Rigorous Analytical Solution for the Linearized Whitham–Broer–Kaup System on the Half‐Line

ABSTRACT The higher‐order Whitham–Broer–Kaup (WBK) system is a pair of coupled nonlinear evolution partial differential equations (PDEs) describing the interaction between long and short dispersive waves in shallow water. We solve explicitly, via a suitable implementation of the Fokas unified transform method, fully nonhomogeneous initial‐boundary‐value problems for the linearized WBK system formulated on a spatiotemporal quarter‐plane. Our novel closed‐form integral representations are expressed as contour integrals in the spectral Fourier plane and verified a posteriori. In addition to solving the canonical problem, analogous analytical formulas are obtained for a problem with nonstandard boundary conditions. Despite the asymmetry of the system, all solutions are shown to have Schrödinger‐type analytic properties, which in turn imply a dynamic behavior analogous to what Chatziafratis, Ozawa, and Tian ( Mathematische Annalen 389, 2023) reported for the solution of the classical free‐space time‐dependent Schrödinger equation. This work is also useful in the study of well‐posedness for the nonlinear analog.