Quasideterminant Darboux solutions of Noncommutative Equations of Langmuir Oscillations

TL;DR

This paper develops quasideterminant Darboux solutions for noncommutative equations modeling Langmuir oscillations, extending classical models and deriving noncommutative nonlinear Schrödinger equations.

Contribution

It constructs Darboux transformations in a noncommutative framework using quasideterminants, including associated discrete Lax systems and derivation of noncommutative nonlinear Schrödinger equations.

Findings

Derived noncommutative Darboux transformations using quasideterminants.

Established noncommutative version of discrete nonlinear Schrödinger equation.

Showed the noncommutative formalism reduces to classical models in the commutative limit.

Abstract

This article encloses some results on nonncommutative analogue of nonabelian equations of Langmuir oscillations. One of the main contributions of this work is to construct the Darbboux transformation for the solution of that equation in noncommutative framework incorporating associated discrete Lax system. Further the standard Darboux transformation on arbitrary eigenfunctions of the Lax system are presented in quasideterminants for few index values. Moreover, these computations include the derivation of noncommutative version of nonabelian discrete nonlinear Schr$\ddot{o}$dinger which coincides with its classical model under commutative limit. The end portion of this article reveals the identity of noncommutative formalism incorporating a derivation of an equation of motion which coincides with its existing commutative form in background zero value of spectral parameter.