Toward a Rational Extended Thermodynamics of dispersive elastic media

Abstract

We develop a one-dimensional theory of dispersive elasticity within Rational Extended Thermodynamics, taking local first-order balance laws rather than higher spatial gradients as the fundamental description. A supplementary mechanical-energy law and the Ruggeri--Strumia main-field principle determine the admissible stress, internal fluxes and production. A canonical two-field hierarchy is symmetric hyperbolic under explicit convexity conditions and contains nonlinear elasticity and a generalized-stress theory as principal subsystems. For the reversible linear singular two-field class, elimination of the fast stress mode yields the Love--Rosenau equation exactly, together with a necessary and sufficient realizability condition. Nonlinear elastic stresses and non-quadratic higher-field energies are compatible with the same RET architecture; for the exact nonlinear Love--Rosenau reduction we retain quadratic higher-field inertia while allowing nonlinear elastic stress. The reduced equation admits a travelling-wave first integral. For the leading cubic elastic correction we obtain an exact parametric smooth solitary pulse in a supersonic velocity window, while the truncated-cosine compacton is excluded. Direct simulations of the hyperbolic parent system show finite-time pulse persistence in the reversible regime and slow decay under weak dissipation. The local parent energy flux is kept distinct from interstitial working in the reduced theory.