We analyse heat transport phenomena in a one-dimensional monatomic lattice with harmonic and quartic interatomic potentials by means of the molecular dynamics technique, paying special attention to non-diffusive heat flow and local temperature profiles in steady nonequilibrium states. The non-diffusive heat current is attributed to modified KDV solitons, which decay through collisions with phonons with short wavelengths. Hence, the non-diffusive heat flow does not propagate beyond a critical distance Icf. We confirm the anomalous diffusion of energy due to non-diffusive heat flow in steady non-equilibrium states for systems smaller than the critical length Icr. The non-diffusive heat current does not contribute to the total heat flow for lattices larger than the critical length Icr. The existence of Fourier's law is confirmed from first principles for the lattices. The temperature profiles become linear, and the resultant thermal conductivity is independent of the local temperatures of the lattice, in accordance with the temperatures profiles, so the local energy conservation law holds.

Heat transport and dynamical properties of a one-dimensional monatomic lattice with large quartic potentials are investigated using the molecular dynamics method. Energy is transported not only through diffusive processes but also nondiffusive processes. The nondiffusive energy flow is assigned to modified KdV solitons, which decrease with propagation distance due to collisions with thermally excited phonons with very short wavelength. The behavior of energy spreading resembles anomalous diffusion due to the nondiffusive current within a critical region N cr or a critical time, τcr=N cr/\tv, where v denotes the velocity of the soliton. The diffusive behavior of energy spreading recovers beyond that region. Fourier's law is confirmed in the regime, and the thermal conductivity is found to be independent of local temperatures.