In earlier papers [cf. for example, R. B. Hetnarski and J. Ignaczak, Proceedings of the First International Symposium on Thermal Stresses, Thermal Stresses '95, Shizuoka University, Hamamatsu, Japan, 5--7 June (1995), pp. 271--274; and R. B. Hetnarski and J. Ignaczak, Int. J. Eng. Sci., to be published in 1996] a low-temperature nonlinear thermoelastic model has been proposed in which one-dimensional solitonlike waves may propagate. Both exact and asymptotic solutions to the nonlinear governing equations have been obtained there. In the present paper a further study of the one-dimensional model is given. A new approximation to the nonlinear governing equations is postulated, and two implicit-form solitonlike solutions valid in a neighborhood of thermodynamical equilibrium are obtained. The two solutions are less restrictive than the asymptotic solutions obtained earlier, and one of them corresponds to a slow quasithermal wave, while the other matches a fast quasimechanical wave. The graphs illustrating the solutions are included.