Integrability of local and nonlocal non-commutative fourth order quintic nonlinear Schrodinger equations

TL;DR

This paper proves the integrability of a generalized non-commutative fourth order quintic nonlinear Schrödinger equation and its nonlocal variants, using a linearisation method and numerical schemes for solution generation.

Contribution

It introduces a succinct proof of integrability for these complex equations using linearisation and extends the approach to nonlocal versions, including numerical implementation.

Findings

Proved integrability of non-commutative fourth order quintic NLS

Extended integrability proof to nonlocal reverse space-time and reverse time versions

Developed a numerical scheme based on the analytical approach

Abstract

We prove integrability of a generalised non-commutative fourth order quintic nonlinear Schrodinger equation. The proof is relatively succinct and rooted in the linearisation method pioneered by Ch. Poppe. It is based on solving the corresponding linearised partial differential system to generate an evolutionary Hankel operator for the `scattering data'. The time-evolutionary solution to the non-commutative nonlinear partial differential system is then generated by solving a linear Fredholm equation which corresponds to the Marchenko equation. The integrability of reverse space-time and reverse time nonlocal versions, in the sense of Ablowitz and Musslimani, of the fourth order quintic nonlinear Schrodinger equation are proved contiguously by the approach adopted. Further, we implement a numerical integration scheme based on the analytical approach above which involves solving the linearised partial differential system followed by numerically solving the linear Fredholm equation to generate the solution at any given time.