Mathematical Sciences: Algebraic Methods in Nonlinear Problems

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The topic of this research is the algebraic and geometric structure of integrable systems and the role of Lie algebras. An analytical interpretation of the bilinear identity, which plays a fundamental role in the theory of tau function, vertex operators, and field theoretic methods of the Kyoto school in integrable systems is sought. The investigator will study the relationship between the tau method and the dressing method of Zakharov and Shabat. Possible applications are an extensions of the theory of tau function to other integrable systems, including multidimensional problems. The study of integrable systems is aimed at improving our understanding the nature of certain physical phenomena that have the properties of energy preservation, such as solitons.