The manuscript investigates the dynamics of solitary wave interactions within the framework of the cubic Whitham equation, a model that extends the classical Whitham equation by including a cubic nonlinear term. The scientific problem addressed in the paper pertains to understanding the interaction dynamics of solitary waves described by the cubic Whitham equation. Solitary waves, or solitons, are fundamental structures in nonlinear wave theory, retaining their shape while propagating over long distances. The cubic Whitham equation, which considers unidirectional wave propagation in a fluid with constant vorticity, presents complex nonlinear behaviours not captured by simpler models such as the Korteweg-de Vries (KdV) equation. This study specifically focuses on overtaking collisions between solitary waves and aims to determine whether the interaction dynamics can be categorized using the Lax geometric framework, which classifies wave interactions based on the number of local maxima during collisions.
The authors employ numerical methods to solve the cubic Whitham equation. They implement a Fourier pseudospectral method combined with an integrating factor technique for temporal evolution, ensuring high accuracy in simulating wave interactions. The study initiates the solitary wave interactions by considering two waves with distinct amplitudes and numerically simulates their collision using the aforementioned methods. The study examines the preservation of solitary wave shapes post-collision, the phase shifts that occur due to the interaction, and the production of dispersive radiation.
The main findings of the manuscript reveal that the cubic Whitham equation does preserve the geometric classification of solitary wave interactions as proposed by Lax for the KdV equation. However, the study finds that an algebraic classification based on the amplitude ratio of the solitary waves is not possible for the cubic Whitham equation, marking a significant departure from the behaviour observed in the KdV and Euler equations. The interactions between solitary waves of different polarities, especially when the amplitudes differ significantly, can lead to wave-breaking phenomena, a behaviour that is not typical in simpler integrable systems. This wave-breaking suggests that the cubic nonlinearity introduces complexities that necessitate a more nuanced understanding of wave interactions in such systems.
In conclusion, the research presented in this manuscript makes a substantial contribution to the study of nonlinear wave dynamics by exploring the intricacies of solitary wave interactions under the cubic Whitham equation. The findings have implications for understanding soliton turbulence and the broader field of nonlinear wave propagation in fluids. The significance of this work lies in its challenge to existing categorizations of wave interactions, highlighting the need for further studies to fully comprehend the implications of higher-order nonlinearities in wave dynamics.