A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

Abstract

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.

A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.

The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.

Keywords:

• recursion operator
• generalized symmetries
• complete integrability
• nonlinear PDEs
• symbolic software

2000 AMS Subject Classifications::

• Primary: 37K10
• 35Q51
• 68W30
• Secondary: 37K05
• 35Q53
• 35Q58

Acknowledgements

This material is based upon work supported by the National Science Foundation (NSF) under Grant Nos. CCF-0830783 and CCR-9901929. This work was also partially supported by a NDSEG Fellowship awarded to DB. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF. WH thanks Jan Sanders (Free University Amsterdam, The Netherlands) and Jing Ping Wang (University of Kent, Canterbury, UK) for many valuable discussions.