Well-posedness of the higher-order nonlinear Schr\"odinger equation on a finite interval

TL;DR

This paper proves the local well-posedness of a third-order nonlinear Schr"odinger equation on a finite interval, establishing existence, uniqueness, and continuous dependence of solutions with boundary conditions, using advanced linear and nonlinear estimates.

Contribution

It introduces new well-posedness results for a higher-order nonlinear Schr"odinger equation on finite intervals, including novel linear estimates and time regularity results.

Findings

Established local well-posedness for the equation.

Derived explicit solution formulas using Fokas' unified transform.

Proved new time regularity results for related half-line problems.

Abstract

We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated on a finite interval with a combination of nonzero Dirichlet and Neumann boundary conditions. Specifically, for initial and boundary data in suitable Sobolev spaces that are related to one another through the time regularity induced by the equation, we prove the existence of a unique solution as well as the continuous dependence of that solution on the data. The precise choice of solution space depends on the value of the Sobolev exponent and is dictated both by the linear estimates associated with the forced linear counterpart of the nonlinear initial-boundary value problem and, in the low-regularity setting below the Sobolev algebra property threshold, by certain nonlinear estimates that control the Sobolev norm of the power nonlinearity. In particular, as usual in Schr\"odinger-type equations, in the case of low regularity it is necessary to derive Strichartz estimates in suitable Lebesgue/Bessel potential spaces. The proof of well-posedness is based on a contraction mapping argument combined with the linear estimates, which are established by employing the explicit solution formula for the forced linear problem derived via the unified transform of Fokas. Due to the nature of the finite interval problem, this formula involves contour integrals in the complex Fourier plane with corresponding integrands that contain differences of exponentials in their denominators, thus requiring delicate handling through appropriate contour deformations. It is worth noting that, in addition to the various linear and nonlinear results obtained for the finite interval problem, novel time regularity results are established here also for the relevant half-line problem.