Soliton-like waves in a low-temperature nonlinear rigid heat conductor

Abstract

A nonlinear rigid heat conductor whose free energy and heat flux depend not only on the absolute temperature but also on the elastic heat flow that satisfies an evolution equation is considered. First, nonlinear field equations coupling the temperature T and elastic heat flow B are scaled to the dimensionless form involving a low temperature parameter ϵ, and the associated nonlinear initial-boundary value problem is formulated. Next, a nonlinear coupled (T; B) problem in which B = ▽Φ, where Φ is a scalar potential, is replaced by a nonlinear Φ-problem alone. Finally, the Φ-problem is used to obtain a number of new properties of one-dimensional soliton-like thermal waves propagating in an infinite rigid heat conductor. In particular, it is shown that (i) a thermal wave propagates with the velocity c = 1/√ϵ along the conductor, (ii) its internal energy is a convex function of scaled temperature, (iii) the associated elastic heat flow is a linear decreasing function of scaled temperature that passes through a zero at which the absolute temperature attains a reference value, and (iv) in a neighborhood of thermodynamical equilibrium the thermal wave can be approximated by a singular soliton with singularities on a plane wave front propagating with velocity c = 1/√ϵ. Graphs illustrating the propagation of a thermal wave along the conductor are presented.