Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations

摘要

Three different hierarchies of integrable two-component nonlinear systems arederived from elastic (stretching) flows of null curves in 3-dimensionalMinkowski space. The derivation uses a Lorentzian version of a well-knowngeometrical moving frame method, combined with a new analog of the Hasimototransformation, which corresponds geometrically to a natural type of movingframe for elastic null curves. This frame is related to a Frenet-type nullframe by a gauge transformation that belongs to the little group contained inthe Lorentz group SO(2,1). The Cartan structure equations of the transformedframe are shown to encode a hereditary recursion operator giving atwo-component generalization of the recursion operator for Burgers equation, aswell as a generalization of the Cole-Hopf transformation. The different typesof integrable systems arise from the various symmetries of this recursionoperator. The first hierarchy contains two-component Burgers-type and nonlinearAiry-type systems; the second hierarchy contains novel quasilinearSchrodinger-type (NLS) systems; and the third hierarchy contains semilinearwave equations (in two-component system form). Each of these integrable systemsis shown to correspond to a geometrical flow of a family of elastic null curvesin 3-dimensional Minkowski space.