Long-time reduction for the biharmonic nonlinear Schrödinger equation

In this article, we study the reduction of the one-dimensional biharmonic nonlinear Schrödinger equation to the cubic nonlinear Schrödinger equation in the vanishing higher-order dispersion limit. In the intermediate regularity regime $0<s<2$, where the energy conservation law does not control the $H^s$–norm, we prove the long-time $L^2$–convergence of $H^s$–solutions with an exponential-in-time approximation bound. The argument relies mainly on persistence of regularity. This result provides a simple example of its use in limit problems without relying on higher-order conservation laws.