Covariant forms of Lax one-field operators

TL;DR

This paper explores covariant forms of Lax operators involving polynomials in differentiation operators, establishing operator forms through covariance and Frechet derivatives, and applies these concepts to generalized Boussinesq equations and non-commutative potentials.

Contribution

It introduces a method to derive covariant operator forms of Lax pairs with non-Abelian entries and constructs integrable non-commutative potentials with explicit dressing formulas.

Findings

Derived covariant operator forms for Lax pairs.

Established bilinear representation in the Abelian case.

Constructed integrable non-commutative potentials.

Abstract

Polynomials in differentiation operators are considered. The Darboux transformations covariance determines non-Abelian entries to form the coefficients of the polynomials. Joint covariance of a pair of such polynomials (Lax pair) as a function of one-field is studied. Methodically, the transforms of the coefficients are equalized to Frechet derivatives (first term of the Taylor series on prolonged space) to establish the operator forms. In the commutative (Abelian) case that results in binary Bell (Faa de Bruno) differential polynomials having natural bilinear representation. The example of generalized Boussinesq equation is studied, the chain equations for the case are derived. A set of integrable non-commutative potentials and hence nonlinear equations is constructed altogether with explicit dressing formulas.