Local well-posedness of the higher order nonlinear Schr\"odinger equation on the half-line: single boundary condition case

Abstract

We establish local well-posedness for the higher-order nonlinear Schr\"odinger equation, formulated on the half-line. We consider the scenario of associated coefficients such that only one boundary condition is required, which is assumed to be Dirichlet type. Our functional framework centers around fractional Sobolev spaces. We treat both high regularity and low regularity solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of initial value problems. The linear analysis, which is the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method. In this connection, we note that the higher-order Schr\"odinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivative. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multi-term linear differential operator.