On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS

We study the periodic cubic derivative non-linear Schr\"odinger equation (dNLS) and (focussing) quintic (NLS). These are both $L^2$ critical dispersive models, which exhibit threshold type behavior, when posed on line ${\mathbb R}$. describe (three parameter) family of non-vanishing bell-shaped solutions for problem, in closed form. The main objective paper is to their stability with respect co-periodic perturbations. analyze these waves framework DNLS. provide a criteria stability, depending sign scalar quantity. proof relies an instability index count, turn critically depends detailed spectral analysis self-adjoint matrix Hill operator. region parameter space, produces spectrally stable waves. also explicit description all traveling NLS, turns out be two subfamily one exhibited give complete - as it some stable, while other unstable,