A Mathematical Justification of the Momentum Density Function Associated to the KdV Equation

Abstract

Consideration is given to the KdV equation as an approximate model for long waves of small amplitude at the free surface of an inviscid fluid. It is shown that there is an approximate momentum density associated to the KdV equation, and the difference between this density and the physical momentum density derived in the context of the full Euler equations can be estimated in terms of the long-wave parameter. Résumé. L'équation de KdV est considérée comme un modèle approximatif pour des ondes longues de faible amplitude à la surface libre d'un fluide non visqueux. On montre qu'il y a une densité de moment approximative associée à l'équation de KdV, et que la différence entre cette densité et la densité de de moment physique dérivée dans le contexte du système d'Euler peut être estimée en fonction du paramètre d'onde longue.

Convergence of mechanical balance laws for water waves: from KdV to Euler

“…Suppose that the assumption of corollary 1 is satisfied. In this section we would like to prove results analogous to the work presented in [16] concerning the energy formulated in the KdV approximation.…”

“…Since the analysis in [4] was based on a formal asymptotic analysis, the question of whether these physical identities can be made mathematically rigorous have so far remained open (note however that in the special case of the momentum density I defined above in (1.4) it was shown in [16] that this expression converges to the corresponding quantity in the full Euler equations if the parameters µ and ε tend to zero. In the present paper, similar convergence results will be proved also for the momentum flux, as well as the energy density and energy flux.…”

A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling

“…4 Root problems defined by the parameters λ, c, ν, q E as functions of m η as functions of m as given by formula ( 24) and (25). Then use a nonlinear solver to find m from equation (22). With m in hand, one can determine the wave height H (m) at the current local depth.…”

“…Then use the assumption of zero set-down to find the roots given by (20). Having the roots as functions of m we may use energy conservation to define a nonlinear equation as done in equation (22). Solving for m we are free to determine the wave height at a specified depth h.…”

A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling

“…The first equation is conservation of frequency ν, while the second equation expresses conservation of energy flux integrated over a period T [4]. Finally, the third equation is (16), and is indeed a formulation of conservation of momentum expressed in terms of radiation stress. In previous work [20], a similar system was found, and then solved for the parameters f 1 , f 2 and f 3 .…”