Mathematical Sciences: Integrable Systems and Their PhysicalApplications

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This project continues studies in near-integrable systems along two main lines: physical applications, and the development of solution methods based on the algebraic structure of the equations. Applications focus on problems related to nonlinear optics. Along the second line attention is concentrated on solving the initial value problem for dispersionless Lax equations, such as the dispersionless KP equation. Integrable systems of partial differential equations often have solutions whose properties can be exactly established. This is quite unlike the usual case for nonlinear partial differential equations. What is important here is that often the properties remain when the system is perturbed. Thus one can tell a great deal about otherwise intractable problems. Such problems arise in nonlinear optics (for example, in the design of long-distance communication networks using fiber optics), in the motion of waves in nonlinear acoustic media, and in flows around thin wings.