摘要:

AbstractIn Part I the scattering transform method is used to study the weak limit of solutions to the initial value problem for the Korteweg‐deVries (KdV) equation as the dispersion tends to zero. In that limit the associated Schrödinger operator becomes semiclassical, so the exact scattering data is replaced by its corresponding WKB expressions. Only nonpositive initial data are considered; in that case the limiting reflection coefficient vanishes. The explicit solution of Kay and Moses for the reflectionless inverse transform is then analyzed, and the weak limit, valid for all time, is characterized by a quadratic minimum problem with constraints. This minimum problem is reduced to a Riemann‐Hilbert problem in function theory.In Parts II and III we use function theoretical methods to solve the Riemann‐Hilbert problem in terms of solutions to an auxiliary initial value problem.The weak limit satisfies the KdV equation with the dispersive term dropped until its derivatives become infinite. Up to that time the weak limit is a strong L2‐limit. At later times the weak limit is locally described by Whitham's averaged equations or, more generally, by the equations found by Flaschka et al. using multiphase averaging. For large times, behavior of the weak limit is studied directly from the minimum

ISSN:0010-3640    

DOI:10.1002/cpa.3160360302

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1983

数据来源: WILEY

ISSN:0010-3640    

DOI:10.1002/cpa.3160360303

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1983

数据来源: WILEY

ISSN:0010-3640    

DOI:10.1002/cpa.3160360304

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1983

数据来源: WILEY

摘要:

AbstractWe present an algorithm that solves a two‐dimensional case of the following problem which arises in robotics: Given a bodyB, and a region bounded by a collection of “walls”, either find a continuous motion connecting two given positions and orientations ofBduring whichBavoids collision with the walls, or else establish that no such motion exists. The algorithm is polynomial in the number of walls (O(n5) ifnis the number of walls), but for typical wall configurations can run more efficiently. It is somewhat related to a technique outlined by

ISSN:0010-3640    

DOI:10.1002/cpa.3160360305

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1983

数据来源: WILEY

ISSN:0010-3640    

DOI:10.1002/cpa.3160360301

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1983

数据来源: WILEY