On the Galilean invariance of some dispersive wave equations

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.

💡 Research Summary

The paper addresses a fundamental shortcoming of many widely used dispersive water‑wave models: the loss of Galilean invariance. Classical approximations such as the Benjamin‑Bona‑Mahony (BBM) equation and the Peregrine (classical Boussinesq) system are derived from the Euler equations by asymptotic expansion, but the resulting equations contain terms that change under a Galilean boost (x → x − Vt, u → u + V). Consequently, the shape and speed of solitary waves depend on the observer’s frame, which is unphysical.

To remedy this, the authors propose a systematic “Galilean‑restoration” mechanism. They first identify the non‑invariant contributions that appear when a boost is applied to the original models. By redefining the nonlinear advection coefficient and adding a higher‑order dispersive term (typically of the form u_{xxt}), they construct modified BBM and Peregrine equations whose structure remains unchanged under any constant velocity shift V. The new equations retain the same order of accuracy as the classical models, preserve the usual conserved quantities (mass, energy), and reproduce the correct linear dispersion relation.

The paper then proves the existence of solitary‑wave solutions for the modified equations and investigates their stability analytically and numerically. Using Gaussian initial data, the authors perform time‑integration with several observer speeds (V = 0, ±0.5, ±1.0). In all cases the solitary wave retains its amplitude, width, and propagation speed, confirming true Galilean invariance.

A key part of the study is a quantitative comparison with fully nonlinear Euler‑equation simulations performed by a high‑resolution spectral method. Metrics such as phase speed, waveform asymmetry, and phase shift after soliton‑soliton collisions are examined. The modified models show markedly reduced errors relative to the classical BBM and Peregrine systems, especially for long‑distance propagation and multi‑soliton interactions.

The authors conclude that preserving Galilean symmetry is not a mere mathematical nicety but a practical requirement for accurate wave modeling. They suggest that the restoration technique can be extended to higher‑dimensional Boussinesq‑type systems and could improve the reliability of engineering applications such as wave‑energy harvesting, coastal protection, and tsunami modeling. Overall, the work provides a clear pathway to reconcile asymptotic wave models with the fundamental invariance properties of the underlying fluid dynamics.