A priori estimates for a quadratic dNLS

In this work we consider integrable PDE with higher dimensional Lax pairs. Our main example is a quadratic dNLS equation with a $3\times 3$ Lax pair. For this equation we show a-priori estimates in Sobolev spaces of negative regularity $H^s(\mathbb{R}),s>-\frac{1}{2}$. We also prove that for general $N\times N$ Lax operators $L$, the transmission coefficient coincides with the $2$-renormalized perturbation determinant.