ht. 1. Engng ScL Vol. 28, No. 8, pp. 821-827, 1990 cmo-7225/90 $3.00 + 0.00 F’rintedin Great Britain. All rights reserved Copyright @ 1990 Pergarnon Pressplc zyxwvu ON SOME FEATURES OF THE WAVE MODULATION IN NONLINEAR MEDIA? D. FUSCO’ and N. MANGANARO’ ’ Dipartimento di Matematica e Applicazioni, Universita di Napoli, via Mezzocannone 8, 80134 Napoli, Italy ’ Dipartimento di Matematica, Universita di Messina, Contrada Papardo, Salita Sperone 31, 98166 Sant’Agata, Messina, Italy zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ Abstract-We consider a dissipative quasilinear first order system of partial differential equations which is compatible with different levels of description of the wave processes. For the model in point we study the propagation of a modulated quasi-monochromatic wave into a constant state and we deduce the evolution equation ruling the wave amplitude. Finally we use the general procedure for investigating wave propagation in nerve fibers. 1. INTRODUCTION Several physical theories such as irreversible thermodynamics, channel flows, active media lead to governing mathematical models for which different levels of description of the wave processes are possible. Within such a context the results obtained in [l, 21 for certain dissipative hyperbolic first order systems of P.D.E. of physical interest point out that, in the far field behaviour, the wave amplitude is ruled by a Burgers-like evolution equation or by a generalized form of it. In this paper we consider the following system of first order zyxwvutsrqponmlkjihgfedcbaZYXWV A”(U, V) d,U + B”(U, V) d,V = C(U, V) (1-l) M”(U, V) d,V + N”(U, V) d&l = P(U, V) (1.2) where d, = a/&q x0 = t and x1 = x are, respectively, space and time coordinates, U and V are column vectors of R" and R", respectively, representing the field; Aa,Be,Ma, N" are suitable matrix coefficients while C and P are column vectors. Moreover a subscript denotes partial derivative with respect to the indicated variable. The use of a well established approach [3] shows that for the system of equations (1.1) and (1.2) the modulation of a quasi-monochromatic wave propagating into a constant state is governed by a Schrodinger-like equation. However a lower order description of the wave process is possible when the left-hand side of the equation (1.2) is negligible so that P(U, V) = 0 (1.3) If IVvPJ #O, Vv= (a/aV), then from the relation (1.3) V= V(U) and consequently (1.1) gives rise to the “reduced” governing model Ayu) a,u = C(u) (1.4) where A”= (B”.VuV+A”); c = C((U, V(U)) (1.5) and vu = (a/au). Here we are interested in the situation where the order of magnitude of the left-hand side of (1.2) is small but not negligible. Namely the following relation holds c2(Ma a,v + N” a,u) = P (1.6) where E << 1 is a real parameter. t This research was supported by Consiglio Nazionale delle Ricerche under Contract No. 88.0185501. 821