New York Journal of Mathematics
Volume 8 (2002) 9-30


Rudi Weikard

On Commuting Matrix Differential Operators

Published: January 26, 2002
Keywords: Meromorphic solutions of differential equations, KdV-hierarchy, AKNS-hierarchy, Gelfand-Dikii-hierarchy
Subject: 34M05, 37K10

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.


Research supported in part by the US National Science Foundation under Grant No. DMS-9970299.

Author information

Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170, USA