Marat Fakhretdinov

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.

The generation and evolution of localized waves on an impurity in the scattering of a kink of the sine-Gordon equation are studied. It is shown that the problem can be considered as excitation of oscillations of a harmonic oscillator by a short external impulse. The external impulse is modeled by the scattering of a kink on an impurity. The influence of the modes of motion of a kink on the excitation energy of localized waves is numerically and analytically studied. The method of collective coordinate for the analytical study is used. The value of this energy is determined by the ratio of the impurity parameters and the initial kink velocity. It is shown that the dependence of the energy (and amplitude) of the generated localized waves on the initial kink velocity has only one maximum. This behavior is observed for the cases of point and extended impurities. Analytical expression for the amplitude of the localized wave in the case of point impurity is obtained. This allows controlling the excitation energy of localized waves using the initial kink velocity and impurity parameters. The study of the evolution of localized impurities under the action of an external force and damping has shown a good agreement with the nondissipative case. It is shown that small values of the external force have no significant effect on the oscillations of localized waves. An analytical expression for the logarithmic decrement of damping is obtained. This study may help to control the parameters of the excited waves in real physical systems.

Discrete breathers and multibreathers are investigated within the Peyrard–Bishop model. Region of existence of discrete breathers and multibreathers is defined. One, two and three site discrete breathers solutions are obtained. Their properties and stability are investigated.