Kevorkian DMS-9525905 This proposal concerns research that builds on the PI's recent work in developing multiple scale asymptotic expansions that approximate solutions over long periods and in the far field for weakly nonlinear conservation laws with source terms. The primary effect of source terms is that the leading approximation in the multiple scale expansion is complicated; in general, it involves an integral representation that has to be analyzed asymptotically for large time before one can derive the appropriate evolution equations. Extensions of previous work to include the effects of variable coefficients, leading order diffusion, and resonant wave interactions are outlined. In particular, it is proposed to study wave propagation problems with axial or radial symmetry where radial dependence occurs in the source terms. A second phase of the research involves a class of problems where the usual Ginzburg-Landau technique does not apply because the linearly most unstable wave number equals zero. Problems with three or more dependent variables involve a further comlication, the possibility of resonant interactions between waves. The combined effect of source terms snd resonant interactions has not been studied, and is one of the main goals of the proposed research. This proposal outlines research in the approximate analytic solution of conservation laws when the desired solution is close to one that is known exactly. Conservation laws that model physical or biological processes typically involve source terms as well as diffusive terms. Source terms usually arise when reactions are present, whereas diffusive terms often account for dissipation. The effect of small diffusion is benign; as in many fluid dynamics problems, small diffusion corresponds to small viscosity and acts to "smooth out" the solution in a thin layer centerd around a shock wave. If, however, diffusion occurs to leading order, as in many biological applications, the structure of the solution is fundamentally altered. This proposal identifies a class of problems involving diffusion to leading order for which the standard approaches do not apply. It is also proposed to study the effect of source terms in systms that have three or more interacting waves, as in gas dynamics for reacting flows. Here, the primary focus is to clarify the role of source terms in the process that governs the resonant interaction of these waves.
