Search for long-living topological solutions of nonlinear $\varphi^4$ field theory

We look for long-living topological solutions of classical nonlinear $(1+1)-$ dimensional $\varphi^4$ field theory. To that effect we use the well-known cut-and-match method. In this framework, new long-living states are obtained in both topological sectors. In particular, in one case a highly excited state of a kink is found. We discover several ways of energy reset. In addition to the expected emission wave packets (with small amplitude), for some selected initial conditions the production of kink-antikink pairs results in a large energy reset. Also, the topological number of a kink in the central region changes in the contrast of saving full topological number. At lower excitation energies there is a long-living excited vibrational state of the kink; this phenomenon is the final stage of all considered initial states. Over time this excited state of the kink changes to a well-known linearized solution - a discrete kink excitation mode. This method yields a qualitatively new way to describe the large-amplitude bion, which was detected earlier in the kink-scattering processes in the nontopological sector.