Local well-posedness in the bi-harmonic NLS with low power nonlinearities

Abstract: In this work we consider one of the variant of the famous Schrödinger equation, where the potential term can be expressed as a power nonlinearity (with any positive power) and the dispersion operator has the higher order, i.e., instead of the Laplacian (or two derivatives in one dimension) we consider a double Laplacian (with four derivatives). This equation was introduced to describe the nonlinear propagation of pulses through optical fibers by taking into account the higher dispersion effects that those given by the standard Schrödinger equation. We investigate solution to this model and show the local in time existence and uniqueness of solutions and continuous dependence on the initial data. For that we construct a new space, based on the so-called weighted Sobolev space, and then using the contraction mapping principle we extract the solutions and their properties. Furthermore we design the fractional weight approach for estimates on the linear part of equation, a new result in the bi-Laplacian setting. This talk is based on the joint work with Marcos Masip (undergrad), Oscar Riaño and Svetlana Roudenko, which was initiated during the REU program “AMRPU @ FIU” in Summer 2021.

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