A Symbolic Algorithm for Computing Recursion Operators of Nonlinear PDEs

TL;DR

This paper introduces a symbolic, algorithmic method implemented in Mathematica for computing recursion operators of nonlinear PDEs, aiding in the analysis of their integrability and symmetry structures.

Contribution

The paper presents a new symbolic algorithm and Mathematica package for efficiently computing polynomial recursion operators of (1+1)-dimensional nonlinear PDEs.

Findings

Successfully computed recursion operators for several well-known PDEs.

The method verifies the complete integrability of polynomial PDEs.

Provides a practical tool for researchers in mathematical physics and soliton theory.

Abstract

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear 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.