Numerical scheme for the solution of the "bad" Boussinesq equation

TL;DR

This work develops a Fourier spectral numerical scheme for the ill-posed 'bad' Boussinesq equation $u_{tt}-u_{xx}-(u^2)_{xx}-u_{xxxx}=0$, designed to capture long-wave, small-amplitude dynamics relevant to water waves while mitigating ill-posedness through spectral damping. The method rewrites the equation as a first-order system, applies a Fourier transform, enforces a careful frequency cutoff, and uses smooth damping near the cutoff to stabilize time integration, ultimately reconstructing the solution via an inverse transform. The authors validate the scheme by comparing with exact one-soliton solutions and with CLmain's detailed long-time asymptotics across multiple initial data, demonstrating good accuracy and stability, especially for moderate to large times. They show that the scheme reproduces bidirectional wave interactions, phase shifts, and soliton dynamics predicted by asymptotic theory, and provide evidence that the leading asymptotics extend robustly across sectors, with practical implications for simulating long-range nonlinear dispersive waves in settings modeled by the bad Boussinesq equation.

Abstract

We present a numerical scheme for the solution of the initial-value problem for the ``bad'' Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic formulas for the solution.

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Theorems & Definitions (1)

• Remark 2.1