Numerical study of the Amick-Schonbek system (Q6590512)

The manuscript addresses a particular Boussinesq-type system in mathematical fluid mechanics known as the Amick-Schonbek system. This system, derived within the abcd-Boussinesq class of models, provides a mathematical framework for studying weakly nonlinear, long surface water waves and is notable for its unique properties among Boussinesq models. Specifically, it exhibits the dispersive perturbation of the Saint-Venant system, which underlies shallow water wave theory. This research contributes to the numerical analysis of this system by examining the behaviour of solitary waves, their stability, and possible blow-up phenomena, offering novel insights into the system's long-term dynamics and stability characteristics.\N\NThe authors employ spectral methods, particularly a Fourier spectral method, for their analysis of the Amick-Schonbek system in one-dimensional space. By numerically constructing solitary waves of this system, the paper investigates their stability properties under perturbations and explores the system's response when initial conditions do not satisfy the so-called noncavitation condition. Through this method, the authors construct a range of solitary waves characterised by different velocities, which they then subject to perturbations to observe the stability behaviour. The time evolution of these solitary wave solutions, as simulated through a fourth-order Runge-Kutta scheme, reveals stable solitary waves under small perturbations, suggesting an asymptotically stable configuration of solitary waves for large time evolution.\N\NThe study's findings indicate that the solitary waves of the Amick-Schonbek system are asymptotically stable for specific initial conditions, as these waves tend to evolve into stable configurations characterised by solitary waves plus radiation. However, when initial conditions violate the noncavitation condition, the system exhibits blow-up behaviour, implying the presence of critical thresholds in the initial data that lead to singularities within finite time. Additionally, the study explores the zero-dispersion limit of the system, highlighting the formation of dispersive shock waves under this condition, which are rapid, oscillatory zones near the shocks of the corresponding dispersionless Saint-Venant system.\N\NIn summary, this paper presents a detailed numerical investigation of the Amick-Schonbek system, elucidating the stable configurations of solitary waves, the conditions leading to blow-up, and the formation of dispersive shock waves. These findings are significant for advancing the theoretical understanding of dispersive wave systems, especially within the context of fluid dynamics, as they provide numerical evidence supporting conjectures on soliton stability and shock formation in Boussinesq-type equations. This work enriches the mathematical theory of wave propagation and stability in nonlinear dispersive systems, offering a valuable contribution to the field.