Mathematical Sciences: Some Questions in Nonlinear Differential Equations

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PI: Rabinowitz DMS-9500570 Rabinowitz will investigate the existence of basic homoclinic solutions of Hamiltonian systems of various types such as singular systems, systems where there seem to be solutions homoclinic to infinity, and systems with almost periodic potentials. Once basic homoclinics have been determined, recent variational methods suggest that one can also find infinitely many so-called multibump solutions which are near sums of translates of the basic solutions. This is another question Rabinowitz will pursue. Finally, for PDE's he will look at some nonlinear wave and Schrodinger equations with almost periodic potentials. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.