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.

The validity of Fourier's law is investigated by taking account of the temperature dependence of the thermal conductivity for the one- and two-dimensional diatomic lattices. The thermal conductivity is found to be proportional to the inverse of temperature for both the one- and two-dimensional lattices. Fourier's law is confirmed by excluding the nondiffusive or ballistic energy flow from the total energy current.

The authors investigate the validity of Fourier's law in a two-dimensional monoatomic Toda lattice using the molecular dynamics method. The temperature profiles in the lattice exhibit an exponential behaviour with lattice position. The temperature dependence of thermal conductivity is necessarily derived from the spatial variation of the local temperature, and found to be inversely proportional to the local temperature. The validity of Fourier's law is confirmed by excluding the non-diffusive heat flow from the total heat currents.

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A theory of the surface acoustic soliton in a semiconductor is presented based on the coherentstate representation of the equation of motion for the surface phonons interacting with the conduction electrons. It is shown that the two-dimensional displacement field satisfies the nonlinear integro-differential equation with a damping term. With the aid of the reductive perturbation method, the equation can be reduced to the nonlinear Schrodinger equation with a damping term whose coefficient is the attenuation rate of the surface phonon. The approximate solution is derived to reveal excellent agreement with the numerical result.