Mathematical Sciences: Aspects of Integrability

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DMS-9500311 Fokas The PI is investigating mathematical aspects arising in the study of integrable equations, as well as their implications for the associated physical problems modeled by such equations. Emphasis will be placed on the following two problems: (a) The solution of initial- boundary value problems on the half-infinite life for decaying initial and boundary data; (b) Investigation of certain new integrable generalizations of the well known integrable equations and their novel mathematical solutions, which include peakons and compactons. In recent years important advances in the study of certain nonlinear equations have occurred. In particular it has been found that large classes of physically important equations support coherent structures. Such coherent structures include the so called solitons, dromions, peakons, and compactons. Most of these studies have concentrated on initial value problems. However, many physical situations, including problems arising in ionosphere and in optical switches, are modeled by initial-boundary value problems. The proposed investigation of these problems will have important physical implications such as the understanding of the interaction of a light beam with an interface separating two media with different dielectrics.