Integrability of Local and Nonlocal Non-commutative Fourth Order Quintic Nonlinear Schrodinger Equations

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.