 

Rudi Weikard
On Commuting Matrix Differential Operators


Published: 
January 26, 2002 
Keywords: 
Meromorphic solutions of differential equations, KdVhierarchy, AKNShierarchy, GelfandDikiihierarchy 
Subject: 
34M05, 37K10 


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 matrixvalued 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.


Acknowledgements
Research supported in part by the US National Science Foundation under Grant No. DMS9970299.


Author information
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 352941170, USA
rudi@math.uab.edu
http://www.math.uab.edu/rudi/

