The behaviour of solutions of NLS equation of derivative type in the semiclassical limit

This paper is devoted to the semiclassical limit of the nonlinear Schrödinger equations of derivative type for initial data with Sobolev regularity, before shocks appear in the limit system. The strict hyperbolicity and genuine nonlinearity is proved for the dispersion limit of the derivative nonlinear Schrödinger equation as long as the Riemann invariants remain distinct.