Integrability of local and non-local non-commutative fourth-order quintic non-linear Schrödinger equations

We prove integrability of a generalized non-commutative fourth-order quintic non-linear Schrödinger equation. The proof is relatively succinct and rooted in the linearization method pioneered by Ch. Pöppe. It is based on solving the corresponding linearized partial differential system to generate an evolutionary Hankel operator for the ‘scattering data’. The time-evolutionary solution to the non-commutative non-linear 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 non-local versions, in the sense of Ablowitz and Musslimani (2017, Integrable nonlocal nonlinear equations, Stud. Appl. Math. 139, 7–59), of the fourth-order quintic non-linear Schrödinger 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 linearized partial differential system followed by numerically solving the linear Fredholm equation to generate the solution at any given time.