Singular Perturbation & Riemann Problems

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9501255 Schecter and Lin The investigators propose to continue their research on composite wave-front solutions for singularly perturbed partial differential equations and on Riemann problems for systems of conservation laws. Lin proposes to extend his earlier work on the construction of asymptotic expansions for composite wave-front solutions of reaction-diffusion equations, and on a shadowing lemma approach to proving that there is a true solution near such an expansion, to more general singularly perturbed partial differential equations and to partially singularly perturbed systems. He also proposes to extend to higher dimensional systems the SLEP method for studying the stability of such solutions in a special case. Schecter proposes to build on earlier work that characterized structurally stable strictly hyperbolic Riemann solutions for systems of two conservation laws in one space dimension. He proposes in particular to extend this work to the non-strictly-hyperbolic case, to find all codimension one Riemann solutions by examining the violation of each of the conditions for structural stability, and to study each codimension one bifurcation in detail. Schecter and Lin together propose to look at whether Lin's approach to singular perturbation problems will enable one to regard a Riemann problem solution as the start of an asymptotic expansion of a solution to an associated parabolic problem. %%% Sharp wave fronts occur in many areas of science. From the mathematical viewpoint, they arise as solutions of partial differential equations that model various physical situations. The investigators propose to continue their research on wave fronts. For one type of partial differential equation, reaction-diffusion equations, Lin has developed a method of calculating formal solutions in which several sharp fronts separate more slowly changing portions of the solution. He has also developed a rigorous method, modeled on the "shadowing lemma " of dynamical systems theory, of showing that there is a true solution near the calculated formal solution. He proposes to extend this work to more general partial differential equations. He also proposes to use his approach to extend a technique that has been used in a special case to study the stability of such solutions. For another type of partial differential equation, systems of two conservation laws in one space dimension, Schecter has studied solutions in which jump discontinuities separate slowly changing portions. These arise as solutions of Riemann problems, in which the initial state of the system consists of two constant values separated by a single jump. Schecter has characterized the Riemann solutions that are structurally stable, in the sense that the basic character of the solution does not change when the initial states are varied slightly. He proposes to extend this work in a number of directions, for example, by identifying the simplest ways in which structural stability can break down. Schecter and Lin together propose to look at whether Lin's approach to calculating formal solutions with sharp wave fronts will enable one to a regard a Riemann solution as the start of such a formal solution to a more realistic partial differential equation. ***