Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas method

We study the inhomogeneous Airy partial differential equation (also called Stokes or linearized Korteweg–de Vries equation with a negative sign) on the half-line with generic initial and boundary data in a classical smooth setting, via the formula provided by the Fokas unified transform method for linear evolution equations. We first present a suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in it, thus securing convergence in a strict sense. Writing the solution in an Ehrenpreis–Palamodov form, our analysis allows for rigorous a posteriori verification of the full initial-boundary-value problem and a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to prescribed values and the solution admits a smooth extension up to the boundary only under certain data compatibility conditions (with implications for well-posedness, control theory and efficient numerical computations). Importantly, based on this analysis, we perform an effective asymptotic study of far-field dynamics. This yields new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the ‘corner’ of the quadrant. In particular, the asymptotic behavior of the solution is sensitive to perturbations of the data at the origin. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property at soon as time becomes positive. Hereby, we report on the discovery of a novel type of a long-range instability phenomenon for linear dispersive differential equations. Our ideas are extendable to other Airy-like and more general problems for dispersive evolution equations.

Keywords

initial-boundary-value problems for dispersive PDE, forced half-line Airy evolution equation, Fokas unified-transform-method formula, Oscillaroty integrals, classical solution, long-space asymptotics and conditional decay, long-range instability

2010 Mathematics Subject Classification

Primary 35A09, 35A22, 35A25, 35C05, 35C15. Secondary 35G05, 35G16, 35Q53, 35S30, 43A32, 44A15.

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Received 2 May 2023

Published 21 May 2024