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Integrable discrete deformations of discrete plane curves and their application

Hold Date
2015-05-12 12:00〜2015-05-12 13:00
Meeting Room (#122), Faculty of Mathematics building, Ito Campus, Kyushu University
Object person
Kenji Kajiwara (Institute of Mathematics for Industry, Kyushu University)

It is well-known that the classical differential geometry is one of the sources of integrable systems which dates back to 19th century, such as Bäcklund transformations for the surfaces with constant negative curvature and Darboux’s theory of surfaces. Studies of the dynamics of space/plane curves has been initiated by the pioneering work of Hasimoto followed by Lamb in 70s, where the connection with the integrable systems has been explicitly established. After the rediscovery of the connection between the differential geometry and the continuous integrable systems in 80s, studies of “discrete differential geometry” started from the mid 90s. One of the themes of this area is to construct the geometric framework consistent with the theory of the discrete integrable systems, with an expectation that discrete systems may be more fundamental and have rich mathematical structures as was clarified in the theory of integrable systems. We hope that this will serve as a theoretical infrastructure for visualization or simulation of large deformation of geometric objects.

In this talk, starting from some historical remarks and basic ideas of integrable systems and their discretization, we present some results on integrable discrete deformations of discrete curves. Focusing on plane curves, we present: (i) isoperimetric deformation described by (discrete) modified KdV equation (ii) conformal deformation described by (discrete) Burgers equation. We also discuss the exact solutions to the dynamics of discrete curves, which is a great advantage of applying the theory of discrete integrable systems to geometry.

As an application, we discuss a construction of self-adaptive moving mesh numerical schemes of some nonlinear wave equations admitting “loop solitons". It is difficult to carry out numerical calculations for this class of equations because of singularities, but our geometric formulation provides us with accurate and stable numerical schemes.