Mathematical Sciences: Differential Equations

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Work will be done on problems involving nonlinear differential equations. The work will focus on three areas. In the first, investigations will be carried out on the relation between Painleve transcendents and certain linear integral equations arising in inverse scattering theory. Certain ordinary differential equations in the complex plane have only poles for singularities. These equations, or transcendents, are related to nonlinear partial differential equations solvable by the method of inverse scattering. At this time only selected examples illustrate the connection but it is clear that an entire class of equations can be treated. A systematic study of the entire relationship will be undertaken. A second line of research deals with the failure or blow-up of solutions of nonlinear diffusion equations. Blow-up of solutions depends heavily on the nature of initial conditions (as well as the nonlinearity term in the equation). Since one is often dealing with functions of several space variables, blow-up can occur at different points at different times. This leads to the main conjecture of the project: blow-up must be confined to a set of space variables of negligible size (measure zero). Third, work will be done on existence, multiplicity and asymptotics for combustion theory models with complex chemistry including radicals. The models represent steady planar flame fronts in premixed combustion. This analysis goes beyond traditional simple reactions. Most models have to confront or hypothesize away what is known as the cold boundary difficulty to support traveling wave solutions. If intermediate species or radicals are introduced into a system of reactions, a scheme of the principal investigator leads to new existence proofs for flame layers. Work will proceed in expanding methods developed to cover other complex models, validate their asymptotic expansions and consider problems of multiplicity or uniqueness.