Generalized Error Bounds in the Recovery of Solitary Wave Profiles

TL;DR

This paper analyzes the stability of a reconstruction formula for water wave profiles, providing error bounds under perturbations and supporting results with numerical simulations.

Contribution

It derives explicit $L^2$ error estimates for the wave profile reconstruction under simultaneous perturbations of key parameters.

Findings

Error bounds depend sublinearly on perturbation size

Numerical simulations confirm theoretical stability estimates

Reconstruction formula remains robust under small perturbations

Abstract

We investigate the robustness of Constantin's explicit reconstruction formula for two-dimensional irrotational solitary water waves. This formula recovers the free-surface profile from the dynamic pressure trace at the bed and depends on both the wave speed and the undisturbed depth. We consider simultaneous perturbations in these three quantities and derive an $L^2$ error estimate for the reconstructed profile. The proof uses the hodograph transform, holomorphic extension arguments, and Paley--Wiener Fourier-decay estimates, yielding stability estimates with sublinear dependence on the perturbation size. We include numerical computations to illustrate the effects of specifically designed perturbations.