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تسجيل مستخدم جديدThe Complex Korteweg-de Vries Equation: A Deeper Theory of Shallow Water Waves
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Using Levi-Civitas theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory, and the long wave, slowly varying velocity approximations for shallow water. The complex KdV equation describes the nontrivial dynamics of all water particles from the surface to the bottom of the water layer. A crucial new step made in our work is the proof that a natural consequence of the complex KdV theory is that the wave elevation is described by the real KdV equation. The complex KdV approach in the theory of shallow fluids is thus more fundamental than the one based on the real KdV equation. We demonstrate how it allows direct calculation of the particle trajectories at any point of the fluid, and that these results agree well with numerical simulations of other authors.
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87 -
R. A. Kraenkel
, M. A. Manna (Physique Mathematique et Theorique
, n URA-CNRS 768 Universite de Montpellier 34095 Montpellier CEDEX 05
1994
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in each order of the perturbative expansion, we show that the Korteweg-de Vries hierarchy equations do appear in the description of such waves. Finally, we show that this procedure of eliminating secularities is closely related to the renormalization technique introduced by Kodama and Taniuti.
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We apply a multiple-time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity-free perturbation theory, we show that the well known
N-soliton dynamics of the shallow water wave equation, in the particular case of $alpha=2 beta$, can be reduced to the N-soliton solution that satisfies simultaneously all equations of the Korteweg-de Vries hierarchy.
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Original Whithams method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg-de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides
in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained for each case. The essential difference between them is illustrated by an example of so-called `generalized Korteweg-de Vries equation. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.
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