Collaborative Research: Nonlinear PDEs and Integro-Differential Equations in the Complex Plane

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NSF Award Abstract - DMS-0103807Mathematical Sciences: Collaborative Research: Nonlinear partial differential equations and integro-differential equations in the complex planeAbstractDMS-0103807CostinThe research focuses on new complex analytic methods exploiting Borel-Laplace duality to investigate regular and singular properties of solutions to certain classes of nonlinear partial differential equations and integro-differential equations in the complex plane. Problems to be studied include equations arising in physical applications that at some critical value of a parameter are either ill-posed in the real domain or are structurally unstable. The methods enable control over sensitive dependence close to a critical point in the parameter space. Many phenomena in nature are modeled by equations that depend sensitively on parameters. Such equations arise in descriptions of fluid mechanics, crystal growth and pattern formation. The underlying sensitivity can make such problems difficult to tackle using computer simulations. It is important to develop theoretical tools based on mathematical analysis to allow precise understanding of such phenomena, and our research is geared towards this goal.