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The “good” Boussinesq equation: long-time asymptotics

Christophe Charlier, Jonatan Lenells and Deng-Shan Wang

Vol. 16 (2023), No. 6, 1351–1388

We consider the initial-value problem for the “good” Boussinesq equation on the line. Using inverse scattering techniques, the solution can be expressed in terms of the solution of a 3 × 3-matrix Riemann–Hilbert problem. We establish formulas for the long-time asymptotics of the solution by performing a Deift–Zhou steepest descent analysis of a regularized version of this Riemann–Hilbert problem. Our results are valid for generic solitonless Schwartz class solutions whose space-average remains bounded as t .

asymptotics, Boussinesq equation, Riemann–Hilbert problem, inverse scattering transform, initial value problem
Mathematical Subject Classification
Primary: 34E05, 35G25, 35Q15, 37K15, 76B15
Received: 19 June 2020
Revised: 2 November 2021
Accepted: 20 December 2021
Published: 23 August 2023
Christophe Charlier
Department of Mathematics
KTH Royal Institute of Technology
Jonatan Lenells
Department of Mathematics
KTH Royal Institute of Technology
Deng-Shan Wang
Laboratory of Mathematics and Complex Systems
School of Mathematical Sciences
Beijing Normal University

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