Mathematical Sciences: Studies in Nonlinear Analysis

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****************************************************************************** STUDIES IN NONLINEAR ANAYSIS: ABSTRACT The proposed research is directed towards two types of problems concerning the behavior of solutions to nonlinear differential equations. The first of these problems concerns the existence and intermediate asymptotic behavior of self-similar solutions to the porous medium equation (a degenerate nonlinear parabolic equation) and to the Navier-Stokes equations of gas dynamics. The objective here is to obtain the asymptotic description of various critical transitions in the flows involved, such as the closing of holes (focusing) in porous medium flow or the implosion of shock waves in gas dynamics. The second problem concerns the study of the dynamics of of load coupled arrays of Josephson point junctions. These arrays are modeled by nonlinear systems of ordinary differential equations, and the object of the study is to understand the onset of stability and instability of various kinds of phase locked solutions. Particular emphasis will be placed on predicting the loss of stability of phase lock and the dynamical consequences of such a loss. Self-similar solutions to partial differential equations are solutions of a particularly simple structure determined by the underlying symmetries of the equation. They often arise in the asymptotic description of critical transitions in the flow, because these transitions are characteristic of the equations themselves and do not depend on the details of the particular flow. An example of this phenomenon occurs in engineering practice in connection with coating flows of very viscous fluids. The process of filling holes in such flows is often described by a porous medium equation, and our results will be useful in developing numerical methods for the study and control of such processes. Large arrays of load coupled Josephson junctions are used in many electronic devices such as microwave generators and parametric amplifiers. In all these devices the most desirable operating condition is one in which all of the junctions oscillate in unison. Unfortunately, for various junction characteristics and load configurations this phase locked state is not stable. It is therefore important to know how to predict the stability and instability of the phase locked state in terms of the parameter values for any particular array.