Integrable reductions of the multi-component Kaup-Newell equations

We investigate the local and nonlocal integrable reductions of the multi-component Kaup-Newell equations and derive a range of novel integrable derivative nonlinear Schrodinger (NLS) equations. Firstly, we demonstrate that the multi-component Kaup-Newell equations can undergo one local reduction and three nonlocal reductions, specifically space-reversal, time-reversal, and space-time-reversal reductions. These reductions result in four multi-component derivative NLS equations. Secondly, focusing on the four-component Kaup- Newell equations, we introduce six joint reductions, which are combinations of the aforementioned local and nonlocal reductions. Through these joint reductions between the local reduction and nonlocal reductions, we discover three scalar nonlocal derivative NLS equations, including a T-reversal derivative NLS equation. Furthermore, through the joint reductions among the nonlocal reductions, we identify three scalar nonlocal derivative NLS equations with four variables, namely (x,t), (-x,t), (x,-t), and (-x,-t). All these reduced systems are integrable systems, meaning they possess infinitely many conserved integrals.