  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.
