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Abstract The asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large timetasymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial locationx, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments. 

Abstract We establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. 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, this is no longer the case and, instead, 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 Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. 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; and (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 multiterm linear differential operator. 

This work studies the initial-boundary value problem (ibvp) of the two-dimensional nonlinear Schrödinger equation on the half-plane with initial data in Sobolev spaces and Neumann or Robin boundary data in appropriate Bourgain spaces. It establishes well-posedness in the sense of Hadamard by using the explicit solution formula for the forced linear ibvp obtained via Fokas’s unified transform, and a contraction mapping argument.