On commuting matrix differential operators

Academic Article

Abstract

  • If the differential expressions P and L are polynomials (over C) of another differential expression they will obviously commute. To have a P which does not arise in this way but satisfies [P,L] = 0 is rare. Yet the question of when it happens has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P,L). In this paper the question is answered in the case where the given expression L has matrix-valued coefficients which are rational functions bounded at infinity or simply periodic functions bounded at the end of the period strip: if Ly = zy has only meromorphic solutions then there exists a P such that [P,L] = 0 while P and L are not both polynomials of any other differential expression. The result is applied to the AKNS hierarchy where L = JD + Q is a first order expression whose coefficients J and Q are 2 × 2 matrices. It is therefore an elementary exercise to determine whether a given matrix Q with rational or simply periodic coefficients is a stationary solution of an equation in the AKNS hierarchy.
  • Authors

    Author List

  • Weikard R
  • Start Page

  • 9
  • End Page

  • 30
  • Volume

  • 8