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

This paper consists of two parts. The first part proposes a new methodological framework within which the heat conductivity in 1D lattices can be studied. The total process of heat conductivity is decomposed into two contributions where the first one is the equilibrium process at equal temperatures T of both lattice ends and the second -- non-equilibrium process with the temperature \Delta T of one end and zero temperature of the other. The heat conductivity in the limit \Delta T \to 0 is reduced to the heat conductivity of harmonic lattice. A threshold temperature T_{thr} scales T_{thr}(N) \sim N^{-3} with the lattice size N. Some unusual properties of heat conductivity can be exhibited on nanoscales at low temperatures. The thermodynamics of the \beta-FPU lattice can be adequately approximated by the harmonic lattice. The second part testifies in the favor of the soliton and breather 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 is reduced to the modified Korteweg - de Vries equation with soliton and breather solutions. Numerical simulations demonstrate their high stability. New method for the visualization of moving solitons and breathers is suggested. An accurate expression for the dependence of the sound velocity on temperature is also obtained. Our results support the conjecture on the solitons and breathers contribution to the heat conductivity.