On the formulation of energy conservation in the eeKdV equation (Q6546916)

The manuscript addresses significant aspects of the extended Korteweg-de Vries (eeKdV) equation, particularly its energy conservation properties. The primary scientific problem is the formulation and verification of energy conservation laws in the context of the eeKdV equation. The Korteweg-de Vries (KdV) equation, traditionally used for modelling unidirectional shallow-water surface waves, incorporates dispersion and weak non-linearity. The eeKdV equation extends the KdV equation by including higher-order terms, which account for situations where the amplitude parameter is somewhat larger relative to the wavelength parameter, reflecting more complex wave dynamics observed in experimental studies such as those conducted by Favre.\N\NThe authors derive the eeKdV equation from the classical surface wave problem by assuming different relative sizes of non-dimensional key parameters compared to those used in the standard KdV equation. This derivation involves an asymptotic expansion of the velocity potential and the application of mechanical balance laws. The resulting eeKdV equation includes additional nonlinear and dispersive terms that account for higher-order effects.\N\NTo address the problem of energy conservation, the authors develop corresponding formulations of energy balance laws for the eeKdV equation. These formulations are presented for an inertial reference frame and involve expressions for energy flux and energy density. The paper also includes a partial verification of the energy flux expressions by examining far-field, uniform flow situations and performing numerical studies to confirm the behavior of the energy balance in the context of undular bore flows.\N\NThe main findings of the manuscript are multifaceted. Firstly, the derived eeKdV equation provides a more comprehensive model for shallow-water wave phenomena, incorporating higher-order terms that are neglected in the standard KdV equation. Secondly, the derived energy flux and energy density expressions for the eeKdV equation are shown to be consistent with the shallow-water theory in the far field. This indicates that the eeKdV equation correctly captures the energy dynamics of the system it models. Thirdly, numerical simulations confirm the theoretical derivations, demonstrating that the error in the energy balance using the derived expressions is within the expected order of magnitude, particularly when the wave train develops into approximations of solitary waves.\N\NThe significance of this research lies in its contribution to the understanding of wave dynamics in shallow water environments. By extending the KdV equation to include higher-order terms, the authors provide a more accurate and robust model for predicting wave behavior in various scenarios. The energy conservation formulations presented in this paper enhance the theoretical framework for analysing wave energy dynamics, which is crucial for applications in coastal engineering, oceanography, and related fields. The numerical verification of the derived expressions further solidifies the practical applicability of the eeKdV equation in simulating and understanding complex wave phenomena.