Quasideterminant Darboux solutions of Noncommutative Equations of Langmuir Oscillations

TL;DR

The paper develops a noncommutative analogue of Langmuir oscillations by formulating $u_{nt}=u_{n-1}u_n-u_n u_{n+1}$ and deriving a Lax pair to establish NC integrability. It constructs a Darboux transformation in additive form and extends it via quasideterminants to generate NC solutions from a seed, including explicit $N$-fold formulas. A NC Riccati equation is derived, producing Backlund transformations that reduce to the classical Darboux structure in the commutative limit. It also presents an exact one-soliton solution on zero background and discusses potential connections to a discrete noncommutative NLS equation and matrix/determinant methods for future work.

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.