Traveling-wave solutions of convection–diffusion systems by center manifold reduction

Abstract

A convection–di$usion system in one space dimension is a partial dierential equation of the form ut + A(u)ux =(B(u)ux)x: (1.1) Here u∈Rn, and A(u) and B(u) are n×n matrices that we shall assume are C2 functions of u. If we ignore diusion, we have the convection system ut + A(u)ux =0: (1.2) If A(u)=Df(u) for some 4ux function f : Rn → Rn, then Eq. (1.2) becomes ut + f(u)x =0; (1.3) a system of conservation laws, while Eq. (1.1) becomes ut + f(u)x =(B(u)ux)x; (1.4) a system of viscous conservation laws. The conservation law case occurs more often in applications and is far better studied; see [5] and [12]. However, equations of the