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Rational Recursion Operators for Integrable Differential-Difference Equations.

Sylvain Carpentier¹, Alexander V Mikhailov², Jing Ping Wang³

¹Mathematics Department, Columbia University, New York, USA.

Communications in Mathematical Physics

|July 18, 2020

Summary

This study introduces preHamiltonian pairs of difference operators, linking them to Nijenhuis operators. This provides criteria for inverse recursion operators in differential-difference equations.

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Area of Science:

Mathematical Physics
Nonlinear Dynamics
Integrable Systems

Background:

Differential-difference equations are crucial in modeling complex systems.
Nijenhuis operators play a key role in the integrability of such equations.
Understanding recursion operators is essential for finding symmetries and solutions.

Purpose of the Study:

Introduce and define preHamiltonian pairs of difference operators.
Establish the connection between preHamiltonian pairs and Nijenhuis operators.
Develop criteria for weakly nonlocal inverse recursion operators for differential-difference equations.

Main Methods:

Rigorous algebraic setup using skew fields and matrix rational operators.
Definition of preHamiltonian operators and pairs based on Lie algebra properties.
Construction and analysis of Nijenhuis operators derived from preHamiltonian pairs.

Main Results:

A criterion for rational operators to be weakly nonlocal is established.
PreHamiltonian pairs are shown to naturally lead to Nijenhuis operators.
A specific Nijenhuis recursion operator is constructed for the Adler-Postnikov equation, generating local commuting symmetries.

Conclusions:

The developed theory provides a systematic method for identifying Nijenhuis operators.
The study offers insights into the integrability of differential-difference equations.
The findings are illustrated with well-known examples like Toda and Ablowitz-Ladik equations.