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

We look for long-living topological solutions of classical nonlinear $(1+1)-$ dimensional $varphi^4$ field theory. As for that the original method cut and match is offered. In the framework of this method new long-living states are obtained in both topological sectors. In particular, a highly excited state of a kink are found in one case. We discover several ways of energy reset. In addition to the expected emission wave packets (with small amplitude) in the case of some selected initial conditions a large energy reset becomes a result of the production of kink-antikink pairs. Besides a topological number of a kink in the central region is changing 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 is changing to linearized well-known solution - a discrete kinks excitation mode. The proposed method yieldes a qualitatively new way of describing the large-amplitude bion, which was detected earlier in the kink scattering processes in the non-topological sector.