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

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