Propagation of heat waves in rigid bodies is investigated. The originality of the approach is that it rests on a revisited version of extended irreversible thermodynamics. In comparison with earlier developments, two innovations are proposed. First, we depart from the linear approach, best illustrated by Cattaneo's relation, to explore the non-linear regime. Second, the extra variables are no longer the usual dissipative fluxes, but renormalized expressions of the fluxes, in order to include the specific material properties of the systems under study. The present model is particularly well suited for studying heat transport at low temperatures in dielectric crystals.

Approximate symmetries of a mathematical model describing one-dimensional motion in a medium with a small nonlinear viscosity are studied. In a physical application, the approximate solution is calculated making use of the approximate generator of the first-order approximate symmetry. MSC: Primary 35J25; 32A37; 43A15; secondary 35A58; 42B20

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results.

The aim of this paper was to propose a systematic study of a ( 1 + 1 ) -dimensional higher order nonlinear Schrödinger equation, arising in two different contexts regarding the biological science and the nonlinear optics. We performed a Lie symmetry analysis and here present exact solutions of the equation.

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