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Register a new userThe Korteweg-de Vries Hierarchy and Long Water-Waves
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By using the multiple scale method with the simultaneous introduction of multiple times, we study the propagation of long surface-waves in a shallow inviscid fluid. As a consequence of the requirements of scale invariance and absence of secular terms 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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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.
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.
We study the Boussinesq equation from the point of view of a multiple-time reductive perturbation method. As a consequence of the elimination of the secular producing terms through the use of the Korteweg--de Vries hierarchy, we show that the solitary--wave of the Boussinesq equation is a solitary--wave satisfying simultaneously all equations of the Korteweg--de Vries hierarchy, each one in an appropriate slow time variable.
With the assistance of one fold Darboux transformation formula, we derive rogue wave solutions of the complex modified Korteweg-de Vries equation on an elliptic function background. We employ an algebraic method to find the necessary squared eigenfunctions and eigenvalues. To begin we construct the elliptic function background. Then, on top of this background, we create a rogue wave. We demonstrate the outcome for three distinct elliptic modulus values. We find that when we increase the modulus value the amplitude of rogue waves on the dn-periodic background decreases whereas it increases in the case of cn-periodic background.
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.
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