+ = = × i i i N N i H t K x t x i H K N N (.) ( ) ∂ ∂ ( ) ∂ ∂ ( ) ( , , ) WEAKLY NON-LINEAR HIGH FREQUENCY WAVES FOR A FIRST ORDER QUASI-LINEAR SYSTEM INVOLVING SOURCE-LIKE TERMS 1. Introduction. W W W W LW W R L R L WEAKLY NON-LINEAR HIGH FREQUENCY WAVES FOR A FIRST ORDER QUASI-LINEAR SYSTEM,... RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo XLIX (2000), pp. 205-220 FIAMMETTA CONFORTO - SEBASTIANO GIAMB ` O Making use of the method of asymptotic expansion of multiple scales, a study of weakly non-linear, high frequency waves, through “homogeneous” media characterized by dissipative or dispersive hyperbolic systems of partial differential equations, is proposed. Within the present theoretical framework, asymptotic waves in a heat- conducting fluid are considered. 205 Let us consider the following dissipative or dispersive hyperbolic system of first order partial differential equations: 11 where and 123 are, respectively, time and space coordinates; is a column vector of denoting the field independent variables, is a column vector of , while and are matrices. Several physical situations involving dissipative or dispersive effects can be described by a governing system of the form (1.1), where the vector plays the role of a source-like term. Many examples are given by Extended Irreversible Thermodynamics (E.I.T.) which distinguishes from other thermodynamical treatments by raising the thermodynamic fluxes (like