. A P ] 2 1 Ju n 20 04 CORRECTIONS TO THE KDV APPROXIMATION FOR WATER WAVES — PREPRINT
نویسنده
چکیده
In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone. AMS classification: 76B15, 35Q51, 35Q53
منابع مشابه
Corrections to the KdV Approximation for Water Waves
In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are e...
متن کاملVariational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves
The unsmooth boundary will greatly affect motion morphology of a shallow water wave, and a fractal space is introduced to establish a generalized KdV-Burgers equation with fractal derivatives. The semi-inverse method is used to establish a fractal variational formulation of the problem, which provides conservation laws in an energy form in the fractal space and possible solution structures of t...
متن کاملSoliton interaction for the extended Korteweg-de Vries equation
Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order ...
متن کاملHigher order corrections for shallow-water solitary waves: elementary derivation and experiments
We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first order equation is shown to be equivalent to the Korteweg−de Vries (KdV) equation, while the second order equation is solved numeric...
متن کاملA numerical study of the Whitham equation as a model for steady surface water waves
The object of this article is the comparison of numerical solutions of the so-called Whitham equation describing wave motion at the surface of a perfect fluid to numerical approximations of solutions of the full Euler free-surface water-wave problem. The Whitham equation ηt + 3 2 c0 h0 ηηx +Kh0∗ ηx = 0 was proposed by Whitham [33] as an alternative to the KdV equation for the description of sur...
متن کامل