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This paper consists of two parts. The first part proposes a newmethodological framework within which the heat conductivity in 1D lattices canbe studied. The total process of heat conductivity is decomposed into twocontributions where the first one is the equilibrium process at equaltemperatures T of both lattice ends and the second -- non-equilibrium processwith the temperature \Delta T of one end and zero temperature of the other. Theheat conductivity in the limit \Delta T \to 0 is reduced to the heatconductivity of harmonic lattice. A threshold temperature T_{thr} scalesT_{thr}(N) \sim N^{-3} with the lattice size N. Some unusual properties of heatconductivity can be exhibited on nanoscales at low temperatures. Thethermodynamics of the \beta-FPU lattice can be adequately approximated by theharmonic lattice. The second part testifies in the favor of the soliton andbreather contribution to the heat conductivity in contrast to [N. Li, B. Li, S.Flach, PRL 105 (2010) 054102]. In the continuum limit the \beta-FPU lattice isreduced to the modified Korteweg - de Vries equation with soliton and breathersolutions. Numerical simulations demonstrate their high stability. New methodfor the visualization of moving solitons and breathers is suggested. Anaccurate expression for the dependence of the sound velocity on temperature isalso obtained. Our results support the conjecture on the solitons and breatherscontribution to the heat conductivity.

Heat conductivity in the beta-FPU lattice. Solitons and breathers as energy carriers