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Department of Chemistry, University of New South Wales, Syndey, Australia
ABSTRACT
The Boussinesq equation, introduced by Joseph Boussinesq in 1872, was the first mathematical model to explain the existence of the solitary wave observed by John Scott Russell in 1834, and it remains the canonical model for bidirectional, weakly nonlinear, weakly dispersive wave propagation in shallow water and other media. Unlike the unidirectional Kortewegde Vries equation, the Boussinesq equation allows waves to travel in both directions and captures the leading-order balance between nonlinearity and dispersion in a single second-order-in-time evolution equation. Over the past century and a half it has grown into an enormous family of models: the classical, improved, regularized, and generalized Boussinesq equations; the Boussinesq systems for surface and internal waves; the Boussinesq paradigm equation in two dimensions; and the Boussinesq-Burgers, logarithmic Boussinesq, fractional, and stochastic variants. This review provides a comprehensive and critical assessment of the literature. We survey the physical derivations from the water-wave problem, the theory of solitary-wave existence, stability, and blow-up, the well-posedness theory of the good and bad Boussinesq equations, numerical methods including conservative finite-difference, finite-element, and spectral schemes, the Boussinesq-type models used in coastal engineering and tsunami modelling, and the many exact-solution families obtained by Hirota, Wazwaz, and others. Recent contributions from 2025 and 2026 are highlighted throughout, and the review concludes with a discussion of open problems and future directions.
Keywords: Boussinesq Equation, Solitary Waves, Shallow-Water Waves, Nonlinear, Dispersive