OSCILLATORY TRAVELING WAVE SOLUTIONS FOR COAGULATION EQUATIONS

We consider Smoluchowski's coagulation equation with kernels of homogeneity one of the form K-epsilon(xi, eta) = (xi(1-epsilon) + eta(1-epsilon)) (xi eta)(epsilon/2). Heuristically, in suitable exponential variables, one can argue that in this case the long-time behaviour of solutions is similar to the inviscid Burgers equation and that for Riemann data solutions converge to a traveling wave for large times. Numerical simulations in a work by Herrmann and the authors indeed support this conjecture, but also reveal that the traveling waves are oscillatory and the oscillations become stronger with smaller epsilon. The goal of this paper is to construct such oscillatory traveling wave solutions and provide details of their shape via formal matched asymptotic expansions.