Well-posedness of a higher-order nonlinear Schrödinger equation on the interval

Abstract

These are the slides from a presentation given at the The Thirteenth International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory held in Athens, Georgia on April 14, 2025.The higher-order nonlinear Schrödinger (HNLS) equation is a more accurate alternative to the standard NLS equation when studying wave pulses in the femtosecond regime. It arises in a variety of applications ranging from optics to water waves to plasmas to Bose-Einstein condensates. In this talk, we consider the initial-boundary value problem for HNLS on a finite interval in the case of a power nonlinearity. We establish the local well-posedness of this problem in the sense of Hadamard (existence and uniqueness of the solution as well as its continuous dependence on the data) for initial data in the Sobolev space $H^s$ on a finite interval and boundary data in suitable Sobolev spaces determined by the regularity of the initial data and the HNLS equation. The proof relies on a combination of estimates for the linear problem and nonlinear estimates, which vary depending on whether $s > 1/2$ or $0 ≤ s < 1/2$. The linear estimates are established by using the explicit solution formula obtained via the unified transform method of Fokas. This is a joint work with Dionyssis Mantzavinos and Türker Özsarı