Theory of integrable hierarchies and its application to mathematical physics

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Integrable hierarchies are a class of classical integrable system. Typical examples are the KP hierarchy, the Toda hierarchy and multi-component versions thereof. This research is focussed on both theoretical aspects and applications to mathematical physics. All materials play an important role in recent progress of mathematical physics. As regards the theoretical aspect, we have obtained some results on the Toda hierarchy, the dispersionless Toda hierarchy and integrable hierarchies of the B and D types. Subjects of applications to mathematical physics range over statistical mechanics, gauge theories, string theories and related mathematical issues. Solvable models in discrete quantum mechanics are also studied in the context of orthogonal polynomials.