Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation

TL;DR

This paper develops a unified approach to Darboux transformations for the super KdV equation using twisted derivations and quasideterminants, enabling explicit solution formulas and generalizations of binary Darboux solutions.

Contribution

It introduces a generalized Darboux transformation framework for twisted derivations, providing unified quasideterminant solution formulas for the super KdV equation.

Findings

Derived a unified quasideterminant solution formula for super KdV.

Expressed solutions as ratios of superdeterminants, including previously unaddressed cases.

Constructed new quasideterminant solutions from binary Darboux transformations.

Abstract

This paper is concerned with a generalized type of Darboux transformations defined in terms of a twisted derivation $D$ satisfying $D(AB)=D(A)+\sigma(A)B$ where $\sigma$ is a homomorphism. Such twisted derivations include regular derivations, difference and $q$-difference operators and superderivatives as special cases. Remarkably, the formulae for the iteration of Darboux transformations are identical with those in the standard case of a regular derivation and are expressed in terms of quasideterminants. As an example, we revisit the Darboux transformations for the Manin-Radul super KdV equation, studied in Q.P. Liu and M. Ma\~nas, Physics Letters B \textbf{396} 133--140, (1997). The new approach we take enables us to derive a unified expression for solution formulae in terms of quasideterminants, covering all cases at once, rather than using several subcases. Then, by using a known relationship between quasideterminants and superdeterminants, we obtain expressions for these solutions as ratios of superdeterminants. This coincides with the results of Liu and Ma\~nas in all the cases they considered but also deals with the one subcase in which they did not obtain such an expression. Finally, we obtain another type of quasideterminant solutions to the Main-Radul super KdV equation constructed from its binary Darboux transformations. These can also be expressed as ratios of superdeterminants and are a substantial generalization of the solutions constructed using binary Darboux transformations in earlier work on this topic.