more slowly. For large layer width the translation mode of breather oscillations is also excited. For certain parameters of extended impurity, a soliton can form. For a special type of variable field with frequency linearly dependent on time, soliton is switched to antisoliton and vice versa.